Acronym	deca
Name	decachoron, bitruncated pentachoron, equatorial cross-section of (vertex first) q-pent, equatorial cross-section of srip-first sibrid, equatorial cross-section of pen-first rin
	©   ©
Cross sections	©
Circumradius	sqrt(2) = 1.414214
Inradius	sqrt(5/8) = 0.790569
Vertex figure	©
Vertex layers	Layer Symmetry Subsymmetries   o3o3o3o o3o3o . o3o . o o . o3o . o3o3o 1 o3x3x3o o3x3x . tut first o3x . o {3} first o . x3o {3} first . x3x3o tut first 2 o3u3o . o3u . x x . u3o . o3u3o 3 x3x3o . opposite tut x3x . u u . x3x . o3x3x opposite tut 4   u3o . x x . o3u   5 x3o . o opposite {3} o . o3x opposite {3}
Lace city in approx. ASCII-art	o3x o3u x3x o3u u3o x3x u3o x3o
	x o u x x u o x o o u u o o o x x u u x x o
Volume	115 sqrt(5)/48 = 5.357246
Surface	115 sqrt(2)/6 = 27.105760
General of army	(is itself convex)
Colonel of regiment	(is itself locally convex – uniform polychoral members: by cells: tut deca 10 )
Dihedral angles	at {6} between tut and tut:   arccos(-1/4) = 104.477512° at {3} between tut and tut:   arccos(1/4) = 75.522488°
Dual	o3m3m3o
Face vector	30, 60, 40, 10
Confer	Grünbaumian relatives: 2deca   compounds: teppix   related CRFs: baudeca   ambification: redeca   general polytopal classes: Wythoffian polychora   noble polytopes   bistratic lace towers   lace simplices   analogs: bitruncated simplex btSn   mid-truncated simplex mtSn
External links

Note that deca can be thought of as the external blend of 1 spid + 10 tetaltut^† + 20 trafs. (^†: itself a degenerate polytope.) This decomposition is described as the degenerate segmentoteron xo3ox3ox3xo&#x.

As can be read from the matrices below, at every edge there are 2 hexagons. Thus we get as pseudo cells something with hexagons only. From the vertex incidence we further read off that this pseudo tiling happens to use 4 hexagons per vertex. From the here truely being used cells (tut) it is deduced, that any straight edge sequence of that seeming x6o4o needs to be mod-wrapped to triangular holes. Therefore those pseudo cells rather are the skew polyhedron x6o4o|3 instead, which here happens to be finite of course (just 20 remaining such hexagons).

By virtue of an outer symmetry this is a non-quasiregular monotoxal polychoron, that is all edges belong to the same equivalence class.

Incidence matrix according to Dynkin symbol

o3x3x3o

. . . . | 30 ♦  2  2 |  1  4  1 | 2 2
--------+----+-------+----------+----
. x . . |  2 | 30  * |  1  2  0 | 2 1
. . x . |  2 |  * 30 |  0  2  1 | 1 2
--------+----+-------+----------+----
o3x . . |  3 |  3  0 | 10  *  * | 2 0
. x3x . |  6 |  3  3 |  * 20  * | 1 1
. . x3o |  3 |  0  3 |  *  * 10 | 0 2
--------+----+-------+----------+----
o3x3x . ♦ 12 | 12  6 |  4  4  0 | 5 *
. x3x3o ♦ 12 |  6 12 |  0  4  4 | * 5

or
. . . .    | 30 ♦  4 |  2  4 |  4
-----------+----+----+-------+---
. x . .  & |  2 | 60 |  1  2 |  3
-----------+----+----+-------+---
o3x . .  & |  3 |  3 | 20  * |  2
. x3x .    |  6 |  6 |  * 20 |  2
-----------+----+----+-------+---
o3x3x .  & ♦ 12 | 18 |  4  4 | 10

snubbed forms: o3β3x3o, o3β3β3o

o3x3x3/2o

. . .   . | 30 ♦  2  2 |  1  4  1 | 2 2
----------+----+-------+----------+----
. x .   . |  2 | 30  * |  1  2  0 | 2 1
. . x   . |  2 |  * 30 |  0  2  1 | 1 2
----------+----+-------+----------+----
o3x .   . |  3 |  3  0 | 10  *  * | 2 0
. x3x   . |  6 |  3  3 |  * 20  * | 1 1
. . x3/2o |  3 |  0  3 |  *  * 10 | 0 2
----------+----+-------+----------+----
o3x3x   . ♦ 12 | 12  6 |  4  4  0 | 5 *
. x3x3/2o ♦ 12 |  6 12 |  0  4  4 | * 5

o3/2x3x3/2o

.   . .   . | 30 ♦  2  2 |  1  4  1 | 2 2
------------+----+-------+----------+----
.   x .   . |  2 | 30  * |  1  2  0 | 2 1
.   . x   . |  2 |  * 30 |  0  2  1 | 1 2
------------+----+-------+----------+----
o3/2x .   . |  3 |  3  0 | 10  *  * | 2 0
.   x3x   . |  6 |  3  3 |  * 20  * | 1 1
.   . x3/2o |  3 |  0  3 |  *  * 10 | 0 2
------------+----+-------+----------+----
o3/2x3x   . ♦ 12 | 12  6 |  4  4  0 | 5 *
.   x3x3/2o ♦ 12 |  6 12 |  0  4  4 | * 5

or
.   . .   .    | 30 ♦  4 |  2  4 |  4
---------------+----+----+-------+---
.   x .   .  & |  2 | 60 |  1  2 |  3
---------------+----+----+-------+---
o3/2x .   .  & |  3 |  3 | 20  * |  2
.   x3x   .    |  6 |  6 |  * 20 |  2
---------------+----+----+-------+---
o3/2x3x   .  & ♦ 12 | 18 |  4  4 | 10

oox3xux3xoo&#xt   → both heights = sqrt(5/8) = 0.790569
(tut || pseudo u-oct || inv tut)

o..3o..3o..     | 12 *  * ♦  2 1  1  0 0  0 | 1 2 1  2 0 0 0 | 1 1 2 0
.o.3.o.3.o.     |  * 6  * ♦  0 0  2  2 0  0 | 0 0 1  4 1 0 0 | 0 2 2 0
..o3..o3..o     |  * * 12 ♦  0 0  0  1 1  2 | 0 0 0  2 1 2 1 | 0 2 1 1
----------------+---------+-----------------+----------------+--------
... x.. ...     |  2 0  0 | 12 *  *  * *  * | 1 1 0  1 0 0 0 | 1 1 1 0
... ... x..     |  2 0  0 |  * 6  *  * *  * | 0 2 1  0 0 0 0 | 1 0 2 0
oo.3oo.3oo.&#x  |  1 1  0 |  * * 12  * *  * | 0 0 1  2 0 0 0 | 0 1 2 0
.oo3.oo3.oo&#x  |  0 1  1 |  * *  * 12 *  * | 0 0 0  2 1 0 0 | 0 2 1 0
..x ... ...     |  0 0  2 |  * *  *  * 6  * | 0 0 0  0 1 2 0 | 0 2 0 1
... ..x ...     |  0 0  2 |  * *  *  * * 12 | 0 0 0  1 0 1 1 | 0 1 1 1
----------------+---------+-----------------+----------------+--------
o..3x.. ...     |  3 0  0 |  3 0  0  0 0  0 | 4 * *  * * * * | 1 1 0 0
... x..3x..     |  6 0  0 |  3 3  0  0 0  0 | * 4 *  * * * * | 1 0 1 0
... ... xo.&#x  |  2 1  0 |  0 1  2  0 0  0 | * * 6  * * * * | 0 0 2 0
... xux ...&#xt |  2 2  2 |  1 0  2  2 0  1 | * * * 12 * * * | 0 1 1 0
.ox ... ...&#x  |  0 1  2 |  0 0  0  2 1  0 | * * *  * 6 * * | 0 2 0 0
..x3..x ...     |  0 0  6 |  0 0  0  0 3  3 | * * *  * * 4 * | 0 1 0 1
... ..x3..o     |  0 0  3 |  0 0  0  0 0  3 | * * *  * * * 4 | 0 0 1 1
----------------+---------+-----------------+----------------+--------
o..3x..3x..     ♦ 12 0  0 | 12 6  0  0 0  0 | 4 4 0  0 0 0 0 | 1 * * *
oox3xux ...&#xt ♦  3 3  6 |  3 0  3  6 3  3 | 1 0 0  3 3 1 0 | * 4 * *
... xux3xoo&#xt ♦  6 3  3 |  3 3  6  3 0  3 | 0 1 3  3 0 0 1 | * * 4 *
..x3..x3..o     ♦  0 0 12 |  0 0  0  0 6 12 | 0 0 0  0 0 4 4 | * * * 1

or
o..3o..3o..      & | 24 * ♦  2  1  1 | 1 2  1  2 | 1 3
.o.3.o.3.o.        |  * 6 ♦  0  0  4 | 0 0  2  4 | 0 4
-------------------+------+----------+-----------+----
... x.. ...      & |  2 0 | 24  *  * | 1 1  0  1 | 1 2
... ... x..      & |  2 0 |  * 12  * | 0 2  1  0 | 1 2
oo.3oo.3oo.&#x   & |  1 1 |  *  * 24 | 0 0  1  2 | 0 3
-------------------+------+----------+-----------+----
o..3x.. ...      & |  3 0 |  3  0  0 | 8 *  *  * | 1 1
... x..3x..      & |  6 0 |  3  3  0 | * 8  *  * | 1 1
... ... xo.&#x   & |  2 1 |  0  1  2 | * * 12  * | 0 2
... xux ...&#xt    |  4 2 |  2  0  4 | * *  * 12 | 0 2
-------------------+------+----------+-----------+----
o..3x..3x..      & ♦ 12 0 | 12  6  0 | 4 4  0  0 | 2 *
oox3xux ...&#xt  & ♦  9 3 |  6  3  9 | 1 1  3  3 | * 8

oxuxo xuxoo3ooxux&#xt   → all heights = sqrt(5/12) = 0.645497
({3} || pseudo (u,1)-trip || pseudo (1,u)-hip || pseudo inv (u,1)-trip || dual {3})

o.... o....3o....     | 3 *  * * * ♦ 2 2 0  0 0 0  0 0 0 0 | 1 1 4 0 0 0 0 0 0 0 | 2 2 0 0
.o... .o...3.o...     | * 6  * * * ♦ 0 1 1  2 0 0  0 0 0 0 | 0 1 2 1 2 0 0 0 0 0 | 1 2 1 0
..o.. ..o..3..o..     | * * 12 * * ♦ 0 0 0  1 1 1  1 0 0 0 | 0 0 1 1 1 1 1 1 0 0 | 1 1 1 1
...o. ...o.3...o.     | * *  * 6 * ♦ 0 0 0  0 0 0  2 1 1 0 | 0 0 0 0 2 0 1 2 1 0 | 0 1 2 1
....o ....o3....o     | * *  * * 3 ♦ 0 0 0  0 0 0  0 0 2 2 | 0 0 0 0 0 0 0 4 1 1 | 0 0 2 2
----------------------+------------+-----------------------+---------------------+--------
..... x.... .....     | 2 0  0 0 0 | 3 * *  * * *  * * * * | 1 0 2 0 0 0 0 0 0 0 | 2 1 0 0
oo... oo...3oo...&#x  | 1 1  0 0 0 | * 6 *  * * *  * * * * | 0 1 2 0 0 0 0 0 0 0 | 1 2 0 0
.x... ..... .....     | 0 2  0 0 0 | * * 3  * * *  * * * * | 0 1 0 0 2 0 0 0 0 0 | 0 2 1 0
.oo.. .oo..3.oo..&#x  | 0 1  1 0 0 | * * * 12 * *  * * * * | 0 0 1 1 1 0 0 0 0 0 | 1 1 1 0
..... ..x.. .....     | 0 0  2 0 0 | * * *  * 6 *  * * * * | 0 0 1 0 0 1 1 0 0 0 | 1 1 0 1
..... ..... ..x..     | 0 0  2 0 0 | * * *  * * 6  * * * * | 0 0 0 1 0 1 0 1 0 0 | 1 0 1 1
..oo. ..oo.3..oo.&#x  | 0 0  1 1 0 | * * *  * * * 12 * * * | 0 0 0 0 1 0 1 1 0 0 | 0 1 1 1
...x. ..... .....     | 0 0  0 2 0 | * * *  * * *  * 3 * * | 0 0 0 0 2 0 0 0 1 0 | 0 1 2 0
...oo ...oo3...oo&#x  | 0 0  0 1 1 | * * *  * * *  * * 6 * | 0 0 0 0 0 0 0 2 1 0 | 0 0 2 1
..... ..... ....x     | 0 0  0 0 2 | * * *  * * *  * * * 3 | 0 0 0 0 0 0 0 2 0 1 | 0 0 1 2
----------------------+------------+-----------------------+---------------------+--------
..... x....3o....     | 3 0  0 0 0 | 3 0 0  0 0 0  0 0 0 0 | 1 * * * * * * * * * | 2 0 0 0
ox... ..... .....&#x  | 1 2  0 0 0 | 0 2 1  0 0 0  0 0 0 0 | * 3 * * * * * * * * | 0 2 0 0
..... xux.. .....&#xt | 2 2  2 0 0 | 1 2 0  2 1 0  0 0 0 0 | * * 6 * * * * * * * | 1 1 0 0
..... ..... .ox..&#x  | 0 1  2 0 0 | 0 0 0  2 0 1  0 0 0 0 | * * * 6 * * * * * * | 1 0 1 0
.xux. ..... .....&#xt | 0 2  2 2 0 | 0 0 1  2 0 0  2 1 0 0 | * * * * 6 * * * * * | 0 1 1 0
..... ..x..3..x..     | 0 0  6 0 0 | 0 0 0  0 3 3  0 0 0 0 | * * * * * 2 * * * * | 1 0 0 1
..... ..xo. .....&#x  | 0 0  2 1 0 | 0 0 0  0 1 0  2 0 0 0 | * * * * * * 6 * * * | 0 1 0 1
..... ..... ..xux&#xt | 0 0  2 2 2 | 0 0 0  0 0 1  2 0 2 1 | * * * * * * * 6 * * | 0 0 1 1
...xo ..... .....&#x  | 0 0  0 2 1 | 0 0 0  0 0 0  0 1 2 0 | * * * * * * * * 3 * | 0 0 2 0
..... ....o3....x     | 0 0  0 0 3 | 0 0 0  0 0 0  0 0 0 3 | * * * * * * * * * 1 | 0 0 0 2
----------------------+------------+-----------------------+---------------------+--------
..... xux..3oox..&#xt ♦ 3 3  6 0 0 | 3 3 0  6 3 3  0 0 0 0 | 1 0 3 3 0 1 0 0 0 0 | 2 * * *
oxux. xuxo. .....&#xt ♦ 2 4  4 2 0 | 1 4 2  4 2 0  4 1 0 0 | 0 2 2 0 2 0 2 0 0 0 | * 3 * *
.xuxo ..... .oxux&#xt ♦ 0 2  4 4 2 | 0 0 1  4 0 2  4 2 4 1 | 0 0 0 2 2 0 0 2 2 0 | * * 3 *
..... ..xoo3..xux&#xt ♦ 0 0  6 3 3 | 0 0 0  0 3 3  6 0 3 3 | 0 0 0 0 0 1 3 3 0 1 | * * * 2

or
o.... o....3o....      & | 6  *  * ♦ 2  2 0  0  0 | 1 1  4  0 0 0 | 2 2
.o... .o...3.o...      & | * 12  * ♦ 0  1 1  2  0 | 0 1  2  1 2 0 | 1 3
..o.. ..o..3..o..        | *  * 12 ♦ 0  0 0  2  2 | 0 0  2  2 1 1 | 2 2
-------------------------+---------+--------------+---------------+----
..... x.... .....      & | 2  0  0 | 6  * *  *  * | 1 0  2  0 0 0 | 2 1
oo... oo...3oo...&#x   & | 1  1  0 | * 12 *  *  * | 0 1  2  0 0 0 | 1 2
.x... ..... .....      & | 0  2  0 | *  * 6  *  * | 0 1  0  0 2 0 | 0 3
.oo.. .oo..3.oo..&#x   & | 0  1  1 | *  * * 24  * | 0 0  1  1 1 0 | 1 2
..... ..x.. .....      & | 0  0  2 | *  * *  * 12 | 0 0  1  1 0 1 | 2 1
-------------------------+---------+--------------+---------------+----
..... x....3o....      & | 3  0  0 | 3  0 0  0  0 | 2 *  *  * * * | 2 0
ox... ..... .....&#x   & | 1  2  0 | 0  2 1  0  0 | * 6  *  * * * | 0 2
..... xux.. .....&#xt  & | 2  2  2 | 1  2 0  2  1 | * * 12  * * * | 1 1
..... ..... .ox..&#x   & | 0  1  2 | 0  0 0  2  1 | * *  * 12 * * | 1 1
.xux. ..... .....&#xt    | 0  4  2 | 0  0 2  4  0 | * *  *  * 6 * | 0 2
..... ..x..3..x..        | 0  0  6 | 0  0 0  0  6 | * *  *  * * 2 | 2 0
-------------------------+---------+--------------+---------------+----
..... xux..3oox..&#xt  & ♦ 3  3  6 | 3  3 0  6  6 | 1 0  3  3 0 1 | 4 *
oxux. xuxo. .....&#xt  & ♦ 2  6  4 | 1  4 3  8  2 | 0 2  2  2 2 0 | * 6