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Acronym
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ek
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Name
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diacosipentacontahexazetton,
octacross (β8),
aeroyott(id),
octaexal antiprism,
vertex figure of vee,
(one of the) Delone cell(s) of lattice E8,
Gosset polytope 51,1,
lattice C8 contact polytope (span of its small roots)
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Circumradius
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1/sqrt(2) = 0.707107
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Inradius
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1/4 = 0.25
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Lace city
in approx. ASCII-art
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 ©
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o -- o3o3o3o3o3o4o (point)
o g o -- x3o3o3o3o3o4o (zee)
o -- o3o3o3o3o3o4o (point)
where:
o = o3o3o3o3o4o (point)
g = x3o3o3o3o4o (gee)
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o3o3o3o3o3o x3o3o3o3o3o
o3o3o3o3o3x o3o3o3o3o3o
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x o3o3o3o3o o x3o3o3o3o
o o3o3o3o3x x o3o3o3o3o
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x3o o3o3o3o o3o x3o3o3o
o3o o3o3o3x o3x o3o3o3o
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x3o3o o3o3o o3o3o x3o3o
o3o3o o3o3x o3o3x o3o3o
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Coordinates
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(1/sqrt(2), 0, 0, 0, 0, 0, 0, 0) & all permutations, all changes of sign
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Volume
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1/2520 = 0.00039683
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Surface
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4/315 = 0.012698
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Rel. Roundness
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105 π4/65536 = 15.606620 %
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Dual
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octo
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Dihedral angles
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- at hop between oca and oca: arccos(-3/4) = 138.590378°
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Face vector
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16, 112, 448, 1120, 1792, 1792, 1024, 256
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Confer
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- more general:
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xPo3o...o3o4o
- related segmentozetta:
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zeepy
- general polytopal classes:
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Wythoffian polyzetta
orthoplex
noble polytopes
segmentozetta
fundamental lace prisms
Coxeter-Elte-Gosset polytopes
- analogs:
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regular orthoplex On
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External
links
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Incidence matrix according to Dynkin symbol
x3o3o3o3o3o3o4o
. . . . . . . . | 16 ♦ 14 | 84 | 280 | 560 | 672 | 448 | 128
----------------+----+-----+-----+------+------+------+------+----
x . . . . . . . | 2 | 112 ♦ 12 | 60 | 160 | 240 | 192 | 64
----------------+----+-----+-----+------+------+------+------+----
x3o . . . . . . | 3 | 3 | 448 ♦ 10 | 40 | 80 | 80 | 32
----------------+----+-----+-----+------+------+------+------+----
x3o3o . . . . . ♦ 4 | 6 | 4 | 1120 ♦ 8 | 24 | 32 | 16
----------------+----+-----+-----+------+------+------+------+----
x3o3o3o . . . . ♦ 5 | 10 | 10 | 5 | 1792 ♦ 6 | 12 | 8
----------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o . . . ♦ 6 | 15 | 20 | 15 | 6 | 1792 | 4 | 4
----------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o3o . . ♦ 7 | 21 | 35 | 35 | 21 | 7 | 1024 | 2
----------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o3o3o . ♦ 8 | 28 | 56 | 70 | 56 | 28 | 8 | 256
x3o3o3o3o3o3o4/3o
. . . . . . . . | 16 ♦ 14 | 84 | 280 | 560 | 672 | 448 | 128
------------------+----+-----+-----+------+------+------+------+----
x . . . . . . . | 2 | 112 ♦ 12 | 60 | 160 | 240 | 192 | 64
------------------+----+-----+-----+------+------+------+------+----
x3o . . . . . . | 3 | 3 | 448 ♦ 10 | 40 | 80 | 80 | 32
------------------+----+-----+-----+------+------+------+------+----
x3o3o . . . . . ♦ 4 | 6 | 4 | 1120 ♦ 8 | 24 | 32 | 16
------------------+----+-----+-----+------+------+------+------+----
x3o3o3o . . . . ♦ 5 | 10 | 10 | 5 | 1792 ♦ 6 | 12 | 8
------------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o . . . ♦ 6 | 15 | 20 | 15 | 6 | 1792 | 4 | 4
------------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o3o . . ♦ 7 | 21 | 35 | 35 | 21 | 7 | 1024 | 2
------------------+----+-----+-----+------+------+------+------+----
x3o3o3o3o3o3o . ♦ 8 | 28 | 56 | 70 | 56 | 28 | 8 | 256
o3o3o *b3o3o3o3o3x
. . . . . . . . | 16 ♦ 14 | 84 | 280 | 560 | 672 | 448 | 64 64
-------------------+----+-----+-----+------+------+------+------+--------
. . . . . . . x | 2 | 112 ♦ 12 | 60 | 160 | 240 | 192 | 32 32
-------------------+----+-----+-----+------+------+------+------+--------
. . . . . . o3x | 3 | 3 | 448 ♦ 10 | 40 | 80 | 80 | 16 16
-------------------+----+-----+-----+------+------+------+------+--------
. . . . . o3o3x ♦ 4 | 6 | 4 | 1120 ♦ 8 | 24 | 32 | 8 8
-------------------+----+-----+-----+------+------+------+------+--------
. . . . o3o3o3x ♦ 5 | 10 | 10 | 5 | 1792 ♦ 6 | 12 | 4 4
-------------------+----+-----+-----+------+------+------+------+--------
. . . o3o3o3o3x ♦ 6 | 15 | 20 | 15 | 6 | 1792 | 4 | 2 2
-------------------+----+-----+-----+------+------+------+------+--------
. o . *b3o3o3o3o3x ♦ 7 | 21 | 35 | 35 | 21 | 7 | 1024 | 1 1
-------------------+----+-----+-----+------+------+------+------+--------
o3o . *b3o3o3o3o3x ♦ 8 | 28 | 56 | 70 | 56 | 28 | 7 | 128 *
. o3o *b3o3o3o3o3x ♦ 8 | 28 | 56 | 70 | 56 | 28 | 7 | * 128
xo3oo3oo3oo3oo3oo3ox&#x → height = 1/2
(oca || dual oca)
o.3o.3o.3o.3o.3o.3o. & | 16 ♦ 7 7 | 21 63 | 35 140 105 | 35 175 350 | 21 126 315 210 | 7 49 147 245 | 1 8 28 56 35
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x. .. .. .. .. .. .. & | 2 | 56 * ♦ 6 6 | 15 30 15 | 20 60 80 | 15 60 105 60 | 6 30 66 90 | 1 6 16 26 15
oo3oo3oo3oo3oo3oo3oo&#x | 2 | * 56 ♦ 0 12 | 0 30 30 | 0 40 120 | 0 30 120 90 | 0 12 60 120 | 0 2 12 30 20
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o. .. .. .. .. .. & | 3 | 3 0 | 112 * ♦ 5 5 0 | 10 20 10 | 10 30 30 10 | 5 20 30 25 | 1 5 10 11 5
xo .. .. .. .. .. ..&#x & | 3 | 1 2 | * 336 ♦ 0 5 5 | 0 10 30 | 0 10 40 30 | 0 5 25 50 | 0 1 6 15 10
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o.3o. .. .. .. .. & ♦ 4 | 6 0 | 4 0 | 140 * * ♦ 4 4 0 | 6 12 6 0 | 4 12 12 4 | 1 4 6 4 1
xo3oo .. .. .. .. ..&#x & ♦ 4 | 3 3 | 1 3 | * 560 * ♦ 0 4 4 | 0 6 12 6 | 0 4 12 16 | 0 1 4 7 4
xo .. .. .. .. .. ox&#x ♦ 4 | 2 4 | 0 4 | * * 420 ♦ 0 0 8 | 0 0 12 12 | 0 0 8 24 | 0 0 2 8 6
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o.3o.3o. .. .. .. & ♦ 5 | 10 0 | 10 0 | 5 0 0 | 112 * * ♦ 3 3 0 0 | 3 6 3 0 | 1 3 3 1 0
xo3oo3oo .. .. .. ..&#x & ♦ 5 | 6 4 | 4 6 | 1 4 0 | * 560 * ♦ 0 3 3 0 | 0 3 6 3 | 0 1 3 3 1
xo3oo .. .. .. .. ox&#x & ♦ 5 | 4 6 | 1 9 | 0 2 3 | * * 1120 ♦ 0 0 3 3 | 0 0 3 9 | 0 0 1 4 3
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o.3o.3o.3o. .. .. & ♦ 6 | 15 0 | 20 0 | 15 0 0 | 6 0 0 | 56 * * * | 2 2 0 0 | 1 2 1 0 0
xo3oo3oo3oo .. .. ..&#x & ♦ 6 | 10 5 | 10 10 | 5 10 0 | 1 5 0 | * 336 * * | 0 2 2 0 | 0 1 2 1 0
xo3oo3oo .. .. .. ox&#x & ♦ 6 | 7 8 | 4 16 | 1 8 6 | 0 2 4 | * * 840 * | 0 0 2 2 | 0 0 1 2 1
xo3oo .. .. .. oo3ox&#x ♦ 6 | 6 9 | 2 18 | 0 6 9 | 0 0 6 | * * * 560 | 0 0 0 4 | 0 0 0 2 2
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o.3o.3o.3o.3o. .. & ♦ 7 | 21 0 | 35 0 | 35 0 0 | 21 0 0 | 7 0 0 0 | 16 * * * | 1 1 0 0 0
xo3oo3oo3oo3oo .. ..&#x & ♦ 7 | 15 6 | 20 15 | 15 20 0 | 6 15 0 | 1 6 0 0 | * 112 * * | 0 1 1 0 0
xo3oo3oo3oo .. .. ox&#x & ♦ 7 | 11 10 | 10 25 | 5 20 10 | 1 10 10 | 0 2 5 0 | * * 336 * | 0 0 1 1 0
xo3oo3oo .. .. oo3ox&#x & ♦ 7 | 9 12 | 5 30 | 1 16 18 | 0 3 18 | 0 0 3 4 | * * * 560 | 0 0 0 1 1
--------------------------+----+-------+---------+-------------+--------------+----------------+----------------+---------------
x.3o.3o.3o.3o.3o.3o. & ♦ 8 | 28 0 | 56 0 | 70 0 0 | 56 0 0 | 28 0 0 0 | 8 0 0 0 | 2 * * * *
xo3oo3oo3oo3oo3oo ..&#x & ♦ 8 | 21 7 | 35 21 | 35 35 0 | 21 35 0 | 7 21 0 0 | 1 7 0 0 | * 16 * * *
xo3oo3oo3oo3oo .. ox&#x & ♦ 8 | 16 12 | 20 36 | 15 40 15 | 6 30 20 | 1 12 15 0 | 0 2 6 0 | * * 56 * *
xo3oo3oo3oo .. oo3ox&#x & ♦ 8 | 13 15 | 11 45 | 5 35 30 | 1 15 40 | 0 3 15 10 | 0 0 3 5 | * * * 112 *
xo3oo3oo .. oo3oo3ox&#x ♦ 8 | 12 16 | 8 48 | 2 32 36 | 0 8 48 | 0 0 12 16 | 0 0 0 8 | * * * * 70
oqo xoo3ooo3ooo3ooo3ooo3oox&#xt → both heights = 1/sqrt(14) = 0.267261
(hop || perp pseudo q-line || dual hop)
...