Acronym | hodip |
Name | hexagon - octagon duoprism |
Circumradius | sqrt[2+1/sqrt(2)] = 1.645329 |
Volume | 3 [1+sqrt(2)] sqrt(3) = 12.544622 |
General of army | (is itself convex) |
Colonel of regiment | (is itself locally convex) |
Dihedral angles | |
Face vector | 48, 96, 62, 14 |
Confer |
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External links |
As abstract polytope hodip is isomorph to histodip, thereby replacing octagons by octagrams, resp. op by stop.
Incidence matrix according to Dynkin symbol
x6o x8o . . . . | 48 | 2 2 | 1 4 1 | 2 2 --------+----+-------+--------+---- x . . . | 2 | 48 * | 1 2 0 | 2 1 . . x . | 2 | * 48 | 0 2 1 | 1 2 --------+----+-------+--------+---- x6o . . | 6 | 6 0 | 8 * * | 2 0 x . x . | 4 | 2 2 | * 48 * | 1 1 . . x8o | 8 | 0 8 | * * 6 | 0 2 --------+----+-------+--------+---- x6o x . ♦ 12 | 12 6 | 2 6 0 | 8 * x . x8o ♦ 16 | 8 16 | 0 8 2 | * 6
x6o x8/7o . . . . | 48 | 2 2 | 1 4 1 | 2 2 ----------+----+-------+--------+---- x . . . | 2 | 48 * | 1 2 0 | 2 1 . . x . | 2 | * 48 | 0 2 1 | 1 2 ----------+----+-------+--------+---- x6o . . | 6 | 6 0 | 8 * * | 2 0 x . x . | 4 | 2 2 | * 48 * | 1 1 . . x8/7o | 8 | 0 8 | * * 6 | 0 2 ----------+----+-------+--------+---- x6o x . ♦ 12 | 12 6 | 2 6 0 | 8 * x . x8/7o ♦ 16 | 8 16 | 0 8 2 | * 6
x6/5o x8o . . . . | 48 | 2 2 | 1 4 1 | 2 2 ----------+----+-------+--------+---- x . . . | 2 | 48 * | 1 2 0 | 2 1 . . x . | 2 | * 48 | 0 2 1 | 1 2 ----------+----+-------+--------+---- x6/5o . . | 6 | 6 0 | 8 * * | 2 0 x . x . | 4 | 2 2 | * 48 * | 1 1 . . x8o | 8 | 0 8 | * * 6 | 0 2 ----------+----+-------+--------+---- x6/5o x . ♦ 12 | 12 6 | 2 6 0 | 8 * x . x8o ♦ 16 | 8 16 | 0 8 2 | * 6
x6/5o x8/7o . . . . | 48 | 2 2 | 1 4 1 | 2 2 ------------+----+-------+--------+---- x . . . | 2 | 48 * | 1 2 0 | 2 1 . . x . | 2 | * 48 | 0 2 1 | 1 2 ------------+----+-------+--------+---- x6/5o . . | 6 | 6 0 | 8 * * | 2 0 x . x . | 4 | 2 2 | * 48 * | 1 1 . . x8/7o | 8 | 0 8 | * * 6 | 0 2 ------------+----+-------+--------+---- x6/5o x . ♦ 12 | 12 6 | 2 6 0 | 8 * x . x8/7o ♦ 16 | 8 16 | 0 8 2 | * 6
x3x x8o . . . . | 48 | 1 1 2 | 1 2 2 1 | 2 1 1 --------+----+----------+-----------+------ x . . . | 2 | 24 * * | 1 2 0 0 | 2 1 0 . x . . | 2 | * 24 * | 1 0 2 0 | 2 0 1 . . x . | 2 | * * 48 | 0 1 1 1 | 1 1 1 --------+----+----------+-----------+------ x3x . . | 6 | 3 3 0 | 8 * * * | 2 0 0 x . x . | 4 | 2 0 2 | * 24 * * | 1 1 0 . x x . | 4 | 0 2 2 | * * 24 * | 1 0 1 . . x8o | 8 | 0 0 8 | * * * 6 | 0 1 1 --------+----+----------+-----------+------ x3x x . ♦ 12 | 6 6 6 | 2 3 3 0 | 8 * * x . x8o ♦ 16 | 8 0 16 | 0 8 0 2 | * 3 * . x x8o ♦ 16 | 0 8 16 | 0 0 8 2 | * * 3
x3x x8/7o . . . . | 48 | 1 1 2 | 1 2 2 1 | 2 1 1 ----------+----+----------+-----------+------ x . . . | 2 | 24 * * | 1 2 0 0 | 2 1 0 . x . . | 2 | * 24 * | 1 0 2 0 | 2 0 1 . . x . | 2 | * * 48 | 0 1 1 1 | 1 1 1 ----------+----+----------+-----------+------ x3x . . | 6 | 3 3 0 | 8 * * * | 2 0 0 x . x . | 4 | 2 0 2 | * 24 * * | 1 1 0 . x x . | 4 | 0 2 2 | * * 24 * | 1 0 1 . . x8/7o | 8 | 0 0 8 | * * * 6 | 0 1 1 ----------+----+----------+-----------+------ x3x x . ♦ 12 | 6 6 6 | 2 3 3 0 | 8 * * x . x8/7o ♦ 16 | 8 0 16 | 0 8 0 2 | * 3 * . x x8/7o ♦ 16 | 0 8 16 | 0 0 8 2 | * * 3
x4x x6o . . . . | 48 | 1 1 2 | 1 2 2 1 | 2 1 1 --------+----+----------+-----------+------ x . . . | 2 | 24 * * | 1 2 0 0 | 2 1 0 . x . . | 2 | * 24 * | 1 0 2 0 | 2 0 1 . . x . | 2 | * * 48 | 0 1 1 1 | 1 1 1 --------+----+----------+-----------+------ x4x . . | 8 | 4 4 0 | 6 * * * | 2 0 0 x . x . | 4 | 2 0 2 | * 24 * * | 1 1 0 . x x . | 4 | 0 2 2 | * * 24 * | 1 0 1 . . x6o | 6 | 0 0 6 | * * * 8 | 0 1 1 --------+----+----------+-----------+------ x4x x . ♦ 16 | 8 8 8 | 2 4 4 0 | 6 * * x . x6o ♦ 12 | 6 0 12 | 0 6 0 2 | * 4 * . x x6o ♦ 12 | 0 6 12 | 0 0 6 2 | * * 4
x4x x6/5o . . . . | 48 | 1 1 2 | 1 2 2 1 | 2 1 1 ----------+----+----------+-----------+------ x . . . | 2 | 24 * * | 1 2 0 0 | 2 1 0 . x . . | 2 | * 24 * | 1 0 2 0 | 2 0 1 . . x . | 2 | * * 48 | 0 1 1 1 | 1 1 1 ----------+----+----------+-----------+------ x4x . . | 8 | 4 4 0 | 6 * * * | 2 0 0 x . x . | 4 | 2 0 2 | * 24 * * | 1 1 0 . x x . | 4 | 0 2 2 | * * 24 * | 1 0 1 . . x6/5o | 6 | 0 0 6 | * * * 8 | 0 1 1 ----------+----+----------+-----------+------ x4x x . ♦ 16 | 8 8 8 | 2 4 4 0 | 6 * * x . x6/5o ♦ 12 | 6 0 12 | 0 6 0 2 | * 4 * . x x6/5o ♦ 12 | 0 6 12 | 0 0 6 2 | * * 4
x3x x4x . . . . | 48 | 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 --------+----+-------------+-----------------+-------- x . . . | 2 | 24 * * * | 1 1 1 0 0 0 | 1 1 1 0 . x . . | 2 | * 24 * * | 1 0 0 1 1 0 | 1 1 0 1 . . x . | 2 | * * 24 * | 0 1 0 1 0 1 | 1 0 1 1 . . . x | 2 | * * * 24 | 0 0 1 0 1 1 | 0 1 1 1 --------+----+-------------+-----------------+-------- x3x . . | 6 | 3 3 0 0 | 8 * * * * * | 1 1 0 0 x . x . | 4 | 2 0 2 0 | * 12 * * * * | 1 0 1 0 x . . x | 4 | 2 0 0 2 | * * 12 * * * | 0 1 1 0 . x x . | 4 | 0 2 2 0 | * * * 12 * * | 1 0 0 1 . x . x | 4 | 0 2 0 2 | * * * * 12 * | 0 1 0 1 . . x4x | 8 | 0 0 4 4 | * * * * * 6 | 0 0 1 1 --------+----+-------------+-----------------+-------- x3x x . ♦ 12 | 6 6 6 0 | 2 3 0 3 0 0 | 4 * * * x3x . x ♦ 12 | 6 6 0 6 | 2 0 3 0 3 0 | * 4 * * x . x4x ♦ 16 | 8 0 8 8 | 0 4 4 0 0 2 | * * 3 * . x x4x ♦ 16 | 0 8 8 8 | 0 0 0 4 4 2 | * * * 3
xux xxx8ooo&#xt → both heights = sqrt(3)/2 = 0.866025 (op || pseudo (x,u)-op || op) o.. o..8o.. | 16 * * | 1 2 1 0 0 0 0 | 2 1 2 1 0 0 0 0 | 1 1 2 0 0 .o. .o.8.o. | * 16 * | 0 0 1 2 1 0 0 | 0 0 2 1 1 2 0 0 | 0 1 2 1 0 ..o ..o8..o | * * 16 | 0 0 0 0 1 1 2 | 0 0 0 1 0 2 2 1 | 0 0 2 1 1 ----------------+----------+--------------------+-------------------+---------- x.. ... ... | 2 0 0 | 8 * * * * * * | 2 0 0 1 0 0 0 0 | 1 0 2 0 0 ... x.. ... | 2 0 0 | * 16 * * * * * | 1 1 1 0 0 0 0 0 | 1 1 1 0 0 oo. oo.8oo.&#x | 1 1 0 | * * 16 * * * * | 0 0 2 1 0 0 0 0 | 0 1 2 0 0 ... .x. ... | 0 2 0 | * * * 16 * * * | 0 0 1 0 1 1 0 0 | 0 1 1 1 0 .oo .oo8.oo&#x | 0 1 1 | * * * * 16 * * | 0 0 0 1 0 2 0 0 | 0 0 2 1 0 ..x ... ... | 0 0 2 | * * * * * 8 * | 0 0 0 1 0 0 2 0 | 0 0 2 0 1 ... ..x ... | 0 0 2 | * * * * * * 16 | 0 0 0 0 0 1 1 1 | 0 0 1 1 1 ----------------+----------+--------------------+-------------------+---------- x.. x.. ... | 4 0 0 | 2 2 0 0 0 0 0 | 8 * * * * * * * | 1 0 1 0 0 ... x..8o.. | 8 0 0 | 0 8 0 0 0 0 0 | * 2 * * * * * * | 1 1 0 0 0 ... xx. ...&#x | 2 2 0 | 0 1 2 1 0 0 0 | * * 16 * * * * * | 0 1 1 0 0 xux ... ...&#xt | 2 2 2 | 1 0 2 0 2 1 0 | * * * 8 * * * * | 0 0 2 0 0 ... .x.8.o. | 0 8 0 | 0 0 0 8 0 0 0 | * * * * 2 * * * | 0 1 0 1 0 ... .xx ...&#x | 0 2 2 | 0 0 0 1 2 0 1 | * * * * * 16 * * | 0 0 1 1 0 ..x ..x ... | 0 0 4 | 0 0 0 0 0 2 2 | * * * * * * 8 * | 0 0 1 0 1 ... ..x8..o | 0 0 8 | 0 0 0 0 0 0 8 | * * * * * * * 2 | 0 0 0 1 1 ----------------+----------+--------------------+-------------------+---------- x.. x..8o.. ♦ 16 0 0 | 8 16 0 0 0 0 0 | 8 2 0 0 0 0 0 0 | 1 * * * * ... xx.8oo.&#x ♦ 8 8 0 | 0 8 8 8 0 0 0 | 0 1 8 0 1 0 0 0 | * 2 * * * xux xxx ...&#xt ♦ 4 4 4 | 2 2 4 2 4 2 2 | 1 0 2 2 0 2 1 0 | * * 8 * * ... .xx8.oo&#x ♦ 0 8 8 | 0 0 0 8 8 0 8 | 0 0 0 0 1 8 0 1 | * * * 2 * ..x ..x8..o ♦ 0 0 16 | 0 0 0 0 0 8 16 | 0 0 0 0 0 0 8 2 | * * * * 1
or o.. o..8o.. & | 32 * | 1 2 1 0 | 2 1 2 1 0 | 1 1 2 .o. .o.8.o. | * 16 | 0 0 2 2 | 0 0 4 1 1 | 0 2 2 -------------------+-------+-------------+-------------+------ x.. ... ... & | 2 0 | 16 * * * | 2 0 0 1 0 | 1 0 2 ... x.. ... & | 2 0 | * 32 * * | 1 1 1 0 0 | 1 1 1 oo. oo.8oo.&#x & | 1 1 | * * 32 * | 0 0 2 1 0 | 0 1 2 ... .x. ... | 0 2 | * * * 16 | 0 0 2 0 1 | 0 2 1 -------------------+-------+-------------+-------------+------ x.. x.. ... & | 4 0 | 2 2 0 0 | 16 * * * * | 1 0 1 ... x..8o.. & | 8 0 | 0 8 0 0 | * 4 * * * | 1 1 0 ... xx. ...&#x & | 2 2 | 0 1 2 1 | * * 32 * * | 0 1 1 xux ... ...&#xt | 4 2 | 2 0 4 0 | * * * n * | 0 0 2 ... .x.8.o. | 0 8 | 0 0 0 8 | * * * * 2 | 0 2 0 -------------------+-------+-------------+-------------+------ x.. x..8o.. & ♦ 16 0 | 8 16 0 0 | 8 2 0 0 0 | 2 * * ... xx.8oo.&#x & ♦ 8 8 | 0 8 8 8 | 0 1 8 0 1 | * 4 * xux xxx ...&#xt ♦ 8 4 | 4 4 8 2 | 2 0 4 2 0 | * * 8
xux xxx4xxx&#xt → both heights = sqrt(3)/2 = 0.866025 (op || pseudo (x,u)-op || op) o.. o..4o.. | 16 * * | 1 1 1 1 0 0 0 0 0 0 | 1 1 1 1 1 1 0 0 0 0 0 0 | 1 1 1 1 0 0 .o. .o.4.o. | * 16 * | 0 0 0 1 1 1 1 0 0 0 | 0 0 0 1 1 1 1 1 1 0 0 0 | 0 1 1 1 1 0 ..o ..o4..o | * * 16 | 0 0 0 0 0 0 1 1 1 1 | 0 0 0 1 0 0 0 1 1 1 1 1 | 0 1 1 0 1 1 ----------------+----------+-----------------------+-------------------------+------------ x.. ... ... | 2 0 0 | 8 * * * * * * * * * | 1 1 0 1 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0 ... x.. ... | 2 0 0 | * 8 * * * * * * * * | 1 0 1 0 1 0 0 0 0 0 0 0 | 1 1 0 1 0 0 ... ... x.. | 2 0 0 | * * 8 * * * * * * * | 0 1 1 0 0 1 0 0 0 0 0 0 | 1 0 1 1 0 0 oo. oo. oo.&#x | 1 1 0 | * * * 16 * * * * * * | 0 0 0 1 1 1 0 0 0 0 0 0 | 0 1 1 1 0 0 ... .x. ... | 0 2 0 | * * * * 8 * * * * * | 0 0 0 0 1 0 1 1 0 0 0 0 | 0 1 0 1 1 0 ... ... .x. | 0 2 0 | * * * * * 8 * * * * | 0 0 0 0 0 1 1 0 1 0 0 0 | 0 0 1 1 1 0 .oo .oo .oo&#x | 0 1 1 | * * * * * * 16 * * * | 0 0 0 1 0 0 0 1 1 0 0 0 | 0 1 1 0 1 0 ..x ... ... | 0 0 2 | * * * * * * * 8 * * | 0 0 0 1 0 0 0 0 0 1 1 0 | 0 1 1 0 0 1 ... ..x ... | 0 0 2 | * * * * * * * * 8 * | 0 0 0 0 0 0 0 1 0 1 0 1 | 0 1 0 0 1 1 ... ... ..x | 0 0 2 | * * * * * * * * * 8 | 0 0 0 0 0 0 0 0 1 0 1 1 | 0 0 1 0 1 1 ----------------+----------+-----------------------+-------------------------+------------ x.. x.. ... | 4 0 0 | 2 2 0 0 0 0 0 0 0 0 | 4 * * * * * * * * * * * | 1 1 0 0 0 0 x.. ... x.. | 4 0 0 | 2 0 2 0 0 0 0 0 0 0 | * 4 * * * * * * * * * * | 1 0 1 0 0 0 ... x..4x.. | 8 0 0 | 0 4 4 0 0 0 0 0 0 0 | * * 2 * * * * * * * * * | 1 0 0 1 0 0 xux ... ...&#xt | 2 2 2 | 1 0 0 2 0 0 2 1 0 0 | * * * 8 * * * * * * * * | 0 1 1 0 0 0 ... xx. ...&#x | 2 2 0 | 0 1 0 2 1 0 0 0 0 0 | * * * * 8 * * * * * * * | 0 1 0 1 0 0 ... ... xx.&#x | 2 2 0 | 0 0 1 2 0 1 0 0 0 0 | * * * * * 8 * * * * * * | 0 0 1 1 0 0 ... .x.4.x. | 0 8 0 | 0 0 0 0 4 4 0 0 0 0 | * * * * * * 2 * * * * * | 0 0 0 1 1 0 ... .xx ...&#x | 0 2 2 | 0 0 0 0 1 0 2 0 1 0 | * * * * * * * 8 * * * * | 0 1 0 0 1 0 ... ... .xx&#x | 0 2 2 | 0 0 0 0 0 1 2 0 0 1 | * * * * * * * * 8 * * * | 0 0 1 0 1 0 ..x ..x ... | 0 0 4 | 0 0 0 0 0 0 0 2 2 0 | * * * * * * * * * 4 * * | 0 1 0 0 0 1 ..x ... ..x | 0 0 4 | 0 0 0 0 0 0 0 2 0 2 | * * * * * * * * * * 4 * | 0 0 1 0 0 1 ... ..x4..x | 0 0 8 | 0 0 0 0 0 0 0 0 4 4 | * * * * * * * * * * * 2 | 0 0 0 0 1 1 ----------------+----------+-----------------------+-------------------------+------------ x.. x..4x.. ♦ 16 0 0 | 8 8 8 0 0 0 0 0 0 0 | 4 4 2 0 0 0 0 0 0 0 0 0 | 1 * * * * * xux xxx ...&#xt ♦ 4 4 4 | 2 2 0 4 2 0 4 2 2 0 | 1 0 0 2 2 0 0 2 0 1 0 0 | * 4 * * * * xux ... xxx&#xt ♦ 4 4 4 | 2 0 2 4 0 2 4 2 0 2 | 0 1 0 2 0 2 0 0 2 0 1 0 | * * 4 * * * ... xx.4xx.&#x ♦ 8 8 0 | 0 4 4 8 4 4 0 0 0 0 | 0 0 1 0 4 4 1 0 0 0 0 0 | * * * 2 * * ... .xx4.xx&#x ♦ 0 8 8 | 0 0 0 0 4 4 8 0 4 4 | 0 0 0 0 0 0 1 4 4 0 0 1 | * * * * 2 * ..x ..x4..x ♦ 0 0 16 | 0 0 0 0 0 0 0 8 8 8 | 0 0 0 0 0 0 0 0 0 4 4 2 | * * * * * 1
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