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User:Tomruen/List of isotoxal polychora and honeycombs

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

From convex self-dual regular and uniform polychora:

class=wikitable

![p,q,p]

!{{CDD|node_1|p|node|q|node|p|node}}
{p,q,p}
{q,p}

!{{CDD|node|p|node_1|q|node|p|node}}
r{p,q,p}
{}x{p}

!{{CDD|node|p|node_1|q|node_1|p|node}}
2t{p,q,p}
s{2,4}

!{{CDD|node_1|p|node|q|node|p|node_1}}
e{p,q,p}
s{2,2q}

align=center

| [3,3,3]

|{{CDD|node_1|3|node|3|node|3|node}}
{3,3,3}
100px

{{CDD|node|3|node_1|3|node|3|node}}
r{3,3,3}
100px		{{CDD|node|3|node_1|3|node_1|3|node}}
2t{3,3,3}
100px		{{CDD|node_1|3|node|3|node|3|node_1}}
e{3,3,3}
100px
align=center

| [3,4,3]

{{CDD|node_1|3|node|4|node|3|node}}
{3,4,3}
100px		{{CDD|node|3|node_1|4|node|3|node}}
r{3,4,3}
100px		{{CDD|node|3|node_1|4|node_1|3|node}}
2t{3,4,3}
100px		{{CDD|node_1|3|node|4|node|3|node_1}}
e{3,4,3}
100px
align=center

|[5/2,5,5/2]

{{CDD|node_1|5|rat|d2|node|5|node|5|rat|d2|node}}
{5/2,5,5/2}		{{CDD|node|5|rat|d2|node_1|5|node|5|rat|d2|node}}
r{5/2,5,5/2}		{{CDD|node|5|rat|d2|node_1|5|node_1|5|rat|d2|node}}
2t{5/2,5,5/2}
DEGENERATE		{{CDD|node_1|5|rat|d2|node|5|node|5|rat|d2|node_1}}
e{5/2,5,5/2}
DEGENERATE
align=center

|[5/2,5,5/2]

{{CDD|node_1|5|rat|d2|node|5|node|5|rat|d2|node}}
{5/2,5,5/2}		{{CDD|node|5|rat|d2|node_1|5|node|5|rat|d2|node}}
r{5/2,5,5/2}		{{CDD|node|5|rat|d2|node_1|5|node_1|5|rat|d2|node}}
2t{5/2,5,5/2}
DEGENERATE		{{CDD|node_1|5|rat|d2|node|5|node|5|rat|d2|node_1}}
e{5/2,5,5/2}
DEGENERATE
align=center

| [4,3,4]

|{{CDD|node_1|4|node|3|node|4|node}}
{4,3,4}
100px

|{{CDD|node|4|node_1|3|node|4|node}}
r{4,3,4}
100px

|{{CDD|node|4|node_1|3|node_1|4|node}}
2t{4,3,4}
100px

|{{CDD|node_1|4|node|3|node|4|node_1}}
e{4,3,4}
100px

align=center

| [3,5,3]

|{{CDD|node_1|3|node|5|node|3|node}}
{3,5,3}
100px

|{{CDD|node|3|node_1|5|node|3|node}}
r{3,5,3}
100px

|{{CDD|node|3|node_1|5|node_1|3|node}}
2t{3,5,3}
100px

|{{CDD|node_1|3|node|5|node|3|node_1}}
e{3,5,3}
100px

align=center

| [3,6,3]

|{{CDD|node_1|3|node|6|node|3|node}}
{3,6,3}

|{{CDD|node|3|node_1|6|node|3|node}}
r{3,6,3}

|{{CDD|node|3|node_1|6|node_1|3|node}}
2t{3,6,3}

|{{CDD|node_1|3|node|6|node|3|node_1}}
e{3,6,3}

align=center

| [5,3,5]

|{{CDD|node_1|5|node|3|node|5|node}}
{5,3,5}
100px

|{{CDD|node|5|node_1|3|node|5|node}}
r{5,3,5}
100px

|{{CDD|node|5|node_1|3|node_1|5|node}}
2t{5,3,5}
100px

|{{CDD|node_1|5|node|3|node|5|node_1}}
e{5,3,5}
100px

align=center

| [6,3,6]

|{{CDD|node_1|6|node|3|node|6|node}}
{6,3,6}

|{{CDD|node|6|node_1|3|node|6|node}}
r{6,3,6}

|{{CDD|node|6|node_1|3|node_1|6|node}}
2t{6,3,6}

|{{CDD|node_1|6|node|3|node|6|node_1}}
e{6,3,6}

From convex regular and uniform polychora:

class=wikitable

![p,q,r]

!{{CDD|node_1|p|node|q|node|r|node}}
{p,q,r}
{q,r}

!{{CDD|node|p|node_1|q|node|r|node}}
r{p,q,r}
{}x{r}

!{{CDD|node|p|node|q|node_1|r|node}}
r{r,q,p}
{p}x{}

!{{CDD|node|p|node|q|node|r|node_1}}
{r,q,p}
{q,p}

align=center

| [4,3,3]

{{CDD|node_1|4|node|3|node|3|node}}
{4,3,3}
100px		{{CDD|node|4|node_1|3|node|3|node}}
r{4,3,3}
100px		{{CDD|node|4|node|3|node_1|3|node}}
r{3,3,4} (24-cell)
100px		{{CDD|node|4|node|3|node|3|node_1}}
{3,3,4}
100px
align=center

| [5,3,3]

{{CDD|node_1|5|node|3|node|3|node}}
{5,3,3}
100px		{{CDD|node|5|node_1|3|node|3|node}}
r{5,3,3}
100px		{{CDD|node|5|node|3|node_1|3|node}}
r{3,3,5}
100px		{{CDD|node|5|node|3|node|3|node_1}}
{3,3,5}
100px
align=center

| [6,3,3]

{{CDD|node_1|6|node|3|node|3|node}}
{6,3,3}		{{CDD|node|6|node_1|3|node|3|node}}
r{6,3,3}		{{CDD|node|6|node|3|node_1|3|node}}
r{3,3,6}		{{CDD|node|6|node|3|node|3|node_1}}
{3,3,6}
align=center

|[5/2,5,3]

{{CDD|node_1|5|rat|d2|node|5|node|3|node}}
{5/2,5,3}		{{CDD|node|5|rat|d2|node_1|5|node|3|node}}
r{5/2,5,3}		{{CDD|node|5|rat|d2|node|5|node_1|3|node}}
r{3,5,5/2}		{{CDD|node|5|rat|d2|node|5|node|3|node_1}}
{3,5,5/2}
align=center

|[5,3,5/2]

{{CDD|node_1|5|node|3|node|5|rat|d2|node}}
{5,3,5/2}		{{CDD|node|5|node_1|3|node|5|rat|d2|node}}
r{5,3,5/2}		{{CDD|node|5|node|3|node_1|5|rat|d2|node}}
r{5/2,3,5}		{{CDD|node|5|node|3|node|5|rat|d2|node_1}}
{5/2,3,5}
align=center

|[3,5/2,5]

{{CDD|node_1|3|node|5|rat|d2|node|5|node}}
{3,5/2,5}		{{CDD|node|3|node_1|5|rat|d2|node|5|node}}
r{3,5/2,5}		{{CDD|node|3|node|5|rat|d2|node_1|5|node}}
r{5,5/2,3}		{{CDD|node|3|node|5|rat|d2|node|5|node_1}}
{5,5/2,3}
align=center

|[3,3,5/2]

{{CDD|node_1|3|node|3|node|5|rat|d2|node}}
{3,3,5/2}		{{CDD|node|3|node_1|3|node|5|rat|d2|node}}
r{3,3,5/2}		{{CDD|node|3|node|3|node_1|5|rat|d2|node}}
r{5/2,3,3}		{{CDD|node|3|node|3|node|5|rat|d2|node_1}}
{5/2,3,3}
align=center

| [5,3,4]

|{{CDD|node_1|5|node|3|node|4|node}}
{5,3,4}
100px

|{{CDD|node|5|node_1|3|node|4|node}}
r{5,3,4}
100px

|{{CDD|node|5|node|3|node_1|4|node}}
r{4,3,5}
100px

|{{CDD|node|5|node|3|node|4|node_1}}
{4,3,5}
100px

align=center

| [6,3,4]

|{{CDD|node_1|6|node|3|node|4|node}}
{6,3,4}

|{{CDD|node|6|node_1|3|node|4|node}}
r{6,3,4}

|{{CDD|node|6|node|3|node_1|4|node}}
r{4,3,6}

|{{CDD|node|6|node|3|node|4|node_1}}
{4,3,6}

align=center

| [6,3,5]

|{{CDD|node_1|6|node|3|node|5|node}}
{6,3,5}

|{{CDD|node|6|node_1|3|node|5|node}}
r{6,3,5}

|{{CDD|node|6|node|3|node_1|5|node}}
r{5,3,6}

|{{CDD|node|6|node|3|node|5|node_1}}
{5,3,6}

Bifurcated graph honeycombs

class=wikitable

!Family

!{{CDD|node_1|p|node|split1|nodes}}
= {{CDD|node_1|p|node|3|node|4|node}}

!{{CDD|node|p|node_1|split1|nodes}}
= {{CDD|node|p|node_1|3|node|4|node}}

!{{CDD|node|p|node|split1|nodes_10lu}}
= {{CDD|node|p|node|3|node|4|node_h}}

!{{CDD|node|p|node|split1|nodes_11}}
= {{CDD|node|p|node|3|node_1|4|node}}

align=center

![3,31,1]

|100px
{{CDD|node_1|3|node|split1|nodes}}

|100px
{{CDD|node|3|node_1|split1|nodes}}

|100px
{{CDD|node|3|node|split1|nodes_10lu}}

|100px
{{CDD|node|3|node|split1|nodes_11}}

align=center

![4,31,1]

|100px
{{CDD|node_1|4|node|split1|nodes}}

|100px
{{CDD|node|4|node_1|split1|nodes}}

|100px
{{CDD|node|4|node|split1|nodes_10lu}}

|100px
{{CDD|node|4|node|split1|nodes_11}}

align=center

![5,31,1]

|100px
{{CDD|node_1|5|node|split1|nodes}}

|100px
{{CDD|node|5|node_1|split1|nodes}}

|100px
{{CDD|node|5|node|split1|nodes_10lu}}

|100px
{{CDD|node|5|node|split1|nodes_11}}

align=center

![6,31,1]

|{{CDD|node_1|6|node|split1|nodes}}

|{{CDD|node|6|node_1|split1|nodes}}

|{{CDD|node|6|node|split1|nodes_10lu}}

|{{CDD|node|6|node|split1|nodes_11}}

class=wikitable

!Family

!{{CDD|node_1|p|node|split1-44|nodes}}
= {{CDD|node_1|p|node|4|node|4|node}}

!{{CDD|node|p|node_1|split1-44|nodes}}
= {{CDD|node|p|node_1|4|node|4|node}}

!{{CDD|node|p|node|split1-44|nodes_10lu}}
= {{CDD|node|p|node|4|node|4|node_h}}

!{{CDD|node|p|node|split1-44|nodes_11}}
= {{CDD|node|p|node|4|node_1|4|node}}

align=center

![3,41,1]

|{{CDD|node_1|3|node|split1-44|nodes}}

|{{CDD|node|3|node_1|split1-44|nodes}}

|{{CDD|node|3|node|split1-44|nodes_10lu}}

|{{CDD|node|3|node|split1-44|nodes_11}}

align=center

![4,41,1]

|{{CDD|node_1|4|node|split1-44|nodes}}

|{{CDD|node|4|node_1|split1-44|nodes}}

|{{CDD|node|4|node|split1-44|nodes_10lu}}

|{{CDD|node|4|node|split1-44|nodes_11}}

Cyclic graph honeycombs

class=wikitable

!Family

!{{CDD|labelp|branch_10r|3ab|branch|labelp}}

!{{CDD|labelp|branch_10r|3ab|branch_01l|labelp}}

!{{CDD|labelp|branch_11|3ab|branch|labelp}}

!{{CDD|labelp|branch_10r|3ab|branch_10l|labelp}}

!{{CDD|labelp|branch_11|3ab|branch_11|labelp}}

[(3,3,3,3)]
{{CDD|branch|3ab|branch}}

!100px
{{CDD|branch_10r|3ab|branch}}

!100px
{{CDD|branch_10r|3ab|branch_01l}}

!100px
{{CDD|branch_11|3ab|branch}}

!

!100px
{{CDD|branch_11|3ab|branch_11}}

align=center

![(4,3,4,3)]
{{CDD|label4|branch|3ab|branch|label4}}

!100px
{{CDD|label4|branch_10r|3ab|branch|label4}}

!100px
{{CDD|label4|branch_10r|3ab|branch_01l|label4}}

!100px
{{CDD|label4|branch_11|3ab|branch|label4}}

!100px
{{CDD|label4|branch_10r|3ab|branch_10l|label4}}

!100px
{{CDD|label4|branch_11|3ab|branch_11|label4}}

[(5,3,5,3)]
{{CDD|label5|branch|3ab|branch|label5}}

!100px
{{CDD|label5|branch_10r|3ab|branch|label5}}

!100px
{{CDD|label5|branch_10r|3ab|branch_01l|label5}}

!100px
{{CDD|label5|branch_11|3ab|branch|label5}}

!100px
{{CDD|label5|branch_10r|3ab|branch_10l|label5}}

!100px
{{CDD|label5|branch_11|3ab|branch_11|label5}}

[(6,3,6,3)]
{{CDD|label6|branch|3ab|branch|label6}}

!{{CDD|label6|branch_10r|3ab|branch|label6}}

!{{CDD|label6|branch_10r|3ab|branch_01l|label6}}

!{{CDD|label6|branch_11|3ab|branch|label6}}

!{{CDD|label6|branch_10r|3ab|branch_10l|label6}}

!{{CDD|label6|branch_11|3ab|branch_11|label6}}

class=wikitable

!Family

!{{CDD|labelp|branch_10r|3ab|branch|labelq}}

!{{CDD|labelp|branch|3ab|branch_10l|labelq}}

!{{CDD|labelp|branch_11|3ab|branch|labelq}}

!{{CDD|labelp|branch|3ab|branch_11|labelq}}

align=center

![(4,3,3,3)]
{{CDD|label4|branch|3ab|branch}}

!100px
{{CDD|label4|branch_10r|3ab|branch}}

!100px
{{CDD|label4|branch|3ab|branch_10l}}

!100px
{{CDD|label4|branch_11|3ab|branch}}

!100px
{{CDD|label4|branch|3ab|branch_11}}

align=center

![(5,3,3,3)]
{{CDD|label5|branch|3ab|branch}}

!100px
{{CDD|label5|branch_10r|3ab|branch}}

!100px
{{CDD|label5|branch|3ab|branch_10l}}

!100px
{{CDD|label5|branch_11|3ab|branch}}

!100px
{{CDD|label5|branch|3ab|branch_11}}

align=center

![(6,3,3,3)]
{{CDD|label6|branch|3ab|branch}}

!{{CDD|label6|branch_10r|3ab|branch}}

!{{CDD|label6|branch|3ab|branch_10l}}

!{{CDD|label6|branch_11|3ab|branch}}

!{{CDD|label6|branch|3ab|branch_11}}

[(5,3,4,3)]
{{CDD|label5|branch|3ab|branch|label4}}

!100px
{{CDD|label5|branch_10r|3ab|branch|label4}}

!100px
{{CDD|label5|branch|3ab|branch_10l|label4}}

!100px
{{CDD|label5|branch_11|3ab|branch|label4}}

!100px
{{CDD|label5|branch|3ab|branch_11|label4}}

[(6,3,5,3)]
{{CDD|label6|branch|3ab|branch|label5}}

!{{CDD|label6|branch_10r|3ab|branch|label5}}

!{{CDD|label6|branch|3ab|branch_10l|label5}}

!{{CDD|label6|branch_11|3ab|branch|label5}}

!{{CDD|label6|branch|3ab|branch_11|label5}}

class=wikitable

!Family

!{{CDD|labelp|branch_10r|4a4b|branch|labelq}}

!{{CDD|labelp|branch_01r|4a4b|branch|labelq}}

!{{CDD|labelp|branch|4a4b|branch_10l|labelq}}

!{{CDD|labelp|branch_10r|4a4b|branch_01l|labelq}}

!{{CDD|labelp|branch_11|4a4b|branch|labelq}}

!{{CDD|labelp|branch_10r|4a4b|branch_10l|labelq}}

!{{CDD|labelp|branch|4a4b|branch_11|labelq}}

!{{CDD|labelp|branch_11|4a4b|branch_11|labelq}}

[(4,4,4,3)]
{{CDD|label4|branch|4a4b|branch}}

!{{CDD|label4|branch_10r|4a4b|branch}}

!

!{{CDD|label4|branch|4a4b|branch_10l}}

!

!{{CDD|label4|branch_11|4a4b|branch}}

!

!{{CDD|label4|branch|4a4b|branch_11}}

!

[(4,4,4,4)]
{{CDD|label4|branch|4a4b|branch|label4}}

!{{CDD|label4|branch_10r|4a4b|branch|label4}}

!

!

!{{CDD|label4|branch_10r|4a4b|branch_01l|label4}}

!{{CDD|label4|branch_11|4a4b|branch|label4}}

!

!

!{{CDD|label4|branch_11|4a4b|branch_11|label4}}