Octaexon Rank 7 Type Regular Space Spherical Notation Bowers style acronym Oca Coxeter diagram x3o3o3o3o3o3o ( Schläfli symbol {3,3,3,3,3,3} Tapertopic notation 16 Elements Exa 8 heptapeta Peta 28 hexatera Tera 56 pentachora Cells 70 tetrahedra Faces 56 triangles Edges 28 Vertices 8 Vertex figure Heptapeton , edge length 1Measures (edge length 1) Circumradius
7
4
≈
0.66144
{\displaystyle \frac{\sqrt7}{4} ≈ 0.66144}
Edge radius
3
4
≈
0.43301
{\displaystyle \frac{\sqrt3}{4} \approx 0.43301}
Face radius
15
12
≈
0.32275
{\displaystyle \frac{\sqrt{15}}{12} ≈ 0.32275}
Cell radius
1
4
=
0.25
{\displaystyle \frac14 = 0.25}
Teron radius
15
20
≈
0.19365
{\displaystyle \frac{\sqrt{15}}{20} ≈ 0.19365}
Peton radius
3
12
≈
0.14434
{\displaystyle \frac{\sqrt3}{12} ≈ 0.14434}
Inradius
7
28
≈
0.094491
{\displaystyle \frac{\sqrt7}{28} ≈ 0.094491}
Hypervolume
1
20160
≈
0.000049603
{\displaystyle \frac{1}{20160} ≈ 0.000049603}
Diexal angle
arccos
(
1
7
)
≈
81.78679
∘
{\displaystyle \arccos\left(\frac17\right) ≈ 81.78679^\circ}
Heights Point atop hop:
2
7
7
≈
0.75593
{\displaystyle \frac{2\sqrt7}{7} ≈ 0.75593}
Dyad atop perp hix:
3
3
≈
0.57735
{\displaystyle \frac{\sqrt3}{3} ≈ 0.57735}
Trig atop perp pen:
2
15
15
≈
0.51640
{\displaystyle \frac{2\sqrt{15}}{15} ≈ 0.51640}
Tet atop perp tet:
1
2
=
0.5
{\displaystyle \frac12 = 0.5}
Central density 1 Number of pieces 8 Level of complexity 1 Related polytopes Army Oca Regiment Oca Dual Octaexon Conjugate None Abstract properties Flag count40320 Euler characteristic 2 Topological properties Orientable Yes Properties Symmetry A7 , order 40320Convex Yes Nature Tame
The octaexon , or oca , also commonly called the 7-simplex , is the simplest possible non-degenerate polyexon . The full symmetry version has 8 regular heptapeta as facets, joining 3 to a pentachoron peak and 7 to a vertex, and is one of the 3 regular polyexa . It is the 7-dimensional simplex . It is also a pyramid based on the heptapeton.
A regular octaexon of edge length 2 can be inscribed in the unit hepteract .[1] dodecadakon .[2]
The vertices of a regular octaexon of edge length 1, centered at the origin, are given by:
(
±
1
2
,
−
3
6
,
−
6
12
,
−
10
20
,
−
15
30
,
−
21
42
,
−
7
28
)
,
{\displaystyle \left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),}
(
0
,
3
3
,
−
6
12
,
−
10
20
,
−
15
30
,
−
21
42
,
−
7
28
)
,
{\displaystyle \left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),}
(
0
,
0
,
6
4
,
−
10
20
,
−
15
30
,
−
21
42
,
−
7
28
)
,
{\displaystyle \left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),}
(
0
,
0
,
0
,
10
5
,
−
15
30
,
−
21
42
,
−
7
28
)
,
{\displaystyle \left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),}
(
0
,
0
,
0
,
0
,
15
6
,
−
21
42
,
−
7
28
)
,
{\displaystyle \left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28}\right),}
(
0
,
0
,
0
,
0
,
0
,
21
7
,
−
7
28
)
,
{\displaystyle \left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28}\right),}
(
0
,
0
,
0
,
0
,
0
,
0
,
7
4
)
.
{\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4}\right).}
Much simpler sets of coordinates can be found by inscribing the octaexon into the hepteract . One such set is given by:[3]
(
1
4
,
1
4
,
1
4
,
1
4
,
1
4
,
1
4
,
1
4
)
,
{\displaystyle \left(\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14,\,\frac14\right),}
(
1
4
,
1
4
,
1
4
,
−
1
4
,
−
1
4
,
−
1
4
,
−
1
4
)
,
{\displaystyle \left(\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,-\frac14\right),}
(
1
4
,
−
1
4
,
−
1
4
,
−
1
4
,
−
1
4
,
1
4
,
1
4
)
,
{\displaystyle \left(\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,\frac14\right),}
(
1
4
,
−
1
4
,
−
1
4
,
1
4
,
1
4
,
−
1
4
,
−
1
4
)
,
{\displaystyle \left(\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14\right),}
(
−
1
4
,
1
4
,
−
1
4
,
1
4
,
−
1
4
,
1
4
,
−
1
4
)
,
{\displaystyle \left(-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,-\frac14\right),}
(
−
1
4
,
1
4
,
−
1
4
,
−
1
4
,
1
4
,
−
1
4
,
1
4
)
,
{\displaystyle \left(-\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14\right),}
(
−
1
4
,
−
1
4
,
1
4
,
1
4
,
−
1
4
,
−
1
4
,
1
4
)
,
{\displaystyle \left(-\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14,\,-\frac14,\,\frac14\right),}
(
−
1
4
,
−
1
4
,
1
4
,
−
1
4
,
1
4
,
1
4
,
−
1
4
)
.
{\displaystyle \left(-\frac14,\,-\frac14,\,\frac14,\,-\frac14,\,\frac14,\,\frac14,\,-\frac14\right).}
Even simpler coordinates can be given in eight dimensions , as all permutations of:
(
2
2
,
0
,
0
,
0
,
0
,
0
,
0
,
0
)
.
{\displaystyle \left(\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,0,\,0,\,0\right).}
An octaexon has the following Coxeter diagrams :
x3o3o3o3o3o3o (full symmetry)
ox3oo3oo3oo3oo3oo&#x (A6 axial, heptapetal pyramid)
xo ox3oo3oo3oo3oo&#x (A5×A1 axial, hexateral scalene)
xo3oo ox3oo3oo&#x (A4×A2 axial, pentachoric tettene)
xo3oo3oo ox3oo3oo&#x (A3×A3 axial, tetrahedral disphenoid)
oxo3ooo oox3ooo3ooo&#x (A3×A2 symmetry, tetrahedral tettene pyramid)
oxo xoo3ooo ooxooo&#x (A2×A2×A1 symmetry, trigonal disphenoid scalene)
xoo oox oxo3ooo3ooo&#x (A3×A1×A1 symmetry, tetrahedral scalenic scalene)
oxoo3oooo ooxo3oooo&#x (A2×A2 symmetry, trigonal pyramidal disphenoid) Klitzing, Richard. "oca" . 
