hext

Acronym hext (alt.: trat gytrat)

Name hexadecachoric tetracomb,
demitesseractic tetracomb,
Delone complex of body-centered tesseractic lattice,
Gosset polytope 1_1,1,1,
trat gyrotratism

Lace city
in approx. ASCII-art

C       C     :		where:
    B       B :		A = x3o3o3*a (trat)
A       A     :		B = o3x3o3*a (gyro trat)
    C       C :		C = o3o3x3*a (alt. gyro trat)
B       B     :
    A       A :
. . . . . . . .		(finite repetition unit only)

Vertex layers
(first ones only)

Layer	Symmetry	Subsymmetries
	`o3o3o4o3o`	`o3o3o4o .`	`. o3o4o3o`
1	`x3o3o4o3o`	`x3o3o4o .` hypercell first	`. o3o4o3o` vertex first
2		`o3o3x4o .`	`. x3o4o3o` vertex figure
3		`x3x3o4o .`	`. o3o4o3q`
4		...	`. o3x4o3o`
...		...	...

Coordinates

1. (i, j, k, l) i.e. all integer touples (inscribed test) and
2. (i+1/2, j+1/2, k+1/2, l+1/2) i.e. all half-integer touples (its body centers, a shifted test)
or just (i/sqrt(2), j/sqrt(2), k/sqrt(2), l/sqrt(2)) for integers i,j,k,l with i+j+k+l even (as being the hemiation of test)

Dual icot

Confer

related tesselations:: Delone complex of primitive tesseractic lattice Voronoi complex of primitive tesseractic lattice Voronoi complex of bct lattice
blend-component:: tratgyt
ambification:: icot
general polytopal classes:: noble polytopes partial Stott expansions Coxeter-Elte-Gosset polytopes

External
links

This euclidean honeycomb uses complete dissecting hyperplanes, which are infinite lace tower stacks of trataps. Thereby it can be dissectet into triangular needles, tratgyts, arranged along an orthogonal trat. This is how hext could be understood to be considered as "trat gytrat".

Incidence matrix according to Dynkin symbol

x3o3o4o3o   (N → ∞)

. . . . . | N ♦  24 |  96 |  96 | 24
----------+---+-----+-----+-----+---
x . . . . | 2 | 12N ♦   8 |  12 |  6
----------+---+-----+-----+-----+---
x3o . . . | 3 |   3 | 32N |   3 |  3
----------+---+-----+-----+-----+---
x3o3o . . ♦ 4 |   6 |   4 | 24N |  2
----------+---+-----+-----+-----+---
x3o3o4o . ♦ 8 |  24 |  32 |  16 | 3N

x3o3o *b3o4o   (N → ∞)

. . .    . . | N ♦  24 |  96 | 32  64 | 16 8
-------------+---+-----+-----+--------+-----
x . .    . . | 2 | 12N ♦   8 |  4   8 |  4 2
-------------+---+-----+-----+--------+-----
x3o .    . . | 3 |   3 | 32N |  1   2 |  2 1
-------------+---+-----+-----+--------+-----
x3o3o    . . ♦ 4 |   6 |   4 | 8N   * |  2 0
x3o . *b3o . ♦ 4 |   6 |   4 |  * 16N |  1 1
-------------+---+-----+-----+--------+-----
x3o3o *b3o . ♦ 8 |  24 |  32 |  8   8 | 2N *
x3o . *b3o4o ♦ 8 |  24 |  32 |  0  16 |  * N

x3o3o *b3o *b3o   (N → ∞)

. . .    .    . | N ♦  24 |  96 | 32 32 32 | 8 8 8
----------------+---+-----+-----+----------+------
x . .    .    . | 2 | 12N ♦   8 |  4  4  4 | 2 2 2
----------------+---+-----+-----+----------+------
x3o .    .    . | 3 |   3 | 32N |  1  1  1 | 1 1 1
----------------+---+-----+-----+----------+------
x3o3o    .    . ♦ 4 |   6 |   4 | 8N  *  * | 1 1 0
x3o . *b3o    . ♦ 4 |   6 |   4 |  * 8N  * | 1 0 1
x3o .    . *b3o ♦ 4 |   6 |   4 |  *  * 8N | 0 1 1
----------------+---+-----+-----+----------+------
x3o3o *b3o    . ♦ 8 |  24 |  32 |  8  8  0 | N * *
x3o3o    . *b3o ♦ 8 |  24 |  32 |  8  0  8 | * N *
x3o . *b3o *b3o ♦ 8 |  24 |  32 |  0  8  8 | * * N

s4o3o3o4o   (N → ∞)

demi( . . . . . ) | N ♦  24 |  96 | 32  64 | 16 8
------------------+---+-----+-----+--------+-----
      s4o . . .   | 2 | 12N ♦   8 |  4   8 |  4 2
------------------+---+-----+-----+--------+-----
sefa( s4o3o . . ) | 3 |   3 | 32N |  1   2 |  2 1
------------------+---+-----+-----+--------+-----
      s4o3o . .   ♦ 4 |   6 |   4 | 8N   * |  2 0
sefa( s4o3o3o . ) ♦ 4 |   6 |   4 |  * 16N |  1 1
------------------+---+-----+-----+--------+-----
      s4o3o3o .   ♦ 8 |  24 |  32 |  8   8 | 2N *
sefa( s4o3o3o4o ) ♦ 8 |  24 |  32 |  0  16 |  * N

starting figure: x4o3o3o4o

s4o3o3o4s   (N → ∞)

demi( . . . . . ) | 8N ♦   6  12   6 |  12  36  36  12 |  4  12  12  4  4  16  24  16  4 | 1  4  6  4 1  8
------------------+----+-------------+-----------------+---------------------------------+----------------
      s4o . . .   |  2 | 24N   *   * ♦   4   4   0   0 |  2   2   0  0  2   4   2   0  0 | 1  2  1  0 0  2
      s . 2 . s   |  2 |   * 48N   * ♦   0   4   4   0 |  0   2   2  0  0   2   4   2  0 | 0  1  2  1 0  2
      . . . o4s   |  2 |   *   * 24N ♦   0   0   4   4 |  0   0   2  2  0   0   2   4  2 | 0  0  1  2 1  2
------------------+----+-------------+-----------------+---------------------------------+----------------
sefa( s4o3o . . ) |  3 |   3   0   0 | 32N   *   *   * |  1   0   0  0  1   1   0   0  0 | 1  1  0  0 0  1
sefa( s4o . 2 s ) |  3 |   1   2   0 |   * 96N   *   * |  0   1   0  0  0   1   1   0  0 | 0  1  1  0 0  1
sefa( s 2 . o4s ) |  3 |   0   2   1 |   *   * 96N   * |  0   0   1  0  0   0   1   1  0 | 0  0  1  1 0  1
sefa( . . o3o4s ) |  3 |   0   0   3 |   *   *   * 32N |  0   0   0  1  0   0   0   1  1 | 0  0  0  1 1  1
------------------+----+-------------+-----------------+---------------------------------+----------------
      s4o3o . .   ♦  4 |   6   0   0 |   4   0   0   0 | 8N   *   *  *  *   *   *   *  * | 1  1  0  0 0  0
      s4o . 2 s   ♦  4 |   2   4   0 |   0   4   0   0 |  * 24N   *  *  *   *   *   *  * | 0  1  1  0 0  0
      s 2 . o4s   ♦  4 |   0   4   2 |   0   0   4   0 |  *   * 24N  *  *   *   *   *  * | 0  0  1  1 0  0
      . . o3o4s   ♦  4 |   0   0   6 |   0   0   0   4 |  *   *   * 8N  *   *   *   *  * | 0  0  0  1 1  0
sefa( s4o3o3o . ) ♦  4 |   6   0   0 |   4   0   0   0 |  *   *   *  * 8N   *   *   *  * | 1  0  0  0 0  1
sefa( s4o3o 2 s ) ♦  4 |   3   3   0 |   1   3   0   0 |  *   *   *  *  * 32N   *   *  * | 0  1  0  0 0  1
sefa( s4o 2 o4s ) ♦  4 |   1   4   1 |   0   2   2   0 |  *   *   *  *  *   * 48N   *  * | 0  0  1  0 0  1
sefa( s 2 o3o4s ) ♦  4 |   0   3   3 |   0   0   3   1 |  *   *   *  *  *   *   * 32N  * | 0  0  0  1 0  1
sefa( . o3o3o4s ) ♦  4 |   0   0   6 |   0   0   0   4 |  *   *   *  *  *   *   *   * 8N | 0  0  0  0 1  1
------------------+----+-------------+-----------------+---------------------------------+----------------
      s4o3o3o .   ♦  8 |  24   0   0 |  32   0   0   0 |  8   0   0  0  8   0   0   0  0 | N  *  *  * *  *
      s4o3o 2 s   ♦  8 |  12  12   0 |   8  24   0   0 |  2   6   0  0  0   8   0   0  0 | * 4N  *  * *  *
      s4o 2 o4s   ♦  8 |   4  16   4 |   0  16  16   0 |  0   4   4  0  0   0   8   0  0 | *  * 6N  * *  *
      s 2 o3o4s   ♦  8 |   0  12  12 |   0   0  24   8 |  0   0   6  2  0   0   0   8  0 | *  *  * 4N *  *
      . o3o3o4s   ♦  8 |   0   0  24 |   0   0   0  32 |  0   0   0  8  0   0   0   0  8 | *  *  *  * N  *
sefa( s4o3o3o4s ) ♦  8 |   6  12   6 |   4  12  12   4 |  0   0   0  0  1   4   6   4  1 | *  *  *  * * 8N

starting figure: x4o3o3o4x

:xoo:3:oxo:3:oox:3*a&#x   (N → ∞)   → pw. heights = sqrt(2/3) = 0.816497

 o.. 3 o.. 3 o.. 3*a     & | N ♦  6 18 |  6  54  36 | 24  72 | 24
---------------------------+---+-------+------------+--------+---
 x..   ...   ...         & | 2 | 3N  * ♦  2   6   0 |  6   6 |  6
 oo. 3 oo. 3 oo. 3*a     & | 2 |  * 9N ♦  0   4   4 |  2  10 |  6
---------------------------+---+-------+------------+--------+---
 x.. 3 o..   ...         & | 3 |  3  0 | 2N   *   * |  3   0 |  3
 xo.   ...   ...     &#x & | 3 |  1  2 |  * 18N   * |  1   2 |  3
 ooo 3 ooo 3 ooo 3*a &#x & | 3 |  0  3 |  *   * 12N |  0   3 |  3
---------------------------+---+-------+------------+--------+---
 xo.   ...   oo. 3*a &#x & ♦ 4 |  3  3 |  1   3   0 | 6N   * |  2
 xoo   ...   ...     &#x & ♦ 4 |  1  5 |  0   2   2 |  * 18N |  2
---------------------------+---+-------+------------+--------+---
:xoo:3:oxo:  ...     &#x & ♦ 8 |  6 18 |  2  18  12 |  4  12 | 3N

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