Truncated tetracontoctachoron

Truncated tetracontoctachoron
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Ticont
Elements
Cells	288 tetragonal disphenoids, 48 ditruncated cubes
Faces	1152 isosceles triangles, 192 ditrigons, 144 dioctagons
Edges	576+576+1152
Vertices	1152
Vertex figure	Sphenoid
Measures (variant with regular hexagons of edge length 1)
Edge lengths	Edges of hexagons (288+288): 1
	Lacing edges (576): ${\sqrt {2+{\sqrt {2}}}}\approx 1.84776$
Circumradius	$2{\sqrt {13+9{\sqrt {2}}}}\approx 10.14454$
Central density	1
Related polytopes
Army	Ticont
Regiment	Ticont
Dual	Tetrakis bitetracontoctachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The truncated tetracontoctachoron or ticont is a convex isogonal polychoron that consists of 48 ditruncated cubes and 288 tetragonal disphenoids. 1 tetragonal disphenoid and 3 ditruncated cubes join at each vertex. It can be formed by truncating the tetracontoctachoron.

It can also be formed as the convex hull of 2 oppositely oriented semi-uniform variants of the great rhombated icositetrachoron, where if the great rhombated icositetrachora are of the form a3b4c3o, then $c=b+(2+{\sqrt {2}})a$ It is one of five polychora (including two transitional cases) formed from two great rhombated icositetrachora, and is the transitional point between the medial bicantitruncatotetracontoctachoron and great bicantitruncatotetracontoctachoron.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {2+{\sqrt {2}}}}$ ≈ 1:1.84776. This variant uses regular hexagons as faces.

Vertex coordinates

The vertices of a truncated tetracontoctachoron with hexagons of edge length 1, centered at the origin, are given by all permutations of:

• $\left(0,\,\pm 3{\frac {2+{\sqrt {2}}}{2}},\,\pm (3+2{\sqrt {2}}),\,\pm {\frac {6+5{\sqrt {2}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {2}}}{2}},\,\pm 3{\frac {1+{\sqrt {2}}}{2}},\,\pm 3{\frac {3+2{\sqrt {2}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {5+3{\sqrt {2}}}{2}},\,\pm {\frac {5+3{\sqrt {2}}}{2}},\,\pm {\frac {7+6{\sqrt {2}}}{2}}\right),$
• $\left(\pm 1,\,\pm {\frac {4+3{\sqrt {2}}}{2}},\,\pm {\frac {4+3{\sqrt {2}}}{2}},\,\pm (4+3{\sqrt {2}})\right),$
• $\left(\pm {\frac {3+{\sqrt {2}}}{2}},\,\pm {\frac {3+2{\sqrt {2}}}{2}},\,\pm 3{\frac {1+{\sqrt {2}}}{2}},\,\pm 3{\frac {3+2{\sqrt {2}}}{2}}\right).$

External links

• Klitzing, Richard. "ticont".