User:Tomruen/convex uniform polyhedron

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A convex uniform polyhedron is a polyhedron with regular polygons faces and identical vertices. Furthermore, for every two vertices there is an isometry mapping one into the other, so there is a high degree of reflectional and rotational symmetry. Convex uniform polyhedra are regular or semi-regular.

See also uniform polyhedron for nonconvex forms.

Categories include:

{}x{p} Infinite set of uniform prisms (including star prisms)
s{}x{p}: Infinite set of uniform antiprisms (including star antiprisms)
{p,q}: 5 Platonic solid regular convex polyhedra
- 13 truncations of the platonic solids: (Archimedean solid)
{p,q}: 3 regular tilings of the plane
- 8 uniform truncations of the regular tilings
{p,q}: infinitely many regular tilings of the hyperbolic plane (and uniform truncations)

The Platonic solids date back to the classical Greeks and were studied by Plato, Theaetetus and Euclid. Johannes Kepler (1571-1630) was the first to publish the complete list of Archimedean solids after the original work of Archimedes was lost.

The convex uniform polyhedra can be named by Wythoff construction operations upon a parent form. You can see in the table below that some of the polyhedra exist from different constructions, and these show different symmetries.

Note: Dihedra are members of an infinite set of two-sided polyhedra (2 identical polygons) which generate the prisms as truncated forms.

Each of these convex forms define set of vertices that can be identified for the nonconvex forms in the next section.

	Parent	Truncated	Rectified	Bitruncated (truncated dual)	Birectified (dual)	Cantellated	Cantitruncated (Omnitruncated)	Snub
Extended Schläfli symbol	$\begin{Bmatrix} p , q \end{Bmatrix}$	$t\begin{Bmatrix} p , q \end{Bmatrix}$	$\begin{Bmatrix} p \\ q \end{Bmatrix}$	$t\begin{Bmatrix} q , p \end{Bmatrix}$	$\begin{Bmatrix} q , p \end{Bmatrix}$	$r\begin{Bmatrix} p \\ q \end{Bmatrix}$	$t\begin{Bmatrix} p \\ q \end{Bmatrix}$	$s\begin{Bmatrix} p \\ q \end{Bmatrix}$
Extended Schläfli symbol	t₀{p,q}	t_0,1{p,q}	t₁{p,q}	t_1,2{p,q}	t₂{p,q}	t_0,2{p,q}	t_0,1,2{p,q}	s{p,q}
Wythoff symbol p-q-2	q \| p 2	2 q \| p	2 \| p q	2 p \| q	p \| q 2	p q \| 2	p q 2 \|	\| p q 2
Coxeter-Dynkin diagram (variations)
	(o)-p-o-q-o	(o)-p-(o)-q-o	o-p-(o)-q-o	o-p-(o)-q-(o)	o-p-o-q-(o)	(o)-p-o-q-(o)	(o)-p-(o)-q-(o)	( )-p-( )-q-( )
	xPoQo	xPxQo	oPxQo	oPxQx	oPoQx	xPoQx	xPxQx	sPsQs
	[p,q]:001	[p,q]:011	[p,q]:010	[p,q]:110	[p,q]:100	[p,q]:101	[p,q]:111	[p,q]:111s
Vertex figure	p^q	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	q^p	(p.4.q.4)	(4.2p.2q)	(3.3.p.3.q)
Dihedral 2-3-2	{2,3}	3.4.4	2.3.2.3	2.6.6	{3,2}	2.4.3.4	4.4.6	3.3.3.3.2
Tetrahedral 3-3-2	{3,3}	(3.6.6)	(3.3.3.3)	(3.6.6)	{3,3}	(3.4.3.4)	(4.6.6)	(3.3.3.3.3)
Octahedral 4-3-2	{4,3}	(3.8.8)	(3.4.3.4)	(4.6.6)	{3,4}	(3.4.4.4)	(4.6.8)	(3.3.3.3.4)
Icosahedral 5-3-2	{5,3}	(3.10.10)	(3.5.3.5)	(5.6.6)	{3,5}	(3.4.5.4)	(4.6.10)	(3.3.3.3.5)
Hexagonal 6-3-2	{6,3}	(3.12.12)	(3.6.3.6)	(6.6.6)	{3,6}	(3.4.6.4)	(4.6.12)	(3.3.3.3.6)
Septagonal (Order 3) 7-3-2	{7,3}	(3.14.14)	(3.7.3.7)	(6.7.7)	{3,7}	(3.4.7.4)	(4.6.14)	(3.3.3.3.7)

	Parent	Truncated	Rectified	Bitruncated (truncated dual)	Birectified (dual)	Cantellated	Cantitruncated (Omnitruncated)	Snub
Vertex figure	p^q	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	q^p	(p.4.q.4)	(4.2p.2q)	(3.3.p.3.q)
Dihedral 4-2-2	{4,2}	2.8.8	2.4.2.4	4.4.4	{2,4}	2.4.4.4	4.4.8	3.3.3.4
Octahedral 4-3-2	{4,3}	(3.8.8)	(3.4.3.4)	(4.6.6)	{3,4}	(3.4.4.4)	(4.6.8)	(3.3.3.3.4)
Square 4-4-2	{4,4}	(4.8.8)	(4.4.4.4)	(4.8.8)	{4,4}	(4.4.4.4)	(4.8.8)	(3.3.4.3.4)
{4,5} hyperbolic 4-5-2	{4,5}	5.8.8	4.5.4.5	4.10.10	{5,4}	5.4.4.4	4.8.10	3.3.4.3.5

	Parent	Truncated	Rectified	Bitruncated (truncated dual)	Birectified (dual)	Cantellated	Cantitruncated (Omnitruncated)	Snub
Vertex figure	p^q	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	q^p	(p.4.q.4)	(4.2p.2q)	(3.3.p.3.q)
Dihedral 5-2-2	{5,2}	2.10.10	2.5.2.5	4.4.5	{2,5}	2.4.5.4	4.4.10	3.3.3.5
Dihedral 6-2-2	{6,2}	2.12.12	2.6.2.6	4.4.6	{2,6}	2.4.6.4	4.4.12	3.3.3.6

[edit] Definition of operations

Fundamental domains and generating points

Example forms from the cube and octahedron

Operation	Extended Schläfli symbols		Description
Parent	t_{0_{p,q}}	$\begin{Bmatrix} p , q \end{Bmatrix}$	Any regular polyhedron or tiling
Rectified	t_{1_{p,q}}	$\begin{Bmatrix} p \\ q \end{Bmatrix}$	The edges are fully-truncated into single points. The polyhedron now has the combined faces of the parent and dual.
Birectified Also Dual	t_{2_{p,q}}	$\begin{Bmatrix} q , p \end{Bmatrix}$	The birectified (dual) is a further truncation so that the original faces are reduced to points. New faces are formed under each parent vertex. The number of edges is unchanged and are rotated 90 degrees. The dual of the regular polyhedron {p, q} is also a regular polyhedron {q, p}.
Truncated	t_{0,1_{p,q}}	$t\begin{Bmatrix} p , q \end{Bmatrix}$	Each original vertex is cut off, with new faces filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated polyhedron. The polyhedron has its original faces doubled in sides, and contains the faces of the dual.
Bitruncated	t_{1,2_{p,q}}	$t\begin{Bmatrix} q , p \end{Bmatrix}$	Same as truncated dual.
Cantellated (or rhombated) (Also expanded)	t_{0,2_{p,q}}	$r\begin{Bmatrix} p \\ q \end{Bmatrix}$	Each original edge is beveled with new rectangular faces appearing in their place, as well as the original vertices are also truncated. A uniform cantellation is half way between both the parent and dual forms.
Omnitruncated (or cantitruncated) (or rhombitruncated)	t_{0,1,2_{p,q}}	$t\begin{Bmatrix} p \\ q \end{Bmatrix}$	Thee runcation and cantellation operations are applied together create an omnitruncated form which has the parent's faces doubled in sides, the duals faces doubled in sides, and squares where the original edges existed.
Snub	s{p,q}	$s\begin{Bmatrix} p \\ q \end{Bmatrix}$	The snub takes the omnitruncated form and rectifies alternate vertices. (This operation is only possible for polyhedra with all even-sided faces.) All the original faces end up with half as many sides, and the square degenerate into edges. Since the omnitruncated forms have 3 faces/vertex, new triangles are formed.

User:Tomruen/convex uniform polyhedron

From Wikipedia, the free encyclopedia

[edit] Definition of operations

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