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User:Tomruen/List of isotoxal polychora and honeycombs

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A vertex transitive polytope is also edge-transitive if its vertex figure is vertex transitive! (Since each vertex in the vertex figure represents an edges in the polytope)

I think this is a complete list of regular convex and uniform 4-polytopes/honeycombs that are isotoxal. (And a subset of nonconvex forms from the nonconvex regulars)

Linear graph polychora/honeycombs

[edit]

From convex self-dual regular and uniform polychora:

[p,q,p]
{p,q,p}
{q,p}

r{p,q,p}
{}x{p}

2t{p,q,p}
s{2,4}

e{p,q,p}
s{2,2q}
[3,3,3]
{3,3,3}

r{3,3,3}

2t{3,3,3}

e{3,3,3}
[3,4,3]
{3,4,3}

r{3,4,3}

2t{3,4,3}

e{3,4,3}
[5/2,5,5/2]
{5/2,5,5/2}

r{5/2,5,5/2}

2t{5/2,5,5/2}
DEGENERATE

e{5/2,5,5/2}
DEGENERATE
[5/2,5,5/2]
{5/2,5,5/2}

r{5/2,5,5/2}

2t{5/2,5,5/2}
DEGENERATE

e{5/2,5,5/2}
DEGENERATE
[4,3,4]
{4,3,4}

r{4,3,4}

2t{4,3,4}

e{4,3,4}
[3,5,3]
{3,5,3}

r{3,5,3}

2t{3,5,3}

e{3,5,3}
[3,6,3]
{3,6,3}

r{3,6,3}

2t{3,6,3}

e{3,6,3}


[5,3,5]
{5,3,5}

r{5,3,5}

2t{5,3,5}

e{5,3,5}
[6,3,6]
{6,3,6}

r{6,3,6}

2t{6,3,6}

e{6,3,6}

From convex regular and uniform polychora:

[p,q,r]
{p,q,r}
{q,r}

r{p,q,r}
{}x{r}

r{r,q,p}
{p}x{}

{r,q,p}
{q,p}
[4,3,3]
{4,3,3}

r{4,3,3}

r{3,3,4} (24-cell)

{3,3,4}
[5,3,3]
{5,3,3}

r{5,3,3}

r{3,3,5}

{3,3,5}
[6,3,3]
{6,3,3}

r{6,3,3}

r{3,3,6}

{3,3,6}
[5/2,5,3]
{5/2,5,3}

r{5/2,5,3}

r{3,5,5/2}

{3,5,5/2}
[5,3,5/2]
{5,3,5/2}

r{5,3,5/2}

r{5/2,3,5}

{5/2,3,5}
[3,5/2,5]
{3,5/2,5}

r{3,5/2,5}

r{5,5/2,3}

{5,5/2,3}
[3,3,5/2]
{3,3,5/2}

r{3,3,5/2}

r{5/2,3,3}

{5/2,3,3}
[5,3,4]
{5,3,4}

r{5,3,4}

r{4,3,5}

{4,3,5}
[6,3,4]
{6,3,4}

r{6,3,4}

r{4,3,6}

{4,3,6}
[6,3,5]
{6,3,5}

r{6,3,5}

r{5,3,6}

{5,3,6}

Bifurcated graph honeycombs

[edit]
Family
=

=

=

=
[3,31,1]



[4,31,1]



[5,31,1]



[6,31,1]
Family
=

=

=

=
[3,41,1]
[4,41,1]

Cyclic graph honeycombs

[edit]
Family
[(3,3,3,3)]




[(4,3,4,3)]





[(5,3,5,3)]





[(6,3,6,3)]
Family
[(4,3,3,3)]




[(5,3,3,3)]




[(6,3,3,3)]
[(5,3,4,3)]




[(6,3,5,3)]
Family
[(4,4,4,3)]
[(4,4,4,4)]