Uniform polytope

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A uniform polytope is a vertex-transitive polytope made from uniform polytope facets and lower elements.

Uniformity is a generalization of the older category semiregular, but also includes the regular polytopes. Further, nonconvex regular faces and vertex figures (star polygons) are allowed, which greatly expand the possible solutions.

A strict definition requires uniform polytopes be finite, while a more expansive definition allows uniform tessellations (tilings and honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well.

Uniform polytopes are created by a Wythoff construction, and each form can be represented by a linear Coxeter-Dynkin diagram.

The terminology for the convex uniform polytopes used in uniform polyhedron, uniform polychoron, and convex uniform honeycomb articles were coined by Norman Johnson.

[edit] Rectification operators

Regular n-polytopes have n+1 orders of rectification. The zeroth rectification is the original form. The nth rectification is the dual. The first rectification reduces edges to vertices. The second rectification reduces faces to vertices. The third rectification reduces cells to vertices, etc.

An extended Schläfli symbol can be used for representing rectified forms, with a single subscript:

n-th rectification = t_n{p,q,...}

[edit] Truncation operators

Regular n-polytopes have n orders of truncations that can be applied in any combination, and which can create new uniform polytopes.

Truncation - applied to polygons and higher. A truncation is a form that exists between adjacent rectified forms.
- Schläfli symbol for the nth truncation is t_n-1,n{p,q,...}
Cantellation - applied to polyhedrons and higher and creates uniform polytopes that exists between alternate rectified forms.
- Schläfli symbol for the n-th cantellation is t_n-1,n+1{p,q,...}
Runcination - applied to polychorons and higher and creates uniform polytopes that exists between third alternate rectified forms.
- Schläfli symbol for the n-th runcination is t_n-1,n+2{p,q,...}
Sterication - applied to 5-polytopes and higher and creates uniform polytopes that exists between fourth alternate rectified forms.
- Schläfli symbol for the n-th sterication is t_n-1,n+3{p,q,...}

In addition combinations of truncations can be performed which also generate new uniform polytopes. For example a cantitruncation is a cantellation and truncation applied together.

If all truncations are applied at once the operation can be more generally called an omnitruncation.

[edit] Alternation

One special operation, called alternation removes alternate vertices on polytope with all even-sided faces. An alternation applied to an omnitruncated polytope is called a snub.

The resulting polytopes always can be constructed, and are not generally reflective, and also do not in general have uniform polytope solutions.

[edit] Classes of polytopes by dimension

Uniform polygons: an infinite set of regular polygons and star polygons (one for each rational number greater than 2).

Uniform polyhedra:
- Convex forms
  - 5 convex regular (Platonic solids);
  - 13 convex semiregular (Archimedean solids);
  - an infinite set of semiregular prisms (one for each convex regular polygon);
  - an infinite set of semiregular antiprisms (one for each convex regular polygon);
- Nonconvex forms
  - 4 nonconvex regular (Kepler-Poinsot polyhedra);
  - 53 other nonconvex forms;
  - an infinite set of nonconvex uniform prisms (one for each regular star polygon);
  - an infinite set of nonconvex uniform antiprisms (one for each noninteger rational number greater than 3/2).

Uniform polychoron:
- Convex forms
  - 6 convex regular polychora
  - 41 convex uniform polychora;
  - 18 convex hyperprisms based on the Platonic and Archimedean solids (including the cube-prism, better known as the regular tesseract);
  - an infinite set of hyperprisms based on the convex antiprisms;
  - an infinite set of convex duoprisms;
- Nonconvex forms
  - 10 nonconvex regular polychora (Schläfli-Hess polychora)
  - 57 nonconvex hyperprisms based on the nonconvex uniform polyhedra;
  - an unknown number of nonconvex nonprismatic uniform polychora (over a thousand have been found);
  - an infinite set of hyperprisms based on the nonconvex antiprisms;
  - an infinite set of nonconvex duoprisms based on star polygons.

Higher dimensional uniform polytopes are not fully known. Most may be generated from a Wythoff construction applied to the regular forms.

Regular n-polytope families include the simplex, hypercube, and cross-polytope.

The demihypercube family, derived from the hypercubes by removing alternate vertices, includes the tetrahedron derived from the cube and the 16-cell derived from the tesseract. Higher members of the family are uniform but not regular.

[edit] Families of convex uniform polytopes

Families of convex uniform polytopes are defined by Coxeter groups. In addition prismatic families exist as products of this groups.

Categorical regular and prismatic family groups, up to 8-polytopes, are given below. Each permutation of indices of regular polytopes defines another family.

The Coxeter-Dynkin diagram is given for the first form in each family. Every combination of rings, with each prismatic group having at least one ring, produces another uniform primatic polytope.

[edit] Convex uniform polytope families by dimension

[edit] 1-polytope

1. A₁: [ ] - digon

[edit] 2-polytope

1. D₂^p: [p]

[edit] 3-polytope

1. A₃: [3,3]
2. C₃: [4,3]
3. G₃: [5,3]
4. D₂^pxA₁: [p] x [ ]

[edit] 4-polytope

1. A₄: [3,3,3]
2. C₄: [4,3,3]
3. F₄: [3,4,3]
4. G₄: [5,3,3]
5. B₄:
6. A₃xA₁: [3,3] x [ ] -
7. C₃xA₁: [4,3] x [ ] -
8. G₃xA₁: [5,3] x [ ] -
9. D₂^pxD₂^q: [p] x [q]

[edit] 5-polytope

1. A₅: [3,3,3,3]
2. C₅: [4,3,3,3]
3. B₅:
4. A₄xA₁: [3,3,3] x [ ]
5. C₄xA₁: [4,3,3] x [ ]
6. F₄xA₁: [3,4,3] x [ ]
7. G₄xA₁: [5,3,3] x [ ]
8. B₄xA₁:
9. A₃xD₂^p: [3,3] x [p] -
10. C₃xD₂^p: [4,3] x [p] -
11. G₃xD₂^p: [5,3] x [p] -
12. D₂^pxD₂^qxA₁: [p] x [q] x [ ]

[edit] 6-polytope

1. A₆:[3,3,3,3,3]
2. C₆:[4,3,3,3,3]
3. B₆:
4. E₆:
5. A₅xA₁: [3,3,3,3] x [ ]
6. C₅xA₁:[4,3,3,3] x [ ]
7. B₅xA₁:
8. A₄xD₂^p: [3,3,3] x [p]
9. C₄xD₂^p: [4,3,3] x [p]
10. F₄xD₂^p: [3,4,3] x [p]
11. G₄xD₂^p: [5,3,3] x [p]
12. B₄xD₂^p:
13. A₃xA₃: [3,3] x [3,3]
14. A₃xC₃: [3,3] x [4,3]
15. A₃xG₃: [3,3] x [5,3]
16. C₃xC₃: [4,3] x [4,3]
17. C₃xG₃: [4,3] x [5,3]
18. G₃xA₃: [5,3] x [5,3]
19. A₃xD₂^pxA₁: [3,3] x [p] x [ ]
20. C₃xD₂^pxA₁: [4,3] x [p] x [ ]
21. G₃xD₂^pxA₁: [5,3] x [p] x [ ]
22. D₂^pxD₂^qxD₂^r: [p] x [q] x [r] - triprism

[edit] 7-polytope

1. A₇: [3,3,3,3,3,3]
2. C₇: [4,3,3,3,3,3]
3. B₇:
4. E₇:
5. A₆xA₁: [3,3,3,3,3,3]
6. C₆xA₁: [4,3,3,3,3,3]
7. B₆xA₁:
8. E₆xA₁:
9. A₅xD₂^p: [3,3,3] x [p]
10. C₅xD₂^p: [4,3,3] x [p]
11. B₅xD₂^p:
12. A₄xA₃: [3,3,3] x [3,3]
13. A₄xC₃: [3,3,3] x [4,3]
14. A₄xG₃: [3,3,3] x [5,3]
15. C₄xA₃: [4,3,3] x [3,3]
16. C₄xC₃: [4,3,3] x [4,3]
17. C₄xG₃: [4,3,3] x [5,3]
18. G₄xA₃: [5,3,3] x [3,3]
19. G₄xC₃: [5,3,3] x [4,3]
20. G₄xG₃: [5,3,3] x [5,3]
21. F₄xA₃: [3,4,3] x [3,3]
22. F₄xC₃: [3,4,3] x [4,3]
23. F₄xG₃: [3,4,3] x [5,3]
24. B₄xA₃:
25. B₄xC₃:
26. B₄xG₃:
27. A₄xD₂^pxA₁: [3,3,3] x [p] x [ ]
28. C₄xD₂^pxA₁: [4,3,3] x [p] x [ ]
29. F₄xD₂^pxA₁: [3,4,3] x [p] x [ ]
30. G₄xD₂^pxA₁: [5,3,3] x [p] x [ ]
31. B₄xD₂^pxA₁:
32. A₃xA₃xA₁: [3,3] x [3,3] x [ ]
33. A₃xC₃xA₁: [3,3] x [4,3] x [ ]
34. A₃xG₃xA₁: [3,3] x [5,3] x [ ]
35. C₃xC₃xA₁: [4,3] x [4,3] x [ ]
36. C₃xG₃xA₁: [4,3] x [5,3] x [ ]
37. G₃xA₃xA₁: [5,3] x [5,3] x [ ]
38. A₃xD₂^pxD₂^q: [3,3] x [p] x [q]
39. C₃xD₂^pxD₂^q: [4,3] x [p] x [q]
40. G₃xD₂^pxD₂^q: [5,3] x [p] x [q]
41. D₂^pxD₂^qxD₂^rA₁: [p] x [q] x [r] x [ ]

[edit] 8-polytope (incomplete)

1. A₈: [3,3,3,3,3,3,3]
2. C₈: [4,3,3,3,3,3,3]
3. B₈:
4. E₈:
5. A₇xA₁: [3,3,3,3,3,3] x [ ]
6. C₇xA₁: [4,3,3,3,3,3] x [ ]
7. B₇xA₁:
8. [p,q,r,s,t] x [u]
9. [p,q,r,s] x [t,u]
10. [p,q,r] x [s,t,u]
11. [p,q,r,s] x [t] x [ ]
12. [p,q,r] x [s,t] x [ ]
13. [p,q,r] x [s] x [t]
14. [p,q] x [r,s] x [t]
15. [p,q] x [r] x [s] x [ ]
16. [p] x [q] x [r] x [s] - tetraprism

Special cases of products become hypercubes:

[ ] x [ ] = [4]
[ ] x [ ] x [ ] = [4,3]
[ ] x [ ] x [ ] = [4,3,3]
[ ] x [ ] x [ ] x [ ] = [4,3,3,3]
...

[edit] Uniform polygons

Regular polygons, represented by Schläfli symbol {p} for a p-gon. Regular polygons are self-dual, so the rectification produces the same polygon. The uniform truncation operation doubles the sides to {2p}. The snub operation, alternatingly truncating the truncation returns it back to the original polygon {p}. Thus all uniform polygons are also regular.

Operation	Extended Schläfli Symbols	Regular result	Coxeter- Dynkin Diagram	Position
Operation	Extended Schläfli Symbols	Regular result	Coxeter- Dynkin Diagram	(1)	(0)
Parent	t₀{p}	{p}		{}	--
Rectified (Dual)	t₁{p}	{p}		--	{}
Truncated	t_0,1{p}	{2p}		{}	{}
Snub	s{p}	{p}		--	--

[edit] Uniform polyhedra and tilings

Every regular polyhedron or tiling {p,q} has these five operations that create semiregular polyhedra. The short-hand notation is equivalent to the longer name. For instance, t{3,3} simply means truncated tetrahedron.

The vertical notation is used for dual-symmetric operations - those that generate the same polyhedron from {p,q} as {q,p}.

A second extended notation, also used by Coxeter applies to all dimensions, and are specified by a t followed by a list of indices corresponding to Wythoff construction mirrors. (They also correspond to ringed nodes in a Coxeter-Dynkin diagram.)

In each a Wythoff construction operational name is given first. Second some have alternate terminology (given in parentheses) apply only for a given dimension. Specifically omnitruncation and expansion, as well as dual relations apply differently in each dimension.

The final columns offer the elements centered on each position. A single positional index is a node. A double positional index is an edge. A triple positional index is the triangle interior.

The symbol -- implies a vertex at the position. The symbol { } implies an edge at that position. The symbol { }x{ } is a square face {4}.

Operation	Extended Schläfli Symbols		Coxeter- Dynkin Diagram	Wythoff symbol	Position
Operation	Extended Schläfli Symbols		Coxeter- Dynkin Diagram	Wythoff symbol	(2)	(1)	(0)	(0,1)	(0,2)	(1,2)
Parent	$\begin{Bmatrix} p , q \end{Bmatrix}$	t₀{p,q}		q \| 2 p	{p}	{}	--	--	--	{}
Rectified	$\begin{Bmatrix} p \\ q \end{Bmatrix}$	t₁{p,q}		2 \| p q	{p}	--	{q}	--	{}	--
Birectified (or dual)	$\begin{Bmatrix} q , p \end{Bmatrix}$	t₂{p,q}		p \| 2 q	--	{}	{q}	{}	--	--
Truncated	$t\begin{Bmatrix} p , q \end{Bmatrix}$	t_0,1{p,q}		2 q \| p	{2p}	{}	{q}	--	{}	{}
Bitruncated (or truncated dual)	$t\begin{Bmatrix} q , p \end{Bmatrix}$	t_1,2{p,q}		2 p \| q	{p}	{}	{2q}	{}	{}	--
Cantellated (or expanded)	$r\begin{Bmatrix} p \\ q \end{Bmatrix}$	t_0,2{p,q}		p q \| 2	{p}	{}x{}	{q}	{}	--	{}
Cantitruncated (or omnitruncated)	$t\begin{Bmatrix} p \\ q \end{Bmatrix}$	t_0,1,2{p,q}		2 p q \|	{2p}	{}x{}	{2q}	{}	{}	{}
Snub	$s\begin{Bmatrix} p \\ q \end{Bmatrix}$	s{p,q}		\| 2 p q	{p}	{3} {3}	{q}	--	--	--

Generating triangles

[edit] Uniform polychora and 3-space honeycombs

Example tetrahedron in cubic honeycomb cell.
There are 3 right dihedral angles (2 intersecting perpendicular mirrors):
Edges 1 to 2, 0 to 2, and 1 to 3.

Summary chart of truncation operations

Every regular polytope can be seen as the images of a fundamental region in a small number of mirrors. In a 4-dimensional polytope (or 3-dimensional cubic honeycomb) the fundamental region is bounded by four mirrors. A mirror in 4-space is a three-dimensional hyperplane, but it is more convenient for our purposes to consider only its two-dimensional intersection with the three-dimensional surface of the hypersphere; thus the mirrors form an irregular tetrahedron.

Each of the sixteen regular polychora is generated by one of four symmetry groups, as follows:

group [3,3,3]: the 5-cell {3,3,3}, which is self-dual;
group [3,3,4]: 16-cell {3,3,4} and its dual tesseract {4,3,3};
group [3,4,3]: the 24-cell {3,4,3}, self-dual;
group [3,3,5]: 600-cell {3,3,5}, its dual 120-cell {5,3,3}, and their ten regular stellations.

(The groups are named in Coxeter notation.)

A set of up to 13 (nonregular) uniform polychora can be generated from each regular polychoron and its dual. Eight of the convex uniform honeycombs in Euclidean 3-space are analogously generated from the cubic honeycomb {4,3,4}.

For a given symmetry simplex, a generating point may be placed on any of the four vertices, 6 edges, 4 faces, or the interior volume. On each of these 15 elements there is a point whose images, reflected in the four mirrors, are the vertices of a uniform polychoron.

The extended Schläfli symbols are made by a t followed by inclusion of one to four subscripts 0,1,2,3. If there's one subscript, the generating point is on a corner of the fundamental region, i.e. a point where three mirrors meet. These corners are notated as

0: vertex of the parent polychoron (center of the dual's cell)
1: center of the parent's edge (center of the dual's face)
2: center of the parent's face (center of the dual's edge)
3: center of the parent's cell (vertex of the dual)

(For the two self-dual polychora, "dual" means a similar polychoron in dual position.) Two or more subscripts mean that the generating point is between the corners indicated.

The following table defines all 15 forms. Each trunction form can have from one to four cell types, located in positions 0,1,2,3 as defined above. The cells are labeled by polyhedral truncation notation.

An n-gonal prism is represented as : {n}x{2}.
The green background is shown on forms that are equivalent from either the parent or dual.
The red background shows truncations of the parent, and blue as truncations of the dual.

Operation	Extended Schläfli symbols	Coxeter- Dynkin Diagram	Position
Operation	Extended Schläfli symbols	Coxeter- Dynkin Diagram	(3)	(2)	(1)	(0)
Parent	t_{0_{p,q,r}}		{p,q}	{p}	{}	--
Rectified	t_{1_{p,q,r}}		t₁{p,q}	{p}	--	{q,r}
Birectified (or rectified dual)	t_{2_{p,q,r}}		{q,p}	--	{r}	t₁{q,r}
Trirectifed (or dual)	t_{3_{p,q,r}}		--	{}	{r}	t₂{q,r}

Truncated	t_{0,1_{p,q,r}}		t_0,1{p,q}	{2p}	{}	{q,r}
Bitruncated	t_{1,2_{p,q,r}}		t_1,2{p,q}	{p}	{r}	t_0,1{q,r}
Tritruncated (or truncated dual)	t_{2,3_{p,q,r}}		{q,p}	{}	{2r}	t_1,2{q,r}
Cantellated	t_{0,2_{p,q,r}}		t_0,2{p,q}	{p}	{}x{r}	t₁{q,r}
Bicantellated (or cantellated dual)	t_{1,3_{p,q,r}}		t₁{p,q}	{p}x{}	{r}	t_0,2{q,r}
Runcinated (or expanded)	t_{0,3_{p,q,r}}		{p,q}	{p}x{}	{}x{r}	t₂{q,r}
Cantitruncated	t_{0,1,2_{p,q,r}}		t_0,1,2{p,q}	{2p}	{}x{r}	t_0,1{q,r}
Bicantitruncated (or cantitruncated dual)	t_{1,2,3_{p,q,r}}		t_1,2{p,q}	{p}x{}	{2r}	t_0,1,2{q,r}
Runcitruncated	t_{0,1,3_{p,q,r}}		t_0,1{p,q}	{2p}x{}	{}x{r}	t_0,2{q,r}
Runcicantellated (or runcitruncated dual)	t_{0,2,3_{p,q,r}}		t_0,1,2{p,q}	{p}x{}	{}x{2r}	t_1,2{q,r}
Runcicantitruncated (or omnitruncated)	t_{0,1,2,3_{p,q,r}}		t_0,1,2{p,q}	{2p}x{}	{}x{2r}	t_0,1,2{q,r}

[edit] See also

[edit] External links

Olshevsky, George, Uniform polytope at Glossary for Hyperspace.

[edit] References

Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
H.S.M. Coxeter:
- H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Coxeter, Longuet-Higgins, Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401-50. (Extended Schläfli notation used)

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Category: Polytopes