Non-expanding horizon

The Taylor polyhedra are the vertex-transitive polyhedra that are not included in the standard list of uniform polyhedra.

Some are omitted because they include the double faces {²3},{²5/2 and {²5} that naturally result from the truncation of the inverse polygons {3/2} and {5/4} and the star polygon {5/2} respectively, all even denominator polygons.

Others are omitted because they include the cross polygon {4/2} and its double faced natural truncation {²4}, used to form the {4,4/2} family of cross polyhedra, analogous to the {5,5/2} family of star polyhedra.

Taken with the accepted uniform polyhedra, the Taylor polyhedra allow a more complete classification to emerge, without the peculiar gaps that currently exist within the uniform polyhedra.

These polyhedra generally have densities greater than 1, the tiling of the sphere that produces them taking place in multiple layers with several visits to any one vertex or edge

List

Name	Wythoff symbol	Schläfli symbol	Taylor reference	Vertex figure	Vertices	Edges	Faces by type
Quasitruncated tetrahedron	3 2 |3/2	t{3’, 3}	{3, 3/2} + 2{3, 3}	23.23.3	4×3	6×3	4×23 4×3
Quasitruncated dodecahedron	3 2 |5/4	t{5’, 3}	{3, 5/2 }+ 2{5/2, 5}	25/2.25/2.3	12×5	30×3	12×25/2 20×3
Quasitruncated octahedron	4 2 |3/2	t{3’, 4}	{4, 4/2} + 2{3, 4}	23.23.4	6×4	12×3	8×23 6×4
Quasitruncated icosahedron	5 2 |3/2	t{3’, 5}	{5, 5/2} + 2{3, 5}	23.23.5	12×5	30×3	20×23 12×5
Triquasitruncated octahedron	3/2 2 3|	t$\left\{\genfrac{}{}{0ex}{}{3^{\prime }}{3}\right\}$	[2.4a]	23.6.4	12×2	12×2 12×1	4×23 4×6 6×4
Pentaquasitruncated icosidodecahedron	3 2 5/4|	t$\left\{\genfrac{}{}{0ex}{}{5^{\prime }}{3}\right\}$	[2.4d]	25/2.6.4	60×2	60×2 60×1	12×25/2 20×6 30×4
Triquasitruncated cuboctahedron	3/2 2 4|	t$\left\{\genfrac{}{}{0ex}{}{3^{\prime }}{4}\right\}$	[2.4b]	23.8.4	24×2	24×2 24×1	8×23 6×8 12×4
Triquasitruncated icosidodecahedron	3/2 2 5|	t$\left\{\genfrac{}{}{0ex}{}{3^{\prime }}{5}\right\}$	[2.4e]	23.10.4	60×2	60×2 60×1	20×23 12×10 30×4
Quasiquasitruncated icosidodecahedron	3/2 2 5/4|	t$\left\{\genfrac{}{}{0ex}{}{3^{\prime }}{5^{\prime }}\right\}$	[2.4f]	23.25/2.4	20×6	60×3	12×25/2 20×23 30×4
Quasiquasitruncated cuboctahedron	3/2 2 4/3|	t$\left\{\genfrac{}{}{0ex}{}{3^{\prime }}{4^{\prime }}\right\}$	[2.4c]	23.8/3.4	24×2	24×2 24×1	6×8/3 8×23 12×4
Quasirhombicosidodecahedron	3/2 5 |2	r'$\left\{\genfrac{}{}{0ex}{}{3}{5}\right\}$	[2.4h]	4.3/2.4.5	20×3	60×2	12×5 20×3 30×4
Quasisnub dodecahedron
Quasisnub tetrahedron
Quasisnub octahedron
Small quasidodecicosidodecahedron
Double octahedron
Double tetrahemihexahedron
Small quasirhombidodecahedron
Stella octangula	3| 4/2 3		[1.4a]	3.4/2.3.4/2.3.4/2	8	24	8×3 6×4/2
Inscribed tetrahedron	3| 3/2 3		{3, 3/2} + {3, 3}	3.3/2.3.3/2.3.3/2	4	6×2	4×3 4×3
Inscribed octahedron	4| 3/2 4		{4, 4/2} + {3, 4}	4.3/2.4.3/2.4.3/2.4.3/2	6	12×2	6×4 8×3
Inscribed icosahedron	5| 3/2 5		{5, 5/2} + {3, 5}	5.3/2.5.3/2.5.3/2.5.3/2.5.3/2	12	30×2	12×5 20×3
Truncated stella octangula	3 4/2 |3		[1.4b]	3.6.4/2.6	24	48	8×3 8×6 6×4/2
Quasiquasitruncated inscribed tetrahedron
Quasiquasitruncated stella octangula
Quasiquasitruncated small ditrigonal icosidodecahedron
Quasiquasitruncated inscribed icosahedron
Double stella octangula
Small quasicosicosidodecahedron
Double tetrahemihexahedron
Octaoctahedron
Snub inscribed tetrahedron listed as Octahedron
Snub stella octangula
Snub inscribed octahedron listed as Cuboctahedron
Snub inscribed icosahedron listed as Icosidodecahedron
Quasisnub stella octangula
Quasisnub icosicosidodecahedron
Double tetrahemihexahedron
Retrosnub stella octangula
Inscribed octahedron
Quasiquasisnub inscribed octahedron listed as Inscribed octahedron
Quasiquasisnub inscribed icosahedron listed as Inscribed icosahedron
Inscribed small stellated dodecahedron
Inscribed dodecadodecahedron
Inscribed icosidodecahedron
Inscribed great icosidodecahedron
Double inscribed icosahedron listed as Inscribed icosahedron'
Quasicosidodecadodecahedron
Inscribed small stellated dodecahedron
Quasisnub icosidodecadodecahedron
Great hexahedron	4/2| 2 4	{4,4/2}	[1.3a]	4.4.4.4	6	12	6×4
Stellated hexahedron	4| 2 4/2	{4/2,4}	[1.3d]	4/2.4/2.4/2.4/2	6	12	6×4/2
Truncated great hexahedron	4/2 2 |4	t{4,4/2}	[1.3b]	8.8.4/2	24	36	6×8 6×4/2
Truncated stellated hexahedron	4 2 |4/2	t{4/2,4}	[1.3e] 3{4,3}	24.24.4	8×3	12×3	6×24 6×4
Truncated small stellated dodecahedron	5 2 |5/2	t{5/2,5}	3{5,3}	25.25.5	20×3	30×3	12×25 12×5
Truncated great stellated dodecahedron	3 2 |5/2	t{5/2,3}	{3,5} + 2{5,5/2}	25.25.3	12×5	30×3	12×25 20×3
Hexahexahedron	2| 4 4/2	$\left\{\genfrac{}{}{0ex}{}{4}{4/2}\right\}$	[1.3f]	4.4/2.4.4/2	12	24	6×4 6×4/2
Truncated hexahexahedron	4 2 4/2|	t$\left\{\genfrac{}{}{0ex}{}{4}{4/2}\right\}$	[1.3g] [2.4m]	8.24.4	24×2	24×2 24×1	6×8 6×24 12×4
Truncated dodecadodecahedron	5 2 5/2|	t$\left\{\genfrac{}{}{0ex}{}{5}{5/2}\right\}$	[2.4j]	10.25.4	60×2	60×2 60×1	12×10 12×25 30×4
Truncated great icosidodecahedron	3 2 5/2|	t$\left\{\genfrac{}{}{0ex}{}{3}{5/2}\right\}$	[2.4g]	6.25.4	60×2	60×2 60×1	20×6 12×25 30×4
Rhombihexahexahedron	4 4/2 |2	r$\left\{\genfrac{}{}{0ex}{}{4}{4/2}\right\}$	[1.3i]	4.4/2.4.4	24	48	6×4 6×4/2 12×4
Great rhombicosidodecahedron	3 5/2 |2	r$\left\{\genfrac{}{}{0ex}{}{3}{5/2}\right\}$		4.5/2.4.3	20×3	60×2	20×3 12×5/2 30×4
Snub hexahexahedron	|4 2 4/2	s$\left\{\genfrac{}{}{0ex}{}{4}{4/2}\right\}$	[1.3j]	3.4/2.3.3.4	24	60	6×4 6×4/2 24×3
Quasitruncated great hexahedron
Quasitruncated great dodecahedron
Quasitruncated great icosahedron
Quasitruncated hexahexahedron
Truncated hexahexahedron
Pentaquasitruncated dodecadodecahedron
Triquasitruncated great icosidodecahedron
Quasiquasitruncated great icosdodecahedron
Quasiquasitruncated dodecadodecahedron
Quasirhombidodecadodecahedron
Small quasisnub icosidodecahedron
Rhombihexahedron
Great quasidodecicosidodecahedron
Great quasirhombidodecahedron
Quasirhombicosahedron
Double stellated hexahedron
Double great dodecahedron
Double small stellated dodecahedron
Double hexahexahedron
Double dodecadodecahedron
Double truncated stellated hexahedron
Double truncated great dodecahedron
Quasiquasitruncated great ditrigonal icosidodecahedron
Double quasitruncated small stellated dodecahedron
Quasiquasitruncated inscribed small stellated dodecahedron
Double Dodecadodecahedron 2 This differs from the double dodecadodecahedron in that the pentagrams are two-fold
Great quasicosicosidodecahedron
Triple stella octangula
Triple inscribed icosahedron
Triple small ditrigonal icosidodecahedron
Great icosidodecahedron
Great quasisnub icosicosidodecahedron
Triple great ditrigonal icosidodecahedron
Great retrosnub icosicosidodecahedron
Triple inscribed small stellated dodecahedron
Great retrosnub dodecicosidodecahedron

References

• Taylor, P. The Simpler? Polyhedra—being the third part of several comprising The Complete? Polyhedra Nattygrafix, 1999
• Taylor, P. The Star & Cross Polyhedra—being the fourth part of several comprising The Complete? Polyhedra Nattygrafix, 2000

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