Uniform 5-polytope

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Graphs of regular and uniform 5-polytopes.
5-simplex t0.svg
5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t1.svg
Rectified 5-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t01.svg
Truncated 5-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t02.svg
Cantellated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t03.svg
Runcinated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t04.svg
Stericated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t4.svg
5-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t34.svg
Truncated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t3.svg
Rectified 5-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t24.svg
Cantellated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t14.svg
Runcinated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
5-cube t02.svg
Cantellated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t03.svg
Runcinated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t04.svg
Stericated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t0.svg
5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t01.svg
Truncated 5-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t1.svg
Rectified 5-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube t0 D5.svg
5-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube t01 D5.svg
Truncated 5-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube t02 D5.svg
Cantellated 5-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-demicube t03 D5.svg
Runcinated 5-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

Contents

The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams.

History of discovery

Regular 5-polytopes

Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} 4-polytope facets around each face. There are exactly three such regular polytopes, all convex:

There are no nonconvex regular polytopes in 5 dimensions or above.

Convex uniform 5-polytopes

Unsolved problem in mathematics:

What is the complete set of convex uniform 5-polytopes? [6]

There are 104 known convex uniform 5-polytopes, plus a number of infinite families of duoprism prisms, and polygon-polyhedron duoprisms. All except the grand antiprism prism are based on Wythoff constructions, reflection symmetry generated with Coxeter groups.[ citation needed ]

Symmetry of uniform 5-polytopes in four dimensions

The 5-simplex is the regular form in the A5 family. The 5-cube and 5-orthoplex are the regular forms in the B5 family. The bifurcating graph of the D5 family contains the 5-orthoplex, as well as a 5-demicube which is an alternated 5-cube.

Each reflective uniform 5-polytope can be constructed in one or more reflective point group in 5 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,b,a], have an extended symmetry, [[a,b,b,a]], like [3,3,3,3], doubling the symmetry order. Uniform polytopes in these group with symmetric rings contain this extended symmetry.

If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an alternation operation can generate a new 5-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.

Coxeter diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence. Coxeter diagram finite rank5 correspondence.png
Coxeter diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.
Fundamental families [7]
Group
symbol
Order Coxeter
graph
Bracket
notation
Commutator
subgroup
Coxeter
number

(h)
Reflections
m=5/2 h [8]
A5720CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png[3,3,3,3][3,3,3,3]+615 CDel node c1.png
D51920CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodeab c1.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png[3,3,31,1][3,3,31,1]+820 CDel node c1.png
B53840CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png[4,3,3,3]105 CDel node c2.png20 CDel node c1.png
Uniform prisms

There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes. There is one infinite family of 5-polytopes based on prisms of the uniform duoprisms {p}×{q}×{ }.

Coxeter
group
Order Coxeter
diagram
Coxeter
notation
Commutator
subgroup
Reflections
A4A1120CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c5.png[3,3,3,2] = [3,3,3]×[ ][3,3,3]+10 CDel node c1.png1 CDel node c5.png
D4A1384CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel nodeab c1.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c5.png[31,1,1,2] = [31,1,1]×[ ][31,1,1]+12 CDel node c1.png1 CDel node c5.png
B4A1768CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c5.png[4,3,3,2] = [4,3,3]×[ ]4 CDel node c2.png12 CDel node c1.png1 CDel node c5.png
F4A12304CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c5.png[3,4,3,2] = [3,4,3]×[ ][3+,4,3+]12 CDel node c2.png12 CDel node c1.png1 CDel node c5.png
H4A128800CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c5.png[5,3,3,2] = [3,4,3]×[ ][5,3,3]+60 CDel node c1.png1 CDel node c5.png
Duoprismatic prisms (use 2p and 2q for evens)
I2(p)I2(q)A18pqCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c2.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c1.pngCDel q.pngCDel node c1.pngCDel 2.pngCDel node c5.png[p,2,q,2] = [p]×[q]×[ ][p+,2,q+]pCDel node c2.pngqCDel node c1.png1 CDel node c5.png
I2(2p)I2(q)A116pqCDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c3.pngCDel 2x.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c1.pngCDel q.pngCDel node c1.pngCDel 2.pngCDel node c5.png[2p,2,q,2] = [2p]×[q]×[ ]p CDel node c3.pngpCDel node c2.pngqCDel node c1.png1 CDel node c5.png
I2(2p)I2(2q)A132pqCDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel node c3.pngCDel 2x.pngCDel p.pngCDel node c2.pngCDel 2.pngCDel node c1.pngCDel 2x.pngCDel q.pngCDel node c4.pngCDel 2.pngCDel node c5.png[2p,2,2q,2] = [2p]×[2q]×[ ]pCDel node c3.pngpCDel node c2.pngqCDel node c1.pngqCDel node c4.png1 CDel node c5.png
Uniform duoprisms

There are 3 categorical uniform duoprismatic families of polytopes based on Cartesian products of the uniform polyhedra and regular polygons: {q,r}×{p}.

Coxeter
group
Order Coxeter
diagram
Coxeter
notation
Commutator
subgroup
Reflections
Prismatic groups (use 2p for even)
A3I2(p)48pCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel p.pngCDel node c3.png[3,3,2,p] = [3,3]×[p][(3,3)+,2,p+]6 CDel node c1.pngpCDel node c3.png
A3I2(2p)96pCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel 2x.pngCDel p.pngCDel node c4.png[3,3,2,2p] = [3,3]×[2p]6 CDel node c1.pngpCDel node c3.pngpCDel node c4.png
B3I2(p)96pCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel p.pngCDel node c3.png[4,3,2,p] = [4,3]×[p]3 CDel node c2.png6CDel node c1.pngpCDel node c3.png
B3I2(2p)192pCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel 2x.pngCDel p.pngCDel node c4.png[4,3,2,2p] = [4,3]×[2p]3 CDel node c2.png6 CDel node c1.pngpCDel node c3.pngpCDel node c4.png
H3I2(p)240pCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel p.pngCDel node c3.png[5,3,2,p] = [5,3]×[p][(5,3)+,2,p+]15 CDel node c1.pngpCDel node c3.png
H3I2(2p)480pCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2x.pngCDel p.pngCDel node.pngCDel node c1.pngCDel 5.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c3.pngCDel 2x.pngCDel p.pngCDel node c4.png[5,3,2,2p] = [5,3]×[2p]15 CDel node c1.pngpCDel node c3.pngpCDel node c4.png

Enumerating the convex uniform 5-polytopes

That brings the tally to: 19+31+8+45+1=104

In addition there are:

The A5 family

There are 19 forms based on all permutations of the Coxeter diagrams with one or more rings. (16+4-1 cases)

They are named by Norman Johnson from the Wythoff construction operations upon regular 5-simplex (hexateron).

The A5 family has symmetry of order 720 (6 factorial). 7 of the 19 figures, with symmetrically ringed Coxeter diagrams have doubled symmetry, order 1440.

The coordinates of uniform 5-polytopes with 5-simplex symmetry can be generated as permutations of simple integers in 6-space, all in hyperplanes with normal vector (1,1,1,1,1,1).

#Base point Johnson naming system
Bowers name and (acronym)
Coxeter diagram
k-face element counts Vertex
figure
Facet counts by location: [3,3,3,3]
43210CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
(6)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
[3,3,2]
(15)
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
[3,2,3]
(20)
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[2,3,3]
(15)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
(6)
Alt
1(0,0,0,0,0,1) or (0,1,1,1,1,1) 5-simplex
hexateron (hix)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
61520156 5-simplex verf.png
{3,3,3}
Schlegel wireframe 5-cell.png
{3,3,3}
----
2(0,0,0,0,1,1) or (0,0,1,1,1,1) Rectified 5-simplex
rectified hexateron (rix)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
1245806015 Rectified 5-simplex verf.png
t{3,3}×{ }
Schlegel half-solid rectified 5-cell.png
r{3,3,3}
--- Schlegel wireframe 5-cell.png
{3,3,3}
3(0,0,0,0,1,2) or (0,1,2,2,2,2) Truncated 5-simplex
truncated hexateron (tix)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
1245807530 Truncated 5-simplex verf.png
Tetrah.pyr
Schlegel half-solid truncated pentachoron.png
t{3,3,3}
--- Schlegel wireframe 5-cell.png
{3,3,3}
4(0,0,0,1,1,2) or (0,1,1,2,2,2) Cantellated 5-simplex
small rhombated hexateron (sarx)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
2713529024060 Cantellated hexateron verf.png
prism-wedge
Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
-- Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid rectified 5-cell.png
r{3,3,3}
5(0,0,0,1,2,2) or (0,0,1,2,2,2) Bitruncated 5-simplex
bitruncated hexateron (bittix)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
126014015060 Bitruncated 5-simplex verf.png Schlegel half-solid bitruncated 5-cell.png
2t{3,3,3}
--- Schlegel half-solid truncated pentachoron.png
t{3,3,3}
6(0,0,0,1,2,3) or (0,1,2,3,3,3) Cantitruncated 5-simplex
great rhombated hexateron (garx)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
27135290300120 Canitruncated 5-simplex verf.png Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
-- Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid truncated pentachoron.png
t{3,3,3}
7(0,0,1,1,1,2) or (0,1,1,1,2,2) Runcinated 5-simplex
small prismated hexateron (spix)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
4725542027060 Runcinated 5-simplex verf.png Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
- 3-3 duoprism.png
{3}×{3}
Octahedral prism.png
{ }×r{3,3}
Schlegel half-solid rectified 5-cell.png
r{3,3,3}
8(0,0,1,1,2,3) or (0,1,2,2,3,3) Runcitruncated 5-simplex
prismatotruncated hexateron (pattix)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
47315720630180 Runcitruncated 5-simplex verf.png Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
- 3-6 duoprism.png
{6}×{3}
Octahedral prism.png
{ }×r{3,3}
Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
9(0,0,1,2,2,3) or (0,1,1,2,3,3) Runcicantellated 5-simplex
prismatorhombated hexateron (pirx)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
47255570540180 Runcicantellated 5-simplex verf.png Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
- 3-3 duoprism.png
{3}×{3}
Truncated tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid bitruncated 5-cell.png
2t{3,3,3}
10(0,0,1,2,3,4) or (0,1,2,3,4,4) Runcicantitruncated 5-simplex
great prismated hexateron (gippix)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
47315810900360 Runcicantitruncated 5-simplex verf.png
Irr.5-cell
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
- 3-6 duoprism.png
{3}×{6}
Truncated tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
11(0,1,1,1,2,3) or (0,1,2,2,2,3) Steritruncated 5-simplex
celliprismated hexateron (cappix)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
62330570420120 Steritruncated 5-simplex verf.png Schlegel half-solid truncated pentachoron.png
t{3,3,3}
Truncated tetrahedral prism.png
{ }×t{3,3}
3-6 duoprism.png
{3}×{6}
Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
12(0,1,1,2,3,4) or (0,1,2,3,3,4) Stericantitruncated 5-simplex
celligreatorhombated hexateron (cograx)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
6248011401080360 Stericanitruncated 5-simplex verf.png Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
Truncated octahedral prism.png
{ }×tr{3,3}
3-6 duoprism.png
{3}×{6}
Cuboctahedral prism.png
{ }×rr{3,3}
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
13(0,0,0,1,1,1) Birectified 5-simplex
dodecateron (dot)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
12601209020 Birectified hexateron verf.png
{3}×{3}
Schlegel half-solid rectified 5-cell.png
r{3,3,3}
--- Schlegel half-solid rectified 5-cell.png
r{3,3,3}
14(0,0,1,1,2,2) Bicantellated 5-simplex
small birhombated dodecateron (sibrid)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
3218042036090 Bicantellated 5-simplex verf.png Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
- 3-3 duoprism.png
{3}×{3}
- Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
15(0,0,1,2,3,3) Bicantitruncated 5-simplex
great birhombated dodecateron (gibrid)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
32180420450180 Bicanitruncated 5-simplex verf.png Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
- 3-3 duoprism.png
{3}×{3}
- Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
16(0,1,1,1,1,2) Stericated 5-simplex
small cellated dodecateron (scad)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6218021012030 Stericated hexateron verf.png
Irr.16-cell
Schlegel wireframe 5-cell.png
{3,3,3}
Tetrahedral prism.png
{ }×{3,3}
3-3 duoprism.png
{3}×{3}
Tetrahedral prism.png
{ }×{3,3}
Schlegel wireframe 5-cell.png
{3,3,3}
17(0,1,1,2,2,3) Stericantellated 5-simplex
small cellirhombated dodecateron (card)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
62420900720180 Stericantellated 5-simplex verf.png Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
Cuboctahedral prism.png
{ }×rr{3,3}
3-3 duoprism.png
{3}×{3}
Cuboctahedral prism.png
{ }×rr{3,3}
Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
18(0,1,2,2,3,4) Steriruncitruncated 5-simplex
celliprismatotruncated dodecateron (captid)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
6245011101080360 Steriruncitruncated 5-simplex verf.png Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
Tetrahedral prism.png
{ }×t{3,3}
6-6 duoprism.png
{6}×{6}
Tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
19(0,1,2,3,4,5) Omnitruncated 5-simplex
great cellated dodecateron (gocad)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
6254015601800720 Omnitruncated 5-simplex verf.png
Irr. {3,3,3}
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
Truncated octahedral prism.png
{ }×tr{3,3}
6-6 duoprism.png
{6}×{6}
Truncated octahedral prism.png
{ }×tr{3,3}
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
Nonuniform Omnisnub 5-simplex
snub dodecateron (snod)
snub hexateron (snix)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
422234040802520360 ht0,1,2,3{3,3,3} ht0,1,2,3{3,3,2} ht0,1,2,3{3,2,3} ht0,1,2,3{3,3,2} ht0,1,2,3{3,3,3} (360)
Schlegel wireframe 5-cell.png
Irr. {3,3,3}

The B5 family

The B5 family has symmetry of order 3840 (5!×25).

This family has 251=31 Wythoffian uniform polytopes generated by marking one or more nodes of the Coxeter diagram. Also added are 8 uniform polytopes generated as alternations with half the symmetry, which form a complete duplicate of the D5 family as CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.png... = CDel nodes 10ru.pngCDel split2.png..... (There are more alternations that are not listed because they produce only repetitions, as CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.png... = CDel nodes 11.pngCDel split2.png.... and CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.png... = CDel nodes.pngCDel split2.png.... These would give a complete duplication of the uniform 5-polytopes numbered 20 through 34 with symmetry broken in half.)

For simplicity it is divided into two subgroups, each with 12 forms, and 7 "middle" forms which equally belong in both.

The 5-cube family of 5-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform 5-polytope. All coordinates correspond with uniform 5-polytopes of edge length 2.

#Base pointName
Coxeter diagram
Element counts Vertex
figure
Facet counts by location: [4,3,3,3]
43210CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[4,3,3]
(10)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
[4,3,2]
(40)
CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
[4,2,3]
(80)
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[2,3,3]
(80)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
(32)
Alt
20(0,0,0,0,1)√2 5-orthoplex
triacontaditeron (tac)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
3280804010 Pentacross verf.png
{3,3,4}
---- Schlegel wireframe 5-cell.png
{3,3,3}
21(0,0,0,1,1)√2 Rectified 5-orthoplex
rectified triacontaditeron (rat)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4224040024040 Rectified pentacross verf.png
{ }×{3,4}
Schlegel wireframe 16-cell.png
{3,3,4}
--- Schlegel half-solid rectified 5-cell.png
r{3,3,3}
22(0,0,0,1,2)√2 Truncated 5-orthoplex
truncated triacontaditeron (tot)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
4224040028080 Truncated pentacross.png
(Octah.pyr)
Schlegel wireframe 16-cell.png
{3,3,4}
--- Schlegel half-solid truncated pentachoron.png
t{3,3,3}
23(0,0,1,1,1)√2 Birectified 5-cube
penteractitriacontaditeron (nit)
(Birectified 5-orthoplex)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4228064048080 Birectified penteract verf.png
{4}×{3}
Schlegel half-solid rectified 16-cell.png
r{3,3,4}
--- Schlegel half-solid rectified 5-cell.png
r{3,3,3}
24(0,0,1,1,2)√2 Cantellated 5-orthoplex
small rhombated triacontaditeron (sart)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
8264015201200240 Cantellated pentacross verf.png
Prism-wedge
Schlegel half-solid rectified 16-cell.png
r{3,3,4}
Octahedral prism.png
{ }×{3,4}
-- Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
25(0,0,1,2,2)√2 Bitruncated 5-orthoplex
bitruncated triacontaditeron (bittit)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
42280720720240 Bitruncated pentacross verf.png Schlegel half-solid truncated 16-cell.png
t{3,3,4}
--- Schlegel half-solid bitruncated 5-cell.png
2t{3,3,3}
26(0,0,1,2,3)√2 Cantitruncated 5-orthoplex
great rhombated triacontaditeron (gart)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
8264015201440480 Canitruncated 5-orthoplex verf.png Schlegel half-solid truncated 16-cell.png
t{3,3,4}
Octahedral prism.png
{ }×{3,4}
-- Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
27(0,1,1,1,1)√2 Rectified 5-cube
rectified penteract (rin)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4220040032080 Rectified 5-cube verf.png
{3,3}×{ }
Schlegel half-solid rectified 8-cell.png
r{4,3,3}
--- Schlegel wireframe 5-cell.png
{3,3,3}
28(0,1,1,1,2)√2 Runcinated 5-orthoplex
small prismated triacontaditeron (spat)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
162120021601440320 Runcinated pentacross verf.png Schlegel half-solid rectified 8-cell.png
r{4,3,3}
Cuboctahedral prism.png
{ }×r{3,4}
3-4 duoprism.png
{3}×{4}
Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
29(0,1,1,2,2)√2 Bicantellated 5-cube
small birhombated penteractitriacontaditeron (sibrant)
(Bicantellated 5-orthoplex)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
12284021601920480 Bicantellated penteract verf.png Schlegel half-solid cantellated 16-cell.png
rr{3,3,4}
- 3-4 duoprism.png
{4}×{3}
- Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
30(0,1,1,2,3)√2 Runcitruncated 5-orthoplex
prismatotruncated triacontaditeron (pattit)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
162144036803360960 Runcitruncated 5-orthoplex verf.png Schlegel half-solid cantellated 16-cell.png
rr{3,3,4}
Cuboctahedral prism.png
{ }×r{3,4}
6-4 duoprism.png
{6}×{4}
- Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
31(0,1,2,2,2)√2 Bitruncated 5-cube
bitruncated penteract (bittin)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
42280720800320 Bitruncated penteract verf.png Schlegel half-solid bitruncated 8-cell.png
2t{4,3,3}
--- Schlegel half-solid truncated pentachoron.png
t{3,3,3}
32(0,1,2,2,3)√2 Runcicantellated 5-orthoplex
prismatorhombated triacontaditeron (pirt)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
162120029602880960 Runcicantellated 5-orthoplex verf.png Schlegel half-solid bitruncated 8-cell.png
2t{4,3,3}
Truncated octahedral prism.png
{ }×t{3,4}
3-4 duoprism.png
{3}×{4}
- Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
33(0,1,2,3,3)√2 Bicantitruncated 5-cube
great birhombated triacontaditeron (gibrant)
(Bicantitruncated 5-orthoplex)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
12284021602400960 Bicantellated penteract verf.png Schlegel half-solid cantitruncated 16-cell.png
tr{3,3,4}
- 3-4 duoprism.png
{4}×{3}
- Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
34(0,1,2,3,4)√2 Runcicantitruncated 5-orthoplex
great prismated triacontaditeron (gippit)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
1621440416048001920 Runcicantitruncated 5-orthoplex verf.png Schlegel half-solid cantitruncated 16-cell.png
tr{3,3,4}
Truncated octahedral prism.png
{ }×t{3,4}
6-4 duoprism.png
{6}×{4}
- Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
35(1,1,1,1,1) 5-cube
penteract (pent)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
1040808032 5-cube verf.png
{3,3,3}
Schlegel wireframe 8-cell.png
{4,3,3}
----
36(1,1,1,1,1)
+ (0,0,0,0,1)√2
Stericated 5-cube
small cellated penteractitriacontaditeron (scant)
(Stericated 5-orthoplex)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
2428001040640160 Stericated penteract verf.png
Tetr.antiprm
Schlegel wireframe 8-cell.png
{4,3,3}
Schlegel wireframe 8-cell.png
{4,3}×{ }
3-4 duoprism.png
{4}×{3}
Tetrahedral prism.png
{ }×{3,3}
Schlegel wireframe 5-cell.png
{3,3,3}
37(1,1,1,1,1)
+ (0,0,0,1,1)√2
Runcinated 5-cube
small prismated penteract (span)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
202124021601440320 Runcinated penteract verf.png Schlegel half-solid runcinated 8-cell.png
t0,3{4,3,3}
- 3-4 duoprism.png
{4}×{3}
Octahedral prism.png
{ }×r{3,3}
Schlegel half-solid rectified 5-cell.png
r{3,3,3}
38(1,1,1,1,1)
+ (0,0,0,1,2)√2
Steritruncated 5-orthoplex
celliprismated triacontaditeron (cappin)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
242152028802240640 Steritruncated 5-orthoplex verf.png Schlegel half-solid runcinated 8-cell.png
t0,3{4,3,3}
Schlegel wireframe 8-cell.png
{4,3}×{ }
6-4 duoprism.png
{6}×{4}
Truncated tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid truncated pentachoron.png
t{3,3,3}
39(1,1,1,1,1)
+ (0,0,1,1,1)√2
Cantellated 5-cube
small rhombated penteract (sirn)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
12268015201280320 Cantellated 5-cube vertf.png
Prism-wedge
Schlegel half-solid cantellated 8-cell.png
rr{4,3,3}
-- Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid rectified 5-cell.png
r{3,3,3}
40(1,1,1,1,1)
+ (0,0,1,1,2)√2
Stericantellated 5-cube
cellirhombated penteractitriacontaditeron (carnit)
(Stericantellated 5-orthoplex)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
242208047203840960 Stericantellated 5-orthoplex verf.png Schlegel half-solid cantellated 8-cell.png
rr{4,3,3}
Rhombicuboctahedral prism.png
rr{4,3}×{ }
3-4 duoprism.png
{4}×{3}
Cuboctahedral prism.png
{ }×rr{3,3}
Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
41(1,1,1,1,1)
+ (0,0,1,2,2)√2
Runcicantellated 5-cube
prismatorhombated penteract (prin)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
202124029602880960 Runcicantellated 5-cube verf.png Runcitruncated 16-cell.png
t0,2,3{4,3,3}
- 3-4 duoprism.png
{4}×{3}
Truncated tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid bitruncated 5-cell.png
2t{3,3,3}
42(1,1,1,1,1)
+ (0,0,1,2,3)√2
Stericantitruncated 5-orthoplex
celligreatorhombated triacontaditeron (cogart)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
2422320592057601920 Stericanitruncated 5-orthoplex verf.png Runcitruncated 16-cell.png
t0,2,3{4,3,3}
Rhombicuboctahedral prism.png
rr{4,3}×{ }
6-4 duoprism.png
{6}×{4}
Truncated octahedral prism.png
{ }×tr{3,3}
Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
43(1,1,1,1,1)
+ (0,1,1,1,1)√2
Truncated 5-cube
truncated penteract (tan)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
42200400400160 Truncated 5-cube verf.png
Tetrah.pyr
Schlegel half-solid truncated tesseract.png
t{4,3,3}
--- Schlegel wireframe 5-cell.png
{3,3,3}
44(1,1,1,1,1)
+ (0,1,1,1,2)√2
Steritruncated 5-cube
celliprismated triacontaditeron (capt)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
242160029602240640 Steritruncated 5-cube verf.png Schlegel half-solid truncated tesseract.png
t{4,3,3}
Truncated cubic prism.png
t{4,3}×{ }
8-3 duoprism.png
{8}×{3}
Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
45(1,1,1,1,1)
+ (0,1,1,2,2)√2
Runcitruncated 5-cube
prismatotruncated penteract (pattin)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
202156037603360960 Runcitruncated 5-cube verf.png Schlegel half-solid runcitruncated 8-cell.png
t0,1,3{4,3,3}
- 8-3 duoprism.png
{8}×{3}
Octahedral prism.png
{ }×r{3,3}
Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
46(1,1,1,1,1)
+ (0,1,1,2,3)√2
Steriruncitruncated 5-cube
celliprismatotruncated penteractitriacontaditeron (captint)
(Steriruncitruncated 5-orthoplex)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
2422160576057601920 Steriruncitruncated 5-cube verf.png Schlegel half-solid runcitruncated 8-cell.png
t0,1,3{4,3,3}
Truncated cubic prism.png
t{4,3}×{ }
8-6 duoprism.png
{8}×{6}
Truncated tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
47(1,1,1,1,1)
+ (0,1,2,2,2)√2
Cantitruncated 5-cube
great rhombated penteract (girn)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
12268015201600640 Canitruncated 5-cube verf.png Schlegel half-solid cantitruncated 8-cell.png
tr{4,3,3}
-- Tetrahedral prism.png
{ }×{3,3}
Schlegel half-solid truncated pentachoron.png
t{3,3,3}
48(1,1,1,1,1)
+ (0,1,2,2,3)√2
Stericantitruncated 5-cube
celligreatorhombated penteract (cogrin)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
2422400600057601920 Stericanitruncated 5-cube verf.png Schlegel half-solid cantitruncated 8-cell.png
tr{4,3,3}
Truncated cuboctahedral prism.png
tr{4,3}×{ }
8-3 duoprism.png
{8}×{3}
Cuboctahedral prism.png
{ }×rr{3,3}
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
49(1,1,1,1,1)
+ (0,1,2,3,3)√2
Runcicantitruncated 5-cube
great prismated penteract (gippin)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2021560424048001920 Runcicantitruncated 5-cube verf.png Schlegel half-solid omnitruncated 8-cell.png
t0,1,2,3{4,3,3}
- 8-3 duoprism.png
{8}×{3}
Truncated tetrahedral prism.png
{ }×t{3,3}
Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
50(1,1,1,1,1)
+ (0,1,2,3,4)√2
Omnitruncated 5-cube
great cellated penteractitriacontaditeron (gacnet)
(omnitruncated 5-orthoplex)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
2422640816096003840 Omnitruncated 5-cube verf.png
Irr. {3,3,3}
Schlegel half-solid omnitruncated 8-cell.png
tr{4,3}×{ }
Truncated cuboctahedral prism.png
tr{4,3}×{ }
8-6 duoprism.png
{8}×{6}
Truncated octahedral prism.png
{ }×tr{3,3}
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
51 5-demicube
hemipenteract (hin)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
261201608016 Demipenteract verf.png
r{3,3,3}
Schlegel wireframe 16-cell.png
h{4,3,3}
----(16)
Schlegel wireframe 5-cell.png
{3,3,3}
52 Cantic 5-cube
Truncated hemipenteract (thin)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
42280640560160 Truncated 5-demicube verf.png Schlegel half-solid truncated 16-cell.png
h2{4,3,3}
---(16)
Schlegel half-solid rectified 5-cell.png
r{3,3,3}
(16)
Schlegel half-solid truncated pentachoron.png
t{3,3,3}
53 Runcic 5-cube
Small rhombated hemipenteract (sirhin)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
42360880720160 Schlegel half-solid rectified 8-cell.png
h3{4,3,3}
---(16)
Schlegel half-solid rectified 5-cell.png
r{3,3,3}
(16)
Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
54 Steric 5-cube
Small prismated hemipenteract (siphin)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
8248072040080 Schlegel wireframe 16-cell.png
h{4,3,3}
Tetrahedral prism.png
h{4,3}×{}
--(16)
Schlegel wireframe 5-cell.png
{3,3,3}
(16)
Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
55 Runcicantic 5-cube
Great rhombated hemipenteract (girhin)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4236010401200480 Schlegel half-solid bitruncated 8-cell.png
h2,3{4,3,3}
---(16)
Schlegel half-solid bitruncated 5-cell.png
2t{3,3,3}
(16)
Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
56 Stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
8272018401680480 Schlegel half-solid truncated 16-cell.png
h2{4,3,3}
Truncated tetrahedral prism.png
h2{4,3}×{}
--(16)
Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
(16)
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
57 Steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
8256012801120320 Schlegel half-solid rectified 8-cell.png
h3{4,3,3}
Tetrahedral prism.png
h{4,3}×{}
--(16)
Schlegel half-solid truncated pentachoron.png
t{3,3,3}
(16)
Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
58 Steriruncicantic 5-cube
Great prismated hemipenteract (giphin)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
8272020802400960 Schlegel half-solid bitruncated 8-cell.png
h2,3{4,3,3}
Truncated tetrahedral prism.png
h2{4,3}×{}
--(16)
Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
(16)
Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
NonuniformAlternated runcicantitruncated 5-orthoplex
Snub prismatotriacontaditeron (snippit)
Snub hemipenteract (snahin)
CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png = CDel nodes hh.pngCDel split2.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
11226240108806720960 Schlegel half-solid alternated cantitruncated 16-cell.png
sr{3,3,4}
sr{2,3,4} sr{3,2,4} - ht0,1,2,3{3,3,3} (960)
Schlegel wireframe 5-cell.png
Irr. {3,3,3}
NonuniformEdge-snub 5-orthoplex
Pyritosnub penteract (pysnan)
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
1202792015360105601920sr3{3,3,4} sr3{2,3,4} sr3{3,2,4} Icosahedral prism.png
s{3,3}×{ }
ht0,1,2,3{3,3,3} (960)
Tetrahedral prism.png
Irr. {3,3}×{ }
NonuniformSnub 5-cube
Snub penteract (snan)
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
2162122402160013440960 ht0,1,2,3{3,3,4} ht0,1,2,3{2,3,4} ht0,1,2,3{3,2,4} ht0,1,2,3{3,3,2} ht0,1,2,3{3,3,3} (1920)
Schlegel wireframe 5-cell.png
Irr. {3,3,3}

The D5 family

The D5 family has symmetry of order 1920 (5! x 24).

This family has 23 Wythoffian uniform polyhedra, from 3×8-1 permutations of the D5 Coxeter diagram with one or more rings. 15 (2×8-1) are repeated from the B5 family and 8 are unique to this family, though even those 8 duplicate the alternations from the B5 family.

In the 15 repeats, both of the nodes terminating the length-1 branches are ringed, so the two kinds of CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png element are identical and the symmetry doubles: the relations are CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.png... = CDel nodes 11.pngCDel split2.png.... and CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.png... = CDel nodes.pngCDel split2.png..., creating a complete duplication of the uniform 5-polytopes 20 through 34 above. The 8 new forms have one such node ringed and one not, with the relation CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.png... = CDel nodes 10ru.pngCDel split2.png... duplicating uniform 5-polytopes 51 through 58 above.

# Coxeter diagram
Schläfli symbol symbols
Johnson and Bowers names
Element counts Vertex
figure
Facets by location: CD B5 nodes.png [31,2,1]
43210CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
(16)
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
[31,1,1]
(10)
CDel nodes.pngCDel split2.pngCDel node.pngCDel 2.pngCDel node.png
[3,3]×[ ]
(40)
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png
[ ]×[3]×[ ]
(80)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
(16)
Alt
[51]CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
h{4,3,3,3}, 5-demicube
Hemipenteract (hin)
261201608016 Demipenteract verf.png
r{3,3,3}
Schlegel wireframe 5-cell.png
{3,3,3}
Schlegel wireframe 16-cell.png
h{4,3,3}
---
[52]CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
h2{4,3,3,3}, cantic 5-cube
Truncated hemipenteract (thin)
42280640560160 Truncated 5-demicube verf.png Schlegel half-solid truncated pentachoron.png
t{3,3,3}
Schlegel half-solid truncated 16-cell.png
h2{4,3,3}
-- Schlegel half-solid rectified 5-cell.png
r{3,3,3}
[53]CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
h3{4,3,3,3}, runcic 5-cube
Small rhombated hemipenteract (sirhin)
42360880720160 Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
Schlegel half-solid rectified 8-cell.png
h3{4,3,3}
-- Schlegel half-solid rectified 5-cell.png
r{3,3,3}
[54]CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
h4{4,3,3,3}, steric 5-cube
Small prismated hemipenteract (siphin)
8248072040080 Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,3}
Schlegel wireframe 16-cell.png
h{4,3,3}
Tetrahedral prism.png
h{4,3}×{}
- Schlegel wireframe 5-cell.png
{3,3,3}
[55]CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
h2,3{4,3,3,3}, runcicantic 5-cube
Great rhombated hemipenteract (girhin)
4236010401200480 Schlegel half-solid bitruncated 5-cell.png
2t{3,3,3}
Schlegel half-solid bitruncated 8-cell.png
h2,3{4,3,3}
-- Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
[56]CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
h2,4{4,3,3,3}, stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
8272018401680480 Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
Schlegel half-solid truncated 16-cell.png
h2{4,3,3}
Truncated tetrahedral prism.png
h2{4,3}×{}
- Schlegel half-solid cantellated 5-cell.png
rr{3,3,3}
[57]CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
h3,4{4,3,3,3}, steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
8256012801120320 Schlegel half-solid runcitruncated 5-cell.png
t0,1,3{3,3,3}
Schlegel half-solid rectified 8-cell.png
h3{4,3,3}
Tetrahedral prism.png
h{4,3}×{}
- Schlegel half-solid truncated pentachoron.png
t{3,3,3}
[58]CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
h2,3,4{4,3,3,3}, steriruncicantic 5-cube
Great prismated hemipenteract (giphin)
8272020802400960 Schlegel half-solid omnitruncated 5-cell.png
t0,1,2,3{3,3,3}
Schlegel half-solid bitruncated 8-cell.png
h2,3{4,3,3}
Truncated tetrahedral prism.png
h2{4,3}×{}
- Schlegel half-solid cantitruncated 5-cell.png
tr{3,3,3}
NonuniformCDel nodes hh.pngCDel split2.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png = CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
ht0,1,2,3{3,3,3,4}, alternated runcicantitruncated 5-orthoplex
Snub hemipenteract (snahin)
11226240108806720960 ht0,1,2,3{3,3,3} Schlegel half-solid alternated cantitruncated 16-cell.png
sr{3,3,4}
sr{2,3,4} sr{3,2,4} ht0,1,2,3{3,3,3} (960)
Schlegel wireframe 5-cell.png
Irr. {3,3,3}

Uniform prismatic forms

There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes. For simplicity, most alternations are not shown.

A4 × A1

This prismatic family has 9 forms:

The A1 x A4 family has symmetry of order 240 (2*5!).

# Coxeter diagram
and Schläfli
symbols
Name
Element counts
FacetsCellsFacesEdgesVertices
59CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = {3,3,3}×{ }
5-cell prism (penp)
720302510
60CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = r{3,3,3}×{ }
Rectified 5-cell prism (rappip)
1250907020
61CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = t{3,3,3}×{ }
Truncated 5-cell prism (tippip)
125010010040
62CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = rr{3,3,3}×{ }
Cantellated 5-cell prism (srippip)
2212025021060
63CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = t0,3{3,3,3}×{ }
Runcinated 5-cell prism (spiddip)
3213020014040
64CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = 2t{3,3,3}×{ }
Bitruncated 5-cell prism (decap)
126014015060
65CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = tr{3,3,3}×{ }
Cantitruncated 5-cell prism (grippip)
22120280300120
66CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = t0,1,3{3,3,3}×{ }
Runcitruncated 5-cell prism (prippip)
32180390360120
67CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = t0,1,2,3{3,3,3}×{ }
Omnitruncated 5-cell prism (gippiddip)
32210540600240

B4 × A1

This prismatic family has 16 forms. (Three are shared with [3,4,3]×[ ] family)

The A1×B4 family has symmetry of order 768 (254!).

The last three snubs can be realised with equal-length edges, but turn out nonuniform anyway because some of their 4-faces are not uniform 4-polytopes.

# Coxeter diagram
and Schläfli
symbols
Name
Element counts
FacetsCellsFacesEdgesVertices
[16]CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = {4,3,3}×{ }
Tesseractic prism (pent)
(Same as 5-cube)
1040808032
68CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = r{4,3,3}×{ }
Rectified tesseractic prism (rittip)
2613627222464
69CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = t{4,3,3}×{ }
Truncated tesseractic prism (tattip)
26136304320128
70CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = rr{4,3,3}×{ }
Cantellated tesseractic prism (srittip)
58360784672192
71CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = t0,3{4,3,3}×{ }
Runcinated tesseractic prism (sidpithip)
82368608448128
72CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = 2t{4,3,3}×{ }
Bitruncated tesseractic prism (tahp)
26168432480192
73CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = tr{4,3,3}×{ }
Cantitruncated tesseractic prism (grittip)
58360880960384
74CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = t0,1,3{4,3,3}×{ }
Runcitruncated tesseractic prism (prohp)
8252812161152384
75CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = t0,1,2,3{4,3,3}×{ }
Omnitruncated tesseractic prism (gidpithip)
8262416961920768
76CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = {3,3,4}×{ }
16-cell prism (hexip)
1864885616
77CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = r{3,3,4}×{ }
Rectified 16-cell prism (icope)
(Same as 24-cell prism)
2614428821648
78CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = t{3,3,4}×{ }
Truncated 16-cell prism (thexip)
2614431228896
79CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = rr{3,3,4}×{ }
Cantellated 16-cell prism (ricope)
(Same as rectified 24-cell prism)
50336768672192
80CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = tr{3,3,4}×{ }
Cantitruncated 16-cell prism (ticope)
(Same as truncated 24-cell prism)
50336864960384
81CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png = t0,1,3{3,3,4}×{ }
Runcitruncated 16-cell prism (prittip)
8252812161152384
82CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png = sr{3,3,4}×{ }
snub 24-cell prism (sadip)
1467681392960192
NonuniformCDel node h.pngCDel 2x.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.png
rectified tesseractic alterprism (rita)
5028846428864
NonuniformCDel node h.pngCDel 2x.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.png
truncated 16-cell alterprism (thexa)
2616838433696
NonuniformCDel node h.pngCDel 2x.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h.png
bitruncated tesseractic alterprism (taha)
50288624576192

F4 × A1

This prismatic family has 10 forms.

The A1 x F4 family has symmetry of order 2304 (2*1152). Three polytopes 85, 86 and 89 (green background) have double symmetry [[3,4,3],2], order 4608. The last one, snub 24-cell prism, (blue background) has [3+,4,3,2] symmetry, order 1152.

# Coxeter diagram
and Schläfli
symbols
Name
Element counts
FacetsCellsFacesEdgesVertices
[77]CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = {3,4,3}×{ }
24-cell prism (icope)
2614428821648
[79]CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = r{3,4,3}×{ }
rectified 24-cell prism (ricope)
50336768672192
[80]CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = t{3,4,3}×{ }
truncated 24-cell prism (ticope)
50336864960384
83CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png = rr{3,4,3}×{ }
cantellated 24-cell prism (sricope)
146100823042016576
84