List of Taylor polyhedra

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} thjat 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 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.

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$\left\{{3'\atop3}\right\}$

[2.4a]

23.6.4

12×2

12×2, 12×1

4×23
4×6
6×4

Pentaquasitruncated icosidodecahedron

3 2 5/4|

t$\left\{{5'\atop3}\right\}$

[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$\left\{{3'\atop4}\right\}$

[2.4b]

23.8.4

24×2

24×2, 24×1

8×23
6×8
12×4

Triquasitruncated icosidodecahedron

3/2 2 5|

t$\left\{{3'\atop5}\right\}$

[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$\left\{{3'\atop5'}\right\}$

[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$\left\{{3'\atop4'}\right\}$

[2.4c]

23.8/3.4

24×2

24×2, 24×1

6×8/3
8×23
12×4

Quasirhombicosidodecahedron

3/2 2 5/4|

t$\left\{{3'\atop5'}\right\}$

[2.4f]

23.25/2.4

20×6

60×3

12×25/2
20×23
30×4

Quasisnub dodecahedron

Quasisnub tetrahedron

Quasisnub octahedron

Small quasidodecicosidodecahedron

Double octahedron

Double tetrahemihexahedron

Small quasirhumbidodecahedron

Inscribed tetrahedron

Stella octangula

Inscribed octahedron

Inscribed icosahedron

Truncate stella octangula

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

Stellated hexahedron

Truncated great hexahedron

Truncated stellated hexahedron

Truncated small stellated dodecahedron

Truncated great stellated dodecahedron

Hexahexahedron

Truncated great icosidodecahedron

Truncated hexahexahedron

Truncated dodecadodecahedron

Great rhombicosidodecahedron

Rhombihexahexahedron

Snub hexahexahedron

Quasitruncated great hexahedron

Quasitruncated great dodecahedron

Quasitrincated 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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