euler characteristic (and the unity of mathematics)olazarev/splash.pdf · oleg lazarev euler...
TRANSCRIPT
![Page 1: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/1.jpg)
Euler Characteristic(and the Unity of Mathematics)
Oleg Lazarev
SPLASH
April 9, 2016
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 2: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/2.jpg)
Graphs
Collection of dots (vertices) connected by lines (edges)
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 3: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/3.jpg)
Planar Graphs
Definition: A graph is planar if there are no edge crossings
I Let V =number of vertices, E = number of edges, F =number of faces (including outside face)
I For Graph A: V = 5,E = 6,F = 3
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 4: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/4.jpg)
Euler characteristic
Definition: the Euler characteristic of a planar graph G is
χ(G ) = V − E + F
I Ex 1. χ(G ) = 4− 6 + 4 = 2.
I Ex 2. χ(G ) = 7− 9 + 4 = 2.
I Ex 3. χ(G ) = 8− 7 + 1 = 2.
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 5: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/5.jpg)
Euler’s Theorem
Euler’s theorem: All planar graphs have Euler characteristic 2!
I Ex. when subdivide an edge by adding a vertex,V ′ − V = 1,E ′ − E = 1,F ′ − F = 0 so χ(G ′) = χ(G )
I Ex: when add an edge connected to two vertices,V ′ − V = 0,E ′ − E = 1,F ′ − F = 1 so χ(G ′) = χ(G )
I Ex: when add an edge connected to one vertex,V ′ − V = 1,E ′ − E = 1,F ′ − F = 0 so χ(G ′) = χ(G )
I So χ(G ) = χ(single vertex) = 1 + 1 = 2
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 6: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/6.jpg)
Non-planar graphs
Planarity is really an intrinsic property: a graph is planar if it canbe draw so that no edge crossings.
Example:
These all have the same underlying graph, with edges redrawn.Hence the graph is planar.
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 7: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/7.jpg)
Non-planar graphs
Planar graphs have e ≤ 3v − 6
I For any planar graph, 3f ≤ 2e
I Euler characteristic 2 = v − e + f
I So 6 = 3v − 3e + 3f ≤ 3v − 3e + 2e = 3v − e
I Hence e ≤ 3v − 6 for planar graphs
Ex 1. K5 graph
V = 5,E = 10 so 10 6≤ 3 · 5− 6 so K5 is not planar!
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 8: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/8.jpg)
Non-Planar Graphs
Ex 2. K3,3 graph
By similar argument, K3,3 is non-planar.
Kuratowski’s Theorem: a graph is planar if and only if it has nosubgraphs that are (expansions of) K3,3 or K5
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 9: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/9.jpg)
Topology
Topology: “the study of geometric properties and spatial relationsunaffected by the continuous change of shape or size of figures.”
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 10: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/10.jpg)
Surfaces
I All (orientable) surfaces without boundary are
i.e. spheres and multi-holed donuts.
I Example of non-orientable surface: Mobius strip
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 11: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/11.jpg)
Euler characteristic of Surfaces
Triangulation of surfaces:
Definition: Euler χ(S ,T ) of a surface S is V − E + F for sometriangulation T of S
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 12: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/12.jpg)
Euler characteristic of Surfaces
I Euler’s Theorem implies that all spheres have Eulercharacteristic 2!
I Similar to proof of Euler’s Theorem: χ(S) is independent oftriangulation, i.e. Euler characteristic is a topologicalinvariant!
I HW: what is Euler characteristic for donuts with g holes?I HW: define Euler characteristic for non-planar graphs
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 13: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/13.jpg)
Platonic Solids
Constructed from congruent regular polygons (equal side lengthand angles) with the same number of faces meeting at each vertex.
I Plato: each is associated to ”classical elements” - Earth, air,water, and fire (fifth one?)
I Ex. Euler characteristic of tetrahedron: 4− 6 + 4 = 2
I Ex. Euler characteristic of octahedron: 6− 12 + 8 = 2
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 14: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/14.jpg)
Classification of Platonic Solids
Theorem: These are the only Platonic solids!
I Suppose faces are n-gons and m edges meet at each vertex
I So nF = 2E and mV = 2E
I Euler characteristic becomes V − E + F = 2Em − E + 2E
n = 2or 1
m + 1n = 1
E + 12
I There are only five such pairs satisfying this equation(m, n) = (3, 3), (3, 4), (3, 5), (4, 3), (5, 3) and these all givePlatonic solids (tetrahedron, hexahedron, dodecahedron,octahedron, icosahedron)
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 15: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/15.jpg)
Hairy Ball Theorem
Theorem: Can’t comb a hairy ball flat (two hairs will always besticking out).
Or there is always a point on Earth where wind velocity is exactlyzero (actually two points)Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 16: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/16.jpg)
Hairy Donut Theorem?
Hairy Donuts can be combed.
”Combability” depends on topology.
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 17: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/17.jpg)
Vector Fields
I Mathematicians use vector fields v
I At each point on a
I Ex. I (S2, v) = 2
I Ex. I (T 2, v) = 0
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 18: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/18.jpg)
Poincare-Hopf Theorem
Poincare-Hopf Theorem: for any vector field v
I
I Links topology and geometry; a priori vector fields has nothingto do with topology; if don’t take alternating signs
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 19: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/19.jpg)
Construction of Vector Fields
Proposition: exists a vector field v on X with I (X , v) = χ(X )
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 20: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/20.jpg)
Geometry
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 21: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/21.jpg)
Curvature
Not a topological invariant!
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 22: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/22.jpg)
Gauss-Bonnet Theorem
X is a surface Gauss-Bonnet Theorem:∫X K (x) = χ(X ).
I Links geometry and topology!
I Right-hand-side seems like it depends on how put surface intospace but actually independent!
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 23: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/23.jpg)
Betti Numbers
I Euler characteristic are a shadow of more invariants
I To any space X , we can assign sequence of ‘Betti numbers’bi (X ) ≥ 0, where i ≥ 0 is an integer; that are invariants
I Then χ(X ) =∑
i≥0(−1)ibi (X )
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)
![Page 24: Euler Characteristic (and the Unity of Mathematics)olazarev/SPLASH.pdf · Oleg Lazarev Euler Characteristic (and the Unity of Mathematics) Hairy Ball Theorem Theorem: Can’t comb](https://reader034.vdocument.in/reader034/viewer/2022042810/5f9e02384da65865921f3ea7/html5/thumbnails/24.jpg)
Thank You!
Oleg Lazarev Euler Characteristic (and the Unity of Mathematics)