Acronym hodip
Name hexagon - octagon duoprism
 
Circumradius sqrt[2+1/sqrt(2)] = 1.645329
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Dihedral angles
  • at {6} between hip and hip:   135°
  • at {8} between op and op:   120°
  • at {4} between hip and op:   90°
Confer
general duoprisms:
n,m-dip   2n,m-dip   2n,2m-dip   6,n-dip   8,n-dip  
general polytopal classes:
bistratic lace towers  
External
links
hedrondude   wikipedia

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

© 2004-2019
top of page