Rectified 5-orthoplex

Rectified 5-orthoplex
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Rat
Coxeter diagram	o4o3o3x3o ()
Elements
Tera	10 hexadecachora, 32 rectified pentachora
Cells	160 tetrahedra, 80 octahedra
Faces	80+320 triangles
Edges	240
Vertices	40
Vertex figure	Octahedral prism, edge length 1
Measures (edge length 1)
Circumradius	1
Hypervolume	${\frac {9{\sqrt {2}}}{10}}\approx 1.27279$
Diteral angles	Rap–oct–rap: $\arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }$
	Rap–tet–hex: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
Central density	1
Number of external pieces	42
Level of complexity	4
Related polytopes
Army	Rat
Regiment	Rat
Dual	Joined penteract
Conjugate	None
Abstract & topological properties
Flag count	15360
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B5, order 3840
Convex	Yes
Nature	Tame

The rectified 5-orthoplex, also called the rectified triacontaditeron or rat, is a convex uniform 5-polytope. It consists of 10 hexadecachora, and 32 rectified pentachora. Two hexadecachora and 8 rectified pentachora join at each octahedral prismatic vertex. As the name suggests, it is the rectification of the 5-orthoplex.

The rectified 5-orthoplex contains the vertices of a square-octahedral duoprism.

Vertex coordinates

The vertices of a rectified 5-orthoplex of edge length 1 are given by all permutations of:

• $\left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,0\right)$.

Representations

A rectified 5-orthoplex has the following Coxeter diagrams:

• o4o3o3x3o () (full symmetry)
• o3o3x3o *b3o () (D₅ symmetry)
• ooo4ooo3oxo3xox&#xt (B₄ axial, hexadecachoronfirst)
• ooo3oxo3ooo *b3xox&#xt (D₄ axial, hexadecachoron-first)
• oxo3xoo3oox3oxo&#xt (A₄ axial, rectified pentachoron-first)
• qo oo4oo3ox3xo&#zx (B₄×A₁ symmetry)
• qoo4oxo ooo4oox3oxo#zx (B₃×B₂ symmetry)
• ox(ou)xo oo(oo)oo4oo(xo)oo3ox(oo)oo&#xt (B₃×A₁ axial, vertex-first)
• ox(ou)xo oo(xo)oo3ox(oo)xo3oo(xo)oo&#xt (A₃×A₁ symmetry, vertex-first)

Gallery

• 

  B₄ or D₅ orthographic projection

• 

  B₃, D₄, or A₂

• 

  B₂

• 

  A₃

Related polytopes

The rectified 5-orthoplex is the colonel of a regiment that includes a total of 44 uniform members, plus one compound. Of the 44 members, 19 have full B₅ symmetry, while the other 25 have D₅ symmetry only.

External links

• Bowers, Jonathan. "Category 4: Rat Regiment" (#48).

• Klitzing, Richard. "rat".
• Wikipedia contributors. "Rectified 5-orthoplex".