space lattices crystal structures symmetry, point groups and space groups geometry of crystals...
TRANSCRIPT
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Space Lattices
Crystal Structures
Symmetry, Point Groups and Space Groups
GEOMETRY OF CRYSTALSGEOMETRY OF CRYSTALS
Acknowledgments: Prof. Rajesh Prasad for a lot of things
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Crystal = Lattice + Motif
Motif or basis: an atom or a group of atoms associated with each lattice point
The language of crystallography is one succinctness
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An array of points such that every point has identical surroundings
In Euclidean space infinite array
We can have 1D, 2D or 3D arrays (lattices)
Space Lattice
Translationally periodic arrangement of points in space is called a lattice
or
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A 2D lattice
a
b
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Translationally periodic arrangement of motifs
Crystal
Translationally periodic arrangement of points
Lattice
Lattice the underlying periodicity of the crystal
Basis atom or group of atoms associated with each lattice points
Lattice how to repeat
Motif what to repeat
Crystal = Lattice + Motif
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+
Lattice
Motif
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Crystal
=
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A cell is a finite representation of the infinite lattice
A cell is a parallelogram (2D) or a parallelopiped (3D) with lattice points at their corners.
If the lattice points are only at the corners, the cell is primitive.
If there are lattice points in the cell other than the corners, the cell is nonprimitive.
Cells
Instead of drawing the whole structure I can draw a representative partand specify the repetition pattern
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Primitivecell
Primitivecell
Nonprimitive cell
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Primitivecell
Primitivecell
Nonprimitive cell
Double
Triple
Symmetry of the Lattice or the crystal is not altered by our choice of unit cell!!
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Primitive cell
Nonprimitive cell
4- fold axes
Centred square lattice = Simple/primitive square lattice
Shortest lattice translation vector ½ [11]
a
b
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Nonprimitive cell
Primitive cell
Lower symmetry than the lattice usually not chosen
Maintains the symmetry of the lattice the usual choice
2- fold axes
Centred rectangular lattice
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Centred rectangular lattice Simple rectangular Crystal
Shortest lattice translation vector [10]
Not a cell
Primitive cell
MOTIF
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In order to define translations in 3-d space, we need 3 non-coplanar vectors
Conventionally, the fundamental translation vector is taken from one lattice point to the next in the chosen direction
With the help of these three vectors, it is possible to construct a parallelopiped called a CELL
Cells- 3D
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Different kinds of CELLS
Unit cell
A unit cell is a spatial arrangement of atoms which is tiled in three-dimensional space to describe the crystal.
Primitive unit cell
For each crystal structure there is a conventional unit cell, usually chosen to make the resulting lattice as symmetric as possible. However, the conventional unit cell is not always the smallest possible choice. A primitive unit cell of a particular crystal structure is the smallest possible unit cell one can construct such that, when tiled, it completely fills space.
Wigner-Seitz cell
A Wigner-Seitz cell is a particular kind of primitive cell
which has the same symmetry as the lattice.
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If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation.
SYMMETRY
Given a general point a symmetry operator leaves a finite set of points in space
A symmetry operator closes space onto itself
SYMMETRY OPERATOR
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Symmetry operators
Symmetries
Type II
Type IRotation
Translation
Inversion
Mirror
Takes object to same form → Proper
Takes object to enantiomorphic form → improper Roto-inversion
Roto-reflection
Classification based on the dimension invariant entity of the symmetry operator
Operator Dimension
Inversion 0D
Rotation 1D
Mirror 2D
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Symmetry operators
Symmetries
Microscopic
Macroscopic
Rotation
Mirror
Glide Reflection
Screw Axes
Inversion
Influence the external shape of the crystal
Do not Influence the external shape of the crystal
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R Rotation G Glide reflection
R Roto-inversion S Screw axis
Ones with built in translationOnes acting at a point
Minimum set of symmetry operators required
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If an object come into self-coincidence through smallest non-zero rotation angle of then it is said to have an n-fold rotation axis where
0360n
=180
Rotation Axis
n=2 2-fold rotation axis
=120 n=3 3-fold rotation axis
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=90 n=4 4-fold rotation axis
=60 n=6 6-fold rotation axis
The rotations compatible with translational symmetry are (1, 2, 3, 4, 6)
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Symmetries actingat a point
R R
R + R → rotations compatible with translational symmetry (1, 2, 3, 4, 6)
32 point groups
Along with symmetrieshaving a translation
G + S
230 space groups
Point group symmetry of Lattices →
7 crystal systems
Space group symmetry of Lattices →
14 Bravais lattices
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Crystal = Lattice (Where to repeat)
+ Motif (What to repeat)
Previously
=
+
a
a
2
a
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Crystal =
Space group (how to repeat)
+ Asymmetric unit (Motif’: what to repeat)
Now
=
+
a
aGlide reflection operator
Usually asymmetric units are regions of space within the unit cell- which contain atoms
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Progressive lowering of symmetry in an 1D lattice illustration using the frieze groups
Consider a 1D lattice with lattice parameter ‘a’
a
Unit cell
Three mirror planes The intersection points of the mirror planesgive rise to redundant inversion centres
mmm
Asymmetric Unit
mirror glide reflection
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Decoration of the lattice with a motif may reduce the symmetry of the crystal
Decoration with a “sufficiently” symmetric motif does not reduce the symmetry of the lattice
1
2
mmm
mm
Loss of 1 mirror plane
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Lattice points
Not a lattice point
g
Presence of 1 mirror plane and 1 glide reflection plane, with a redundant inversion centrethe translational symmetry has been reduced to ‘2a’
2 inversion centres
ii
mg3
4
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g
1 mirror plane
m
g
1 glide reflection translational symmetry of ‘2a’
No symmetry except translation
5
6
7
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Effect of the decoration a 2D example
4mmRedundant inversion centre
Decoration retaining the symmetry
4mm
Can be a unit cell for a 2D crystal
Two kinds of decoration are shown (i) for an isolated object, (ii) an object which can be an unit cell.
Redundant mirrors
which need not be drawn
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mm m m
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No symmetry
4
If this is an unit cell of a crystal → then the crystal would still have translational symmetry
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Lattices have the highest symmetry Crystals based on the lattice
can have lower symmetry
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Unit cell ofTriclinic crystal
Amorphous arrangementNo unit cell
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Positioning a object with respect to the symmetry elements
Three mirror planes The intersection points of the mirror planesgive rise to redundant inversion centres
mmm
Right handed object
Left handed object
Object with bilateral symmetry
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Positioning a object with respect to the symmetry elements
Note: this is for a point group and not for a lattice the black lines are not unit cells
General site 8 identiti-points
On mirror plane (m) 4 identiti-points
On mirror plane (m) 4 identiti-points
Site symmetry 4mm 1 identiti-point
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Positioning of a motif w.r.t to the symmetry elements of a lattice Wyckoff positions
A 2D lattice with symmetry elements
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Multi-plicity
Wyckoff
letter
Site symmetry
Coordinates
8 g
Area 1
(x,y) (-x,-y) (-y,x) (y,-x)
(-x,y) (x,-y) (y,x) ((-y,-x)
4 f
Lines
..m (x,x) (-x,-x) (x,-x) (-x,x)
4 e .m. (x,½) (-x, ½) (½,x) (½,-x)
4 d .m. (x,0) (-x,0) (0,x) (0,-x)
2 c
Points
2mm. (½,0) (0,½)
1 b 4mm (½,½)
1 a 4mm (0,0)
a
b
c
d
ef
Number of Identi-points
Any site of lower symmetry should exclude site(s) of higher symmetry [e.g. (x,x) in site f cannot take values (0,0) or (½, ½)]
g
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a
b
c
d
ef
d
Exclude thesepoints
g
Exclude thesepoints
f
Exclude thesepoints
e
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Bravais Space Lattices some other view points
Conventionally, the finite representation of space lattices is done using unit cells which show maximum possible symmetries with the smallest size.
Or the technical definition
There are 14 Bravais Lattices which are the space group symmetries of lattices
Considering
1. Maximum Symmetry, and
2. Minimum Size
Bravais concluded that there are only 14 possible Space Lattices (or Unit Cells to represent them).
These belong to 7 Crystal systems
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Bravais Lattice
A lattice is a set of points constructed by translating a single point in discrete steps by a set of basis vectors. In three dimensions, there are 14 unique Bravais lattices (distinct from one another in that they have different space groups) in three dimensions. All crystalline materials recognized till now fit in one of these arrangements.
or
In geometry and crystallography, a Bravais lattice is an infinite set of points generated by a set of discrete translation operations. A Bravais lattice looks exactly the same no matter from which point one views it.
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Arrangement of lattice points in the unit cell& No. of Lattice points / cell
Position of lattice pointsEffective number of Lattice points / cell
1 P 8 Corners = 8 x (1/8) = 1
2 I8 Corners + 1 body centre
= 1 (for corners) + 1 (BC)
3 F8 Corners +
6 face centres
= 1 (for corners) + 6 x (1/2) = 4
4
A/
B/
C
8 corners +2 centres of opposite faces
= 1 (for corners) + 2x(1/2)= 2
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14 Bravais lattices divided into seven crystal systems
Crystal system Bravais lattices
1. Cubic P I F
2. Tetragonal P I
3. Orthorhombic P I F C
4. Hexagonal P
5. Trigonal P
6. Monoclinic P C
7. Triclinic P
Courtesy Dr. Rajesh Prasad
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14 Bravais lattices divided into seven crystal systems
Crystal system Bravais lattices
1. Cubic P I F C
2. Tetragonal P I
3. Orthorhombic P I F C
4. Hexagonal P
5. Trigonal P
6. Monoclinic P C
7. Triclinic P
Courtesy Dr. Rajesh Prasad
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Cubic F Tetragonal I
The symmetry of the unit cell is lower than that of the crystal
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14 Bravais lattices divided into seven crystal systems
Crystal system Bravais lattices
1. Cubic P I F C
2. Tetragonal P I F
3. Orthorhombic P I F C
4. Hexagonal P
5. Trigonal P
6. Monoclinic P C
7. Triclinic P
Courtesy Dr. Rajesh Prasad
x
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The following 4 things are different
Symmetry of theMotif
Crystal
Lattice
Unit Cell
Eumorphic crystal (equilibrium shape and growth shape of the crystal)
The shape of the crystal corresponds to the point group symmetry of the crystal
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FCT = BCT
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Crystal system
The crystal system is the point group of the lattice (the set of rotation and reflection symmetries which leave a lattice point fixed), not including the positions of the atoms in the unit cell.
There are seven unique crystal systems.
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Concept of symmetry and choice of axes
(a,b)
222 )()( rbyax
222 )()( ryx
Polar coordinates (, )
r
The centre of symmetry of the object does not coincide with the origin
The type of coordinate system chosen is notaccording to the symmetryof the object
Centre of Inversion
Mirror
Our choice of coordinate axis does not alter the symmetry of the object (or the lattice)
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THE 7 CRYSTAL SYSTEMSTHE 7 CRYSTAL SYSTEMS
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TRICLINIC MONOCLINIC ORTHORHOMBIC TRIGONAL TETRAGONAL HEXAGONAL
N 2 3 3 5 7 7
X 1 2 3 4 6
X 1 m2 3 4 6
m
3
1X m
2 3
m
4
m
6
)2(2X 222 32 422 622
)(mXm 2mm 3m 4mm 6mm
mX 2mm 3m
2 m3 m24 26m
12 X m
2
m
2
m
2 3
m
2
m
4
m
2
m
2
m
6
m
2
m
2
Increasing Symmetry
In
crea
sing
Sym
met
ry
CUBIC = ISOMETRIC N = 5
X = 2 X = 4 X = 4
3X 23 m34 432
)13(3 XX m
23 m 3
m
43
m
2
m
43
m
2 (m3m)
N is the number of point groups for a crystal system
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1. Cubic Crystals
a = b= c = = = 90º
• Simple Cubic (P)
• Body Centred Cubic (I) – BCC
• Face Centred Cubic (F) - FCC
FluoriteOctahedron
PyriteCube
m
23
m
4 432, ,3m 3m,4 23, groupsPoint
[1] http://www.yourgemologist.com/crystalsystems.html[2] L.E. Muir, Interfacial Phenomenon in Metals, Addison-Wesley Publ. co.
[1] [1]GarnetDodecahedron
[1]
Vapor grown NiO crystal
[2]
Tetrakaidecahedron(Truncated Octahedron)
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2. Tetragonal Crystalsa = b c = = = 90º
• Simple Tetragonal
• Body Centred Tetragonal
m
2
m
2
m
42m,4 4mm, 422, ,
m
4 ,4 4, groupsPoint
[1] http://www.yourgemologist.com/crystalsystems.html
[1] [1] [1]
Zircon
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3. Orthorhombic Crystalsa b c = = = 90º
• Simple Orthorhombic
• Body Centred Orthorhombic
• Face Centred Orthorhombic
• End Centred Orthorhombic
m
2
m
2
m
2 2mm, 222, groupsPoint
[1] http://www.yourgemologist.com/crystalsystems.html
Topaz
[1]
[1]
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4. Hexagonal Crystalsa = b c = = 90º = 120º
• Simple Hexagonal
m
2
m
2
m
6 m2,6 6mm, 622, ,
m
6 ,6 6, groupsPoint
[1] http://www.yourgemologist.com/crystalsystems.html
[1] Corundum
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5. Rhombohedral Crystalsa = b = c = = 90º
• Rhombohedral (simple)
m
23 3m, 32, ,3 3, groupsPoint
[1] http://www.yourgemologist.com/crystalsystems.html
Tourmaline[1] [1]
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6. Monoclinic Crystalsa b c = = 90º
• Simple Monoclinic• End Centred (base centered) Monoclinic (A/C)
m
2 ,2 2, groupsPoint
[1] http://www.yourgemologist.com/crystalsystems.html
Kunzite
[1]
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7. Triclinic Crystalsa b c
• Simple Triclinic
1 1, groupsPoint
[1] http://www.yourgemologist.com/crystalsystems.html
Amazonite[1]
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Concept of symmetry and choice of axes
(a,b)
222 )()( rbyax
222 )()( ryx
Polar coordinates (, )
r
The centre of symmetry of the object does not coincide with the origin
The type of coordinate system chosen is notaccording to the symmetryof the object
Centre of Inversion
Mirror
Our choice of coordinate axis does not alter the symmetry of the object (or the lattice)
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Alternate choice of unit cells for Orthorhombic lattices
Alternate choice of unit cell for “C”(C-centred orthorhombic) case. The new (orange) unit cell is a rhombic prism with
(a = b c, = = 90o, 90o, 120o) Both the cells have the same symmetry (2/m 2/m 2/m) In some sense this is the true Ortho-”rhombic” cell
m
2
m
2
m
2
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1/2
1/2
Note: All spheres represent lattice points. They are coloured differently but are the same
z = 0 &z = 1
z = ½
Conventional
Alternate choice
(“ortho-rhombic”)
P C2ce the size
IF
2ce the size
FI 1/2 the size
CP 1/2 the size
A consistent alternate set of axis can be chosen for Orthorhombic lattices
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Intuitively one might feel that the orthogonal cell has a higher symmetry is there some reason for this?
Artificially introduced 2-folds (not the operations of the lattice)
2x
2y
2d
2d produces this additional pointnot part of the original lattice
The 2x and 2y axes move lattice points out the plane of the sheet in a semi-circle to other points of the lattice (without introducing any new points) The 2d axis introduces new points which are not lattice points of the original lattice
The motion of the lattice points under the effect of the artificially introduced 2-folds is shown as dashed lines (---)
This is in addition to our liking for 90!
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Cubic48
Tetragonal16
Triclinic2
Monoclinic4
Orthorhombic8
Progressive lowering of symmetry amongst the 7 crystal systems
Hexagonal24
Trigonal12
Incr
easi
ng s
ymm
etry
Superscript to the crystal system is the order of the lattice point group
Arrow marks lead from supergroups to subgroups
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Cubic (p = 2, c = 1, t = 1)a = b = c
= = = 90º
Tetragonal (p = 3, c = 1 , t = 2) a = b c
= = = 90º
Triclinic (p = 6, c = 0 , t = 6) a b c
90º
Monoclinic (p = 5, c = 1 , t = 4)a b c
= = 90º, 90º
Orthorhombic1 (p = 4, c = 1 , t = 3) a b c
= = = 90º
Progressive relaxation of the constraints on the lattice parameters amongst the 7 crystal systems
Hexagonal (p = 4, c = 2 , t = 2)a = b c
= = 90º, = 120º
Trigonal (p = 2, c = 0 , t = 2)a = b = c
= = 90º
Orthorhombic2 (p = 4, c = 1 , t = 3) a = b c
= = 90º, 90º
Incr
easi
ng n
umbe
r t
• p = number of independent parameters = (p e)
• c = number of constraints (positive “=“)
• t = terseness = (p c) (is a measure of the ‘expenditure’ on the parameters
Orthorhombic1 and Orthorhombic2 refer to the two types of cells
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Minimum symmetry requirement for the 7 crystal systems
Crystal system
Characteric symmetry Point groups Comment
Cubic Four 3-fold rotation axes m
23
m
4 432, ,3m 3m,4 23,
3 or 3 in the second place Two 3-fold axes will generate the other two 3-fold axes
Hexagonal One 6-fold rotation axis (or roto-inversion axis) m
2
m
2
m
6 m2,6 6mm, 622, ,
m
6 ,6 6,
6 in the first place
Tetragonal (Only) One 4-fold rotation axis (or roto-inversion axis)
m
2
m
2
m
42m,4 4mm, 422, ,
m
4 ,4 4,
4 in first place but no 3 in second place
Trigonal (Only) One 3-fold rotation axis (or roto-inversion axis)
m
23 3m, 32, ,3 3,
3 or 3 in the first place
Orthorhombic (Only) Three 2-fold rotation axes (or roto-inversion axis)
m
2
m
2
m
2 2mm, 222,
Monoclinic (Only) One 2-fold rotation axis (or roto-inversion axis)
m
2 ,2 2,
Triclinic None 1 1, 1 could be present
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nD No. SYMBOL
0 1
1 1/
2 {p}
3 5 {p, q}
4 6 {p, q, r}
5 3 {p, q, r, s}
POINT
LINE SEGMENT
TRIANGLE {3} SQUARE {4} PENTAGON {5} HEXAGON {6}
TETRAHEDRON {3, 3}
OCTAHEDRON {3, 4}
DODECAHEDRON {5, 3}
ICOSAHEDRON {3, 5}
SIMPLEX {3, 3, 3}
16-CELL {3, 3, 4}
120-CELL {5, 3, 3}
600-CELL {3, 3, 5}
HYPERCUBE {4, 3, 3}
DRP
CRN24-CELL {3, 4, 3}
CUBE {4, 3}
REGULAR SIMPLEX {3, 3, 3, 3}
CROSS POLYTOPE {3, 3, 3, 4}
MEASURE POLYTOPE {4, 3, 3, 3}
REGULAR SOLIDS IN VARIOUS DIMENSIONS
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