geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09d.pdf ·...
TRANSCRIPT
![Page 1: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/1.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
MathViz Champaign-Urbana March 28 2009
Geometric generation of permutation sequences
Dennis Roseman
University of [email protected]
March 26, 2009
![Page 2: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/2.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Overture
Music
Original motivation: apply mathematics to the composition ofmusic.
Mathematics Focus:
Some geometry of the n-dimensional permutahedron.
Visualization Focus
Higher dimensional visualization including braids used as avisualization tool.
![Page 3: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/3.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
From Abstract Mathematics to MusicalComposition
Two very different musical examples:
change ringing
Nomos Alpha of Xenankis
![Page 4: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/4.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
From Change Ringing by Wilfrid G. Wilson
![Page 5: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/5.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
From Change Ringing by Wilfrid G. Wilson
![Page 6: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/6.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
From Change Ringing by Wilfrid G. Wilson
![Page 7: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/7.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
From Change Ringing by Wilfrid G. Wilson
![Page 8: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/8.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
From Change Ringing by Wilfrid G. Wilson
![Page 9: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/9.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
From Formal Music by I. Xenakis—Nomos Alpha
![Page 10: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/10.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
From Formal Music by I. Xenakis—Nomos Alpha
![Page 11: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/11.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
From Formal Music by I. Xenakis—Nomos Alpha
![Page 12: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/12.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
From Formal Music by I. Xenakis—Nomos Alpha
![Page 13: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/13.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Music and Mathematics
Definition
A musical composition is a family of sequences of relatedmusical events.
Time and voices
Progression in time is related to succession in a sequence; eachsequence represents a “voice”.
The mathematical objects we chose are permutations.
Construct families of sequences length k of permutations oforder n , where k and n are independent.
![Page 14: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/14.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Permutations Geometrically: the Permutahedron
The Permutahedron
1 Take the n! permutations Sn to be all permutations of(1, 2, . . . , n)
2 They are n-tuples—plot them as points in Rn.
3 Take the convex hull.
4 The resulting polytope is the permutahedron P(n)
![Page 15: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/15.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Permutations Geometrically: the Permutahedron
The Permutahedron
1 Take the n! permutations Sn to be all permutations of(1, 2, . . . , n)
2 They are n-tuples—plot them as points in Rn.
3 Take the convex hull.
4 The resulting polytope is the permutahedron P(n)
![Page 16: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/16.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Permutations Geometrically: the Permutahedron
The Permutahedron
1 Take the n! permutations Sn to be all permutations of(1, 2, . . . , n)
2 They are n-tuples—plot them as points in Rn.
3 Take the convex hull.
4 The resulting polytope is the permutahedron P(n)
![Page 17: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/17.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Permutations Geometrically: the Permutahedron
The Permutahedron
1 Take the n! permutations Sn to be all permutations of(1, 2, . . . , n)
2 They are n-tuples—plot them as points in Rn.
3 Take the convex hull.
4 The resulting polytope is the permutahedron P(n)
![Page 18: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/18.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Permutations Geometrically: the Permutahedron
The Permutahedron
1 Take the n! permutations Sn to be all permutations of(1, 2, . . . , n)
2 They are n-tuples—plot them as points in Rn.
3 Take the convex hull.
4 The resulting polytope is the permutahedron P(n)
![Page 19: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/19.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Permutations Geometrically: the Permutahedron
The Permutahedron
1 Take the n! permutations Sn to be all permutations of(1, 2, . . . , n)
2 They are n-tuples—plot them as points in Rn.
3 Take the convex hull.
4 The resulting polytope is the permutahedron P(n)
![Page 20: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/20.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Permutations Geometrically: the Permutahedron
The Permutahedron
1 Take the n! permutations Sn to be all permutations of(1, 2, . . . , n)
2 They are n-tuples—plot them as points in Rn.
3 Take the convex hull.
4 The resulting polytope is the permutahedron P(n)
![Page 21: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/21.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Low Dimensional Cases
Some Examples:
1 P(2) is the line segment in R2 with endpoints (1, 2) and(2, 1) .
2 P(3) is a hexagon in R3 in the plane .
3 P(4) is a truncated octahedron in R3 .
![Page 22: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/22.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Low Dimensional Cases
Some Examples:
1 P(2) is the line segment in R2 with endpoints (1, 2) and(2, 1) , a subset of the line x + y = 1 + 2.
2 P(3) is a hexagon in R3 in the plane .
3 P(4) is a truncated octahedron in R3 .
![Page 23: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/23.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Low Dimensional Cases
Some Examples:
1 P(2) is the line segment in R2 with endpoints (1, 2) and(2, 1) , a subset of the line x + y = 3.
2 P(3) is a hexagon in R3 in the plane .
3 P(4) is a truncated octahedron in R3 .
![Page 24: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/24.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Low Dimensional Cases
Some Examples:
1 P(2) is the line segment in R2 with endpoints (1, 2) and(2, 1) .
2 P(3) is a hexagon in R3 in the plane .
3 P(4) is a truncated octahedron in R3 .
![Page 25: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/25.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Low Dimensional Cases
Some Examples:
1 P(2) is the line segment in R2 with endpoints (1, 2) and(2, 1) .
2 P(3) is a hexagon in R3 in the plane subset of the planex + y + z = 6.
3 P(4) is a truncated octahedron in R3 .
![Page 26: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/26.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Low Dimensional Cases
Some Examples:
1 P(2) is the line segment in R2 with endpoints (1, 2) and(2, 1) .
2 P(3) is a hexagon in R3 in the plane .
3 P(4) is a truncated octahedron in R3 .
![Page 27: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/27.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Low Dimensional Cases
Some Examples:
1 P(2) is the line segment in R2 with endpoints (1, 2) and(2, 1) .
2 P(3) is a hexagon in R3 in the plane .
3 P(4) is a truncated octahedron in R3 subset of thehyperplane x + y + z + w = 10.
![Page 28: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/28.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
The Permutahedron of order 2
Figure: The two permutations (1, 2) and (2, 1) : a line segment in R2
![Page 29: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/29.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
The Permutahedron of order 3
Figure: Hexagon in R3 of the six permutations of order 3:(1, 2, 3), (2, 1, 3), (3, 1, 2), (3, 2, 1), (2, 3, 1), (1, 3, 2), (1, 2, 3)
![Page 30: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/30.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
The Permutahedron of order 4
Figure: The 24 permutations of order 4 determine a truncatedoctahedron in R4 which we show in R3
![Page 31: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/31.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Change Ringing n bells
Definition
A sequence Σ of permutations is a change ringingcomposition if
Σ begins and ends with the identity permutation of Sn
Otherwise each of the n! order n permutations occursexactly one time
Two consecutive permutations of Σ differ by switchingtwo consecutive integers.
![Page 32: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/32.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Ringing Changes on Three Bells
1 2 32 1 33 1 23 2 12 3 11 3 21 2 3
Table: One way to ring changes on 3 bells; the second reverses theorder.
![Page 33: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/33.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Ringing Changes Geometrically
Figure: The change Double Canterbury Pleasure Minimuscorresponds to a Hamiltonian path in the edge set of P(4)
![Page 34: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/34.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Change Ringing
Critique of Change Ringing
Change ringing is very limited—hard to get non-trivialexamples.
There is no relationship between one change and another
Each permutation is treated equally. Musically one expectsto make choices.
There is a fixed length to a ring of changes
The difficulty of calculation increases rapidly with n
![Page 35: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/35.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Change Ringing
Critique of Change Ringing
Change ringing is very limited—hard to get non-trivialexamples.
There is no relationship between one change and another
Each permutation is treated equally. Musically one expectsto make choices.
There is a fixed length to a ring of changes
The difficulty of calculation increases rapidly with n
![Page 36: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/36.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Change Ringing
Critique of Change Ringing
Change ringing is very limited—hard to get non-trivialexamples.
There is no relationship between one change and another
Each permutation is treated equally. Musically one expectsto make choices.
There is a fixed length to a ring of changes
The difficulty of calculation increases rapidly with n
![Page 37: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/37.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Change Ringing
Critique of Change Ringing
Change ringing is very limited—hard to get non-trivialexamples.
There is no relationship between one change and another
Each permutation is treated equally. Musically one expectsto make choices.
There is a fixed length to a ring of changes
The difficulty of calculation increases rapidly with n
![Page 38: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/38.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Change Ringing
Critique of Change Ringing
Change ringing is very limited—hard to get non-trivialexamples.
There is no relationship between one change and another
Each permutation is treated equally. Musically one expectsto make choices.
There is a fixed length to a ring of changes
The difficulty of calculation increases rapidly with n
![Page 39: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/39.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A new way to get a sequences of permutations
Bouncing a light in a mirrored P(4)
Build room in the shape of P(4) with all walls made ofmirror.
From inside the room shine a “generic” laser beam frompoint x0 in direction λ0.
The beam as it reflects will hit successive walls giving asequence of points x1, x2, . . ..
Since the beam is generic there will be a unique vertex(permutation) πi of P(4) nearest to xi .
Thus we generate our sequence of permutationsS(x0, λ0) = (π1, π2, . . .)
![Page 40: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/40.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A new way to get a sequences of permutations
Bouncing a light in a mirrored P(4)
Build room in the shape of P(4) with all walls made ofmirror.
From inside the room shine a “generic” laser beam frompoint x0 in direction λ0.
The beam as it reflects will hit successive walls giving asequence of points x1, x2, . . ..
Since the beam is generic there will be a unique vertex(permutation) πi of P(4) nearest to xi .
Thus we generate our sequence of permutationsS(x0, λ0) = (π1, π2, . . .)
![Page 41: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/41.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A new way to get a sequences of permutations
Bouncing a light in a mirrored P(4)
Build room in the shape of P(4) with all walls made ofmirror.
From inside the room shine a “generic” laser beam frompoint x0 in direction λ0.
The beam as it reflects will hit successive walls giving asequence of points x1, x2, . . ..
Since the beam is generic there will be a unique vertex(permutation) πi of P(4) nearest to xi .
Thus we generate our sequence of permutationsS(x0, λ0) = (π1, π2, . . .)
![Page 42: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/42.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A new way to get a sequences of permutations
Bouncing a light in a mirrored P(4)
Build room in the shape of P(4) with all walls made ofmirror.
From inside the room shine a “generic” laser beam frompoint x0 in direction λ0.
The beam as it reflects will hit successive walls giving asequence of points x1, x2, . . ..
Since the beam is generic there will be a unique vertex(permutation) πi of P(4) nearest to xi .
Thus we generate our sequence of permutationsS(x0, λ0) = (π1, π2, . . .)
![Page 43: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/43.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A new way to get a sequences of permutations
Bouncing a light in a mirrored P(4)
Build room in the shape of P(4) with all walls made ofmirror.
From inside the room shine a “generic” laser beam frompoint x0 in direction λ0.
The beam as it reflects will hit successive walls giving asequence of points x1, x2, . . ..
Since the beam is generic there will be a unique vertex(permutation) πi of P(4) nearest to xi .
Thus we generate our sequence of permutationsS(x0, λ0) = (π1, π2, . . .)
![Page 44: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/44.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A new way to get a sequences of permutations
Bouncing a light in a mirrored P(4)
Build room in the shape of P(4) with all walls made ofmirror.
From inside the room shine a “generic” laser beam frompoint x0 in direction λ0.
The beam as it reflects will hit successive walls giving asequence of points x1, x2, . . ..
Since the beam is generic there will be a unique vertex(permutation) πi of P(4) nearest to xi .
Thus we generate our sequence of permutationsS(x0, λ0) = (π1, π2, . . .)
![Page 45: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/45.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bounce points
Definition
The points x1, x2, . . . are called either intersection points(they are calculated as an intersection of a ray and ∂P(n)) orbounce points (since our beam bounces there).
Bouncing for P(n)
Cleary we can define this process for permutations of order n.
![Page 46: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/46.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Lets bounce
Figure: An example of 16 bounces
![Page 47: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/47.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Corresponding Sequence of Permutations
2 1 4 32 1 4 31 3 2 42 4 3 13 1 2 43 2 1 42 3 1 42 4 1 32 3 4 14 2 1 33 4 1 23 4 1 21 3 4 21 3 4 24 1 3 21 2 3 41 4 2 3
1 4 2 33 4 2 14 2 1 31 2 4 31 3 4 23 2 4 14 1 3 23 1 2 43 1 2 42 4 3 11 4 2 31 4 2 32 1 3 42 1 3 44 1 3 22 4 3 1
![Page 48: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/48.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Lets bounce
Figure: Another example of 16 bounces
![Page 49: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/49.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
The numbers we use are not necessarily pitches
Figure: Any knob or input/output on the Control Panel of this Moogcorresponds to a number. Photo by Kevin Lightner
![Page 50: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/50.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Change ringing: finite number of possibilities
![Page 51: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/51.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Bouncing:infinite number of possibilities
![Page 52: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/52.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Bouncing:infinite number of possibilities
Change ringing is hard is it to calculate.
![Page 53: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/53.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Bouncing:infinite number of possibilities
Bouncing is easy to calculate
![Page 54: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/54.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Bouncing:infinite number of possibilities
Bouncing is easy to calculate as we will see
![Page 55: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/55.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Bouncing:infinite number of possibilities
Bouncing is easy to calculate
Change Ringing—no relationship between one changeand another
![Page 56: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/56.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Bouncing:infinite number of possibilities
Bouncing is easy to calculate
Bouncing: if one varies x and λ one gets relatedpermutation sequences S(x , λ)
![Page 57: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/57.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Bouncing:infinite number of possibilities
Bouncing is easy to calculate
Bouncing: if one varies x and λ one gets relatedpermutation sequences S(x , λ)
Change Ringing: each permutation is treated equally
![Page 58: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/58.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Bouncing:infinite number of possibilities
Bouncing is easy to calculate
Bouncing: if one varies x and λ one gets relatedpermutation sequences S(x , λ)
Bouncing: distinct sequences have distinct characteristics
![Page 59: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/59.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Bouncing:infinite number of possibilities
Bouncing is easy to calculate
Bouncing: if one varies x and λ one gets relatedpermutation sequences S(x , λ)
Bouncing: distinct sequences have distinct characteristics
Change Ringing: difficulty of calculation increases rapidlywith n
![Page 60: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/60.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bouncing, Ringing
Comparing Bouncing to Change Ringing
Bouncing:infinite number of possibilities
Bouncing is easy to calculate
Bouncing: if one varies x and λ one gets relatedpermutation sequences S(x , λ)
Bouncing: distinct sequences have distinct characteristics
Bouncing: calculation is quadratic with respect to n
![Page 61: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/61.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
The rest of the talk: mathematics and visualization
Questions we now address
1 How fast can we calculate the bouncing path for fairlyhigh orders—8, 12, 16, 32?
2 How can we visualize the calculational process and theresults?
3 What is the geometry of a high dimensionalpermutahedron?
4 How does the geometry of the permutahedron changewith n?
![Page 62: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/62.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Basic approach
Change Ringing
1 Construct a special polygonal path in the wireframe of apermutahedron.
2 The sequence of vertices on that path is the desiredpermutation sequence.
Bouncing, an alternative
1 Take a generically generated generic path in Rn thatavoids the wireframe
2 Obtain the sequence of permutations by “digitizing” topermutations near the path.
![Page 63: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/63.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Vertices of the Permutahedron
Theorem
There are n! vertices in P(n).
All vertices of P(n) all lie on an (n − 2)-sphere with center Cn,the centroid of P(n), and radius ρn.
Definition
This sphere is the permutahedral sphere of order n and ρn
the permutahedrdal radius. The distance between Cn and thecentroid of Yα is the inner permutahedrdal radius.
![Page 64: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/64.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Generators of the Symmetric Group
Definition
An elementary transposition is a permutation thatinterchanges consecutive integers,
Note:
This is interchange of consecutive integers (wherever they are)not interchange of integers in consecutive positions (whateverthe integers are).
![Page 65: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/65.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Edges of the Permutahedron
Definition
The union of edges of P(n) is called the wireframe of P(n)
Example
The four edges from (1, 2, 3, 4, 5) go to (2, 1, 3, 4, 5),(1, 3, 2, 4, 5), (1, 2, 4, 3, 5), and (1, 2, 3, 5, 4).
Basic general edge facts
Two permutations are connected by an edge if and only ifcoordinates differ by a switch of two coordinates ofconsecutive value.
Thus any edge corresponds to an elementary transposition.
Thus all edges have length√
2.
The order of any vertex is (n − 1).
![Page 66: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/66.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Edges of the Permutahedron
Definition
The union of edges of P(n) is called the wireframe of P(n)
Example
The four edges from (1, 2, 3, 4, 5) go to (2, 1, 3, 4, 5),(1, 3, 2, 4, 5), (1, 2, 4, 3, 5), and (1, 2, 3, 5, 4).
Basic general edge facts
Two permutations are connected by an edge if and only ifcoordinates differ by a switch of two coordinates ofconsecutive value.
Thus any edge corresponds to an elementary transposition.
Thus all edges have length√
2.
The order of any vertex is (n − 1).
![Page 67: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/67.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Edges of the Permutahedron
Definition
The union of edges of P(n) is called the wireframe of P(n)
Example
The four edges from (1, 2, 3, 4, 5) go to (2, 1, 3, 4, 5),(1, 3, 2, 4, 5), (1, 2, 4, 3, 5), and (1, 2, 3, 5, 4).
Basic general edge facts
Two permutations are connected by an edge if and only ifcoordinates differ by a switch of two coordinates ofconsecutive value.
Thus any edge corresponds to an elementary transposition.
Thus all edges have length√
2.
The order of any vertex is (n − 1).
![Page 68: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/68.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Edges of the Permutahedron
Definition
The union of edges of P(n) is called the wireframe of P(n)
Example
The four edges from (1, 2, 3, 4, 5) go to (2, 1, 3, 4, 5),(1, 3, 2, 4, 5), (1, 2, 4, 3, 5), and (1, 2, 3, 5, 4).
Basic general edge facts
Two permutations are connected by an edge if and only ifcoordinates differ by a switch of two coordinates ofconsecutive value.
Thus any edge corresponds to an elementary transposition.
Thus all edges have length√
2.
The order of any vertex is (n − 1).
![Page 69: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/69.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Edges of the Permutahedron
Definition
The union of edges of P(n) is called the wireframe of P(n)
Example
The four edges from (1, 2, 3, 4, 5) go to (2, 1, 3, 4, 5),(1, 3, 2, 4, 5), (1, 2, 4, 3, 5), and (1, 2, 3, 5, 4).
Basic general edge facts
Two permutations are connected by an edge if and only ifcoordinates differ by a switch of two coordinates ofconsecutive value.
Thus any edge corresponds to an elementary transposition.
Thus all edges have length√
2.
The order of any vertex is (n − 1).
![Page 70: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/70.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Visualizing the edges
Rotate and project to low dimensions
We can generically rotate a wireframe of any orderpermutahedron then project homemorphically into R3.
We can generically rotate a wireframe of any orderpermutahedron then project non-homemorphically into R2 andstill get a meaningful image.
Color the edges
We can use (n − 1) colors on the edges to code thecorresponding transpositions.
![Page 71: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/71.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Coloring Edges: Order 4
![Page 72: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/72.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Coloring Edges: Order 5
![Page 73: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/73.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Coloring Edges: Order 6
![Page 74: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/74.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Cells of the Permutahedron
Proposition
An k-cell of P(n) is either subgroup which is a product of ksymmetric groups or a coset of one of such subgroup. HereP(0) = {1}.
Definition
Let Yα = {(x1, . . . , xn) ∈ Sn : x1 = 1} andYω = {(x1, . . . , xn) ∈ Sn : xn = n}. We call Yα the first Youngsubgroup of Sn,Yω the last Young subgroup of Sn.
Remark
Yα and Yω are isomorphic to Sn−1
![Page 75: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/75.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Re-examining P(4)
Figure: The edges of one color are all the cosets of a Young subgroupof order two. The hexagons are all cosets of the two Young subgroupsisomorphic to P(3). The squares are cosets of P(2)× P(2).
![Page 76: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/76.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Re-examining P(4)
Figure: The edges of one color are all the cosets of a Young subgroupof order two. The hexagons are all cosets of the two Young subgroupsisomorphic to P(3). The squares are cosets of P(2)× P(2).
![Page 77: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/77.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Re-examining P(4)
Figure: The edges of one color are all the cosets of a Young subgroupof order two. The hexagons are all cosets of the two Young subgroupsisomorphic to P(3). The squares are cosets of P(2)× P(2).
![Page 78: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/78.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Facets of the Permutahedron
Definition
The facets are the the (n − 2)-cells of P(n) .
Proposition
P(n) has 2n − 2 facets
Example
So P(8) has 254 facets and P(12) has 4094.
Implication
The number of facets is exponential in n. Our light beamcalculation should not be based on examination of all facets.
![Page 79: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/79.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A duality of facets and edges
Transposition colors for facets
At a vertex v of facet F you see (n − 1) edges all of distinctcolors.
One of these colors is not an edge of F .
This color will identify our corresponding elementarytransposition.
![Page 80: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/80.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Coloring the facets
Figure: Here we color the three generators: σ1 σ2 σ3
The facetcolor is the unique color not an edge color of the facet.
![Page 81: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/81.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Coloring the facets
Figure: Here we color the three generators: red green blue
Thefacet color is the unique color not an edge color of the facet.
![Page 82: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/82.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Coloring the facets
Figure: Here we color the three generators: σ1 σ2 σ3
The facetcolor is the unique color not an edge color of the facet.
![Page 83: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/83.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Coloring the facets
Figure: Here we color the three generators: σ1 σ2 σ3 The facetcolor is the unique color not an edge color of the facet.
![Page 84: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/84.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Two group presentations:
Presentation of Order n Braid Group
Generators: σ1, . . . , σn−1
Relations:
σiσj = σjσi if j 6= i ± 1
σiσi+1σi = σi+1σiσi+1
Presentation of Order n Symmetric Group
Generators: σ1, . . . , σn−1
Relations:
σi = σ−1i
σiσj = σjσi if j 6= i ± 1
σiσi+1σi = σi+1σiσi+1
![Page 85: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/85.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Inverses and elementary transpositions
From the symmetric group to the braid group
A finite sequence of elementary transpositions τ1, τ2, . . . , τncorresponds to a word in the symmetric group:
τ1 τ2 · · · τn.
But if (somehow) we can distinguish elementary transpositionsfrom their inverses we would obtain a word in the braid group:
τ ε11 τ ε22 · · · τεnn .
![Page 86: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/86.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Signs for transpositions: one of many methods
Definition
A braid sign convention is a function that associates to anybounce point xi of any bouncing path ε(xi ) = ±1.
Example
Let−→N be the vector from the identity permutation to the
reverse of the identity. Define the sign at xi to be the sign of
the dot product −−−→xi−1xi ·−→N . Think of the identity as the “south
pole ”. Positive means we were heading north before we“bounced”; negative means heading “south”.
![Page 87: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/87.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A bouncing path braid
Definition
Given a bouncing path x0, x1, x2, x3 . . . and a braid signconvention we obtain a bounce path braid— the braid givenby the word in the braid group:
σ(x1)ε1 σ(x2)ε2 σ(x3)ε3 · · ·
![Page 88: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/88.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Example of Bounce Braid for Order 4
![Page 89: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/89.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Example of Bounce Braid for Order 8
![Page 90: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/90.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Example of Bounce Braid for Order 12
![Page 91: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/91.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Example of Bounce Braid for Order 16
![Page 92: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/92.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Bounce Braids
Braid as an Aid
In general we need to look at the bounce permutationstogether with the bounce braid.
The bounce path indicates where the bounce occurs, the braidtells us something about how the “type” of bounce.
![Page 93: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/93.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
The nearest permutation to a point
Definition
A generic point z = (z1, . . . , zn) of Rn will have n distinctcoordinate values.
The rank of zi , denoted r(zi ), is one plus the number ofcoordinates of z smaller than zi .
Definition
The rank vector ρ(z) = (r(z1), . . . , r(zn))
In other words:
Simply Put: The rank of z is the closest permutation to z .
Or not: A generic point is mapped to a chamber of the realn-braid arrangement
![Page 94: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/94.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Finding the intersection of a ray and ∂P(n)
The key
Focus on the plane P that contains the three points
1 the centroid C of P(n),
2 the initial point x0
3 the tip of our vector x0 + λ.
We then project the wireframe of P(n) onto this plane.
![Page 95: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/95.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(5)
![Page 96: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/96.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(5)
![Page 97: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/97.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(5)
![Page 98: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/98.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(5)
![Page 99: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/99.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(5)
![Page 100: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/100.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(5)
![Page 101: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/101.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(6)
![Page 102: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/102.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(6)
![Page 103: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/103.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(7)
![Page 104: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/104.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(7)
![Page 105: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/105.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(7)
![Page 106: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/106.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(7)
![Page 107: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/107.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(7)
![Page 108: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/108.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Projection wireframe P(7)
![Page 109: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/109.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Observations
The higher permutahedra are not round.
(This is important since if we do our bouncing inside around ball, the path will be planar)
There seems to be some structure there that is evidentfrom the projections.(Some will be clearer with a colored wireframe)
The “central” portion of figures is hard to understand butthe area around the edge is much clearer.(In fact the bounding polygonal path of the projection isthe projection of a simple closed polygonal path of P(n)edges )
![Page 110: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/110.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Observations
The higher permutahedra are not round.(This is important since if we do our bouncing inside around ball, the path will be planar)
There seems to be some structure there that is evidentfrom the projections.(Some will be clearer with a colored wireframe)
The “central” portion of figures is hard to understand butthe area around the edge is much clearer.(In fact the bounding polygonal path of the projection isthe projection of a simple closed polygonal path of P(n)edges )
![Page 111: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/111.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Observations
The higher permutahedra are not round.(This is important since if we do our bouncing inside around ball, the path will be planar)
There seems to be some structure there that is evidentfrom the projections.
(Some will be clearer with a colored wireframe)
The “central” portion of figures is hard to understand butthe area around the edge is much clearer.(In fact the bounding polygonal path of the projection isthe projection of a simple closed polygonal path of P(n)edges )
![Page 112: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/112.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Observations
The higher permutahedra are not round.(This is important since if we do our bouncing inside around ball, the path will be planar)
There seems to be some structure there that is evidentfrom the projections.(Some will be clearer with a colored wireframe)
The “central” portion of figures is hard to understand butthe area around the edge is much clearer.(In fact the bounding polygonal path of the projection isthe projection of a simple closed polygonal path of P(n)edges )
![Page 113: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/113.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Observations
The higher permutahedra are not round.(This is important since if we do our bouncing inside around ball, the path will be planar)
There seems to be some structure there that is evidentfrom the projections.(Some will be clearer with a colored wireframe)
The “central” portion of figures is hard to understand butthe area around the edge is much clearer.
(In fact the bounding polygonal path of the projection isthe projection of a simple closed polygonal path of P(n)edges )
![Page 114: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/114.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Observations
The higher permutahedra are not round.(This is important since if we do our bouncing inside around ball, the path will be planar)
There seems to be some structure there that is evidentfrom the projections.(Some will be clearer with a colored wireframe)
The “central” portion of figures is hard to understand butthe area around the edge is much clearer.(In fact the bounding polygonal path of the projection isthe projection of a simple closed polygonal path of P(n)edges )
![Page 115: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/115.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Creating great path: projection to the plane P
![Page 116: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/116.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Same as previous figure after rotation in R3
![Page 117: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/117.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A Great Path Method
1 Let P be the plane through points: x0, x0 + λ and thecentroid of P(n).
2 Let π0 = ρ(x0).
3 Consider the projection φ of the wireframe W of P(n)onto P. Let D be the convex hull of φ(W ).
4 Each edge of ∂D is a projection of a single edge of W .
5 A union of these edges which form a polygonal arc in P(n)is called a great path.
6 From π0, follow this great path in the general direction ofλ obtaining vertex sequence p0, p1, . . ..
7 At each pi find the intersection point of the ray withhyperplanes determined by facets at pi .
8 By convexity of P(n) the closest such intersection point isour bounce point.
![Page 118: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/118.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A Great Path Method
1 Let P be the plane through points: x0, x0 + λ and thecentroid of P(n).
2 Let π0 = ρ(x0).
3 Consider the projection φ of the wireframe W of P(n)onto P. Let D be the convex hull of φ(W ).
4 Each edge of ∂D is a projection of a single edge of W .
5 A union of these edges which form a polygonal arc in P(n)is called a great path.
6 From π0, follow this great path in the general direction ofλ obtaining vertex sequence p0, p1, . . ..
7 At each pi find the intersection point of the ray withhyperplanes determined by facets at pi .
8 By convexity of P(n) the closest such intersection point isour bounce point.
![Page 119: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/119.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A Great Path Method
1 Let P be the plane through points: x0, x0 + λ and thecentroid of P(n).
2 Let π0 = ρ(x0).
3 Consider the projection φ of the wireframe W of P(n)onto P. Let D be the convex hull of φ(W ).
4 Each edge of ∂D is a projection of a single edge of W .
5 A union of these edges which form a polygonal arc in P(n)is called a great path.
6 From π0, follow this great path in the general direction ofλ obtaining vertex sequence p0, p1, . . ..
7 At each pi find the intersection point of the ray withhyperplanes determined by facets at pi .
8 By convexity of P(n) the closest such intersection point isour bounce point.
![Page 120: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/120.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A Great Path Method
1 Let P be the plane through points: x0, x0 + λ and thecentroid of P(n).
2 Let π0 = ρ(x0).
3 Consider the projection φ of the wireframe W of P(n)onto P. Let D be the convex hull of φ(W ).
4 Each edge of ∂D is a projection of a single edge of W .
5 A union of these edges which form a polygonal arc in P(n)is called a great path.
6 From π0, follow this great path in the general direction ofλ obtaining vertex sequence p0, p1, . . ..
7 At each pi find the intersection point of the ray withhyperplanes determined by facets at pi .
8 By convexity of P(n) the closest such intersection point isour bounce point.
![Page 121: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/121.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A Great Path Method
1 Let P be the plane through points: x0, x0 + λ and thecentroid of P(n).
2 Let π0 = ρ(x0).
3 Consider the projection φ of the wireframe W of P(n)onto P. Let D be the convex hull of φ(W ).
4 Each edge of ∂D is a projection of a single edge of W .
5 A union of these edges which form a polygonal arc in P(n)is called a great path.
6 From π0, follow this great path in the general direction ofλ obtaining vertex sequence p0, p1, . . ..
7 At each pi find the intersection point of the ray withhyperplanes determined by facets at pi .
8 By convexity of P(n) the closest such intersection point isour bounce point.
![Page 122: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/122.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A Great Path Method
1 Let P be the plane through points: x0, x0 + λ and thecentroid of P(n).
2 Let π0 = ρ(x0).
3 Consider the projection φ of the wireframe W of P(n)onto P. Let D be the convex hull of φ(W ).
4 Each edge of ∂D is a projection of a single edge of W .
5 A union of these edges which form a polygonal arc in P(n)is called a great path.
6 From π0, follow this great path in the general direction ofλ obtaining vertex sequence p0, p1, . . ..
7 At each pi find the intersection point of the ray withhyperplanes determined by facets at pi .
8 By convexity of P(n) the closest such intersection point isour bounce point.
![Page 123: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/123.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A Great Path Method
1 Let P be the plane through points: x0, x0 + λ and thecentroid of P(n).
2 Let π0 = ρ(x0).
3 Consider the projection φ of the wireframe W of P(n)onto P. Let D be the convex hull of φ(W ).
4 Each edge of ∂D is a projection of a single edge of W .
5 A union of these edges which form a polygonal arc in P(n)is called a great path.
6 From π0, follow this great path in the general direction ofλ obtaining vertex sequence p0, p1, . . ..
7 At each pi find the intersection point of the ray withhyperplanes determined by facets at pi .
8 By convexity of P(n) the closest such intersection point isour bounce point.
![Page 124: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/124.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A Great Path Method
1 Let P be the plane through points: x0, x0 + λ and thecentroid of P(n).
2 Let π0 = ρ(x0).
3 Consider the projection φ of the wireframe W of P(n)onto P. Let D be the convex hull of φ(W ).
4 Each edge of ∂D is a projection of a single edge of W .
5 A union of these edges which form a polygonal arc in P(n)is called a great path.
6 From π0, follow this great path in the general direction ofλ obtaining vertex sequence p0, p1, . . ..
7 At each pi find the intersection point of the ray withhyperplanes determined by facets at pi .
8 By convexity of P(n) the closest such intersection point isour bounce point.
![Page 125: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/125.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Yet more braids!
A braid from edges, not facets
1 Put together the great paths used in calculating a bouncesequence.
2 This sequence of edges of ∂P(n) gives a sequence ofelementary transpositions.
3 There are ways to define signs to this sequence giving yetmore braids.
![Page 126: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/126.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Very briefly—one way to get signs
Example
1 Consider the polygonal path β = P ∩ ∂P(n)
2 orient β using λ
3 orient the (n − 3)-cells of ∂P(n)
4 use intersection numbers of the β with those cells to getour sign
Note:
There is a quick indirect way to calculate this from theconstruction of the great path.
![Page 127: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/127.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Sorting networks
A connection to topic in computer science
There is a relationship between a colored great path whichjoins two antipodal permutations and the concept of a sortingnetwork.
![Page 128: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/128.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A different bouncing braid
A braid based on edges near bounce
1 Take bounce points x1, x2, . . . xn where xi lies in facet Fi
2 Let ei be the edge of Fi closest to xi with associated anelementary transposition Σ(xi )
3 There are a number of ways to assign a braid signconvention εi
4 This gives a braid word
Σ(x1)ε1Σ(x2)ε2 . . .Σ(xn)εn
![Page 129: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/129.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A different bouncing braid
There is not time to go into detail . . .
To give a sense that the color of an edge tells something aboutthe nature of the bounce consider the following graphics.
![Page 130: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/130.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A projection of labeled wireframe of P(5)
![Page 131: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/131.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A projection of labeled wireframe of P(5)
![Page 132: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/132.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A projection of labeled wireframe of P(6)
![Page 133: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/133.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A projection of labeled wireframe of P(7)
![Page 134: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/134.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
An alternative to bouncing: permutahedral tiles
Theorem
Rn can be tiled with translated copies P(n)
![Page 135: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/135.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A black and white tiling of the plane by hexagons
![Page 136: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/136.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Fitting two adjacent P(4)s
![Page 137: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/137.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Fitting three adjacent P(4)s
![Page 138: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/138.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Fitting four adjacent P(4)s
![Page 139: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/139.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A “black and white tiling of R3 by P(4)s
![Page 140: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/140.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
A “black and white tiling of R3 by P(4)s
![Page 141: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/141.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Review
Our Questions:
and some answers
1 How fast can we calculate the bouncing path for fairlyhigh orders—8, 12, 16, 32?
Quadratic, not exponential.
2 How can we visualize the calculational process and theresults?
Projections, color code, braids.
3 What is the geometry of a high dimensionalpermutahedron?
Related to Young subgroups and cosets
.
4 How does the geometry of the permutahedron changewith n?
It does not become rounder. In fact in somedirections it “flattens out”.
![Page 142: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/142.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Review
Our Questions: and some answers
1 How fast can we calculate the bouncing path for fairlyhigh orders—8, 12, 16, 32?
Quadratic, not exponential.
2 How can we visualize the calculational process and theresults?
Projections, color code, braids.
3 What is the geometry of a high dimensionalpermutahedron?
Related to Young subgroups and cosets
.
4 How does the geometry of the permutahedron changewith n?
It does not become rounder. In fact in somedirections it “flattens out”.
![Page 143: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/143.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Review
Our Questions: and some answers
1 How fast can we calculate the bouncing path for fairlyhigh orders—8, 12, 16, 32?
Quadratic, not exponential.
2 How can we visualize the calculational process and theresults?
Projections, color code, braids.
3 What is the geometry of a high dimensionalpermutahedron?
Related to Young subgroups and cosets
.
4 How does the geometry of the permutahedron changewith n?
It does not become rounder. In fact in somedirections it “flattens out”.
![Page 144: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/144.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Review
Our Questions: and some answers
1 How fast can we calculate the bouncing path for fairlyhigh orders—8, 12, 16, 32? Quadratic, not exponential.
2 How can we visualize the calculational process and theresults?
Projections, color code, braids.
3 What is the geometry of a high dimensionalpermutahedron?
Related to Young subgroups and cosets
.
4 How does the geometry of the permutahedron changewith n?
It does not become rounder. In fact in somedirections it “flattens out”.
![Page 145: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/145.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Review
Our Questions: and some answers
1 How fast can we calculate the bouncing path for fairlyhigh orders—8, 12, 16, 32? Quadratic, not exponential.
2 How can we visualize the calculational process and theresults? Projections, color code, braids.
3 What is the geometry of a high dimensionalpermutahedron?
Related to Young subgroups and cosets
.
4 How does the geometry of the permutahedron changewith n?
It does not become rounder. In fact in somedirections it “flattens out”.
![Page 146: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/146.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Review
Our Questions: and some answers
1 How fast can we calculate the bouncing path for fairlyhigh orders—8, 12, 16, 32? Quadratic, not exponential.
2 How can we visualize the calculational process and theresults? Projections, color code, braids.
3 What is the geometry of a high dimensionalpermutahedron? Related to Young subgroups and cosets.
4 How does the geometry of the permutahedron changewith n?
It does not become rounder. In fact in somedirections it “flattens out”.
![Page 147: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/147.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Review
Our Questions: and some answers
1 How fast can we calculate the bouncing path for fairlyhigh orders—8, 12, 16, 32? Quadratic, not exponential.
2 How can we visualize the calculational process and theresults? Projections, color code, braids.
3 What is the geometry of a high dimensionalpermutahedron? Related to Young subgroups and cosets.
4 How does the geometry of the permutahedron changewith n? It does not become rounder. In fact in somedirections it “flattens out”.
![Page 148: Geometric generation of permutation sequenceshomepage.divms.uiowa.edu/~roseman/mathviz09D.pdf · 2009-03-26 · The mathematical objects we chose are permutations. Construct families](https://reader034.vdocument.in/reader034/viewer/2022050220/5f663394daa7ba38e43a8b4c/html5/thumbnails/148.jpg)
Geometricgeneration ofpermutationsequences
DennisRoseman
Permutahedron
ChangeRinging
Bouncing
Problem List
Cell Structure
coloring edges
coloring facets
Braids
BeamCalculation
Edges in layers
Tiling
Thank you