Elementary Honeycombs: Incidence Matrices
The following Incidence Matrices for the elementary honeycombs were kindly provided by Dr Richard Klitzing. Clicking on the links 'Description' leads to my images and VRML models.
10Y4-8T-n
Description
N | 8 4 | 24 4 | 8 10
--+-------+------+------
2 | 4N * | 4 0 | 2 2 lacings
2 | * 2N | 4 2 | 2 4 square grid
--+-------+------+------
3 | 2 1 | 8N * | 1 1
4 | 0 4 | * N | 0 2
--+-------+------+------
4 | 4 2 | 4 0 | 2N * tetrahedra
5 | 4 4 | 4 1 | * 2N 4-pyramids
5Y4-4T-4P4 Description
2N | 4 1 4 | 12 4 4 | 4
5 4
---+---------+----------+--------
2 | 4N * * | 4 0 0 | 2 2 0 obl.
lacings
2 | * N * | 0 4 0 | 0 0 4 prism
lacings
2 | * * 4N | 2 1 2 | 1 2 2 square grid
---+---------+----------+--------
3 | 2 0 1 | 8N * * | 1 1 0
4 | 0 2 2 | * 2N * | 0 0 2 prism
latterals
4 | 0 0 4 | * * 2N | 0 1 1 square grid
---+---------+----------+--------
4 | 4 0 2 | 4 0 0 | 2N * * tetrahedra
5 | 4 0 4 | 4 0 1 | * 2N * 4-pyramids
8 | 0 4 8 | 0 4 2 | * * N cubes
5Y4-4T-6P3-sq-ortho
Description
N | 4 2 2 2 | 6 6 3 4 4 | 4 5 6
--+----------+-------------+------
2 | 2N * * * | 2 2 0 0 0 | 2 2 0 pyr. lacings
2 | * N * * | 0 0 2 2 0 | 0 0 4 wedge lacings
2 | * * N * | 2 0 0 2 2 | 1 2 3 parallels
2 | * * * N | 0 2 1 0 2 | 1 2 2 orthogonals
--+----------+-------------+------
3 | 2 0 1 0 | 2N * * * * | 1 1 0 obl. triangles
3 | 2 0 0 1 | * 2N * * * | 1 1 0 obl. triangles
3 | 0 2 0 1 | * * N * * | 0 0 2 upr. triangles
4 | 0 2 2 0 | * * * N * | 0 0 2 wedge latterals
4 | 0 0 2 2 | * * * * N | 0 1 1 square grid
--+----------+-------------+------
4 | 4 0 1 1 | 2 2 0 0 0 | N * * tetrahedra
5 | 4 0 2 2 | 2 2 0 0 1 | * N * 4-pyramids
6 | 0 4 3 2 | 0 0 2 2 1 | * * N prisms
5Y4-4T-6P3-sq-gyro Description
2N | 4 2 2 2 | 6 6
3 4 4 | 2 2 5 6
---+-------------+----------------+----------
2 | 4N * * * | 2 2 0 0 0 |
1 1 2 0 pyr. lac.
2 | * 2N * * | 0 0 2 2 0 |
0 0 0 4 wedge lac.
2 | * * 2N * | 2 0 0 2 2 |
1 0 2 3 parallels
2 | * * * 2N | 0 2 1 0 2 |
0 1 2 2 orthogonals
---+-------------+----------------+----------
3 | 2 0 1 0 | 4N * * * * |
1 0 1 0 obl. tri.
3 | 2 0 0 1 | * 4N * * * |
0 1 1 0 obl. tri.
3 | 0 2 0 1 | * * 2N * * |
0 0 0 2 upr. tri.
4 | 0 2 2 0 | * * * 2N * |
0 0 0 2 wedge lac.
4 | 0 0 2 2 | * * * * 2N |
0 0 1 1 square grid
---+-------------+----------------+----------
4 | 4 0 2 0 | 4 0 0 0
0 | N * * * para. tet
4 | 4 0 0 2 | 0 4 0 0
0 | * N * * ortho. tet
5 | 4 0 2 2 | 2 2 0 0
1 | * * 2N * 4-pyramid
6 | 0 4 3 2 | 0 0 2 2
1 | * * * 2N prisms
10Y4-8T-ortho Description
N | 4 2 2 4 | 4 3 3 6 6 6 |
10 8
--+-----------+----------------+------
2 | 2N * * * | 1 1 1 1 1 0 | 3 2 plane,
square-inc.
2 | * N * * | 0 1 1 0 0 2 | 2 2 plane,
not sq.-inc.
2 | * * N * | 2 0 0 2 0 2 | 4 2 obl.,
square-inc.
2 | * * * 2N | 0 0 0 1 2 1 | 2 2 obl., not
sq.-inc.
--+-----------+----------------+------
4 | 2 0 2 0 | N * * * * * | 2 0
squares
3 | 2 1 0 0 | * N * * * * | 2 0 plane,
4pyr-4pyr
3 | 2 1 0 0 | * * N * * * | 0 2 plane,
tet-tet
3 | 1 0 1 1 | * * * 2N * * | 1 1 obl.
3 | 1 0 0 2 | * * * * 2N * | 1 1 obl.
3 | 0 1 1 1 | * * * * * 2N | 1 1 obl.
--+-----------+----------------+------
5 | 3 1 2 2 | 1 1 0 1 1 1 | 2N * 4pyramids
4 | 2 1 1 2 | 0 0 1 1 1 1 | * 2N tetrahedra
10Y4-8T-meta Description
N | 4 2 2 2 2 | 4 6 6 6 6 |
10 8
--+------------+---------------+------
2 | 2N * * * * | 1 2 1 0 1 | 3 2 plane,
square-inc.
2 | * N * * * | 0 2 0 2 0 | 2 2 plane,
not sq.-inc.
2 | * * N * * | 0 0 2 2 0 | 2 2 obl.,
not sq.-inc.
2 | * * * N * | 0 0 2 0 2 | 2 2 obl.,
not sq.-inc.
2 | * * * * N | 2 0 0 2 2 | 4 2 obl.,
square-inc.
--+------------+---------------+------
4 | 2 0 0 0 2 | N * * * * | 2 0
squares
3 | 2 1 0 0 0 | * 2N * * * | 1 1 plane
3 | 1 0 1 1 0 | * * 2N * * | 1 1 obl.
3 | 0 1 1 0 1 | * * * 2N * | 1 1 obl.
3 | 1 0 0 1 1 | * * * * 2N | 1 1 obl.
--+------------+---------------+------
5 | 3 1 1 1 2 | 1 1 1 1 1 | 2N * 4-pyramids
4 | 2 1 1 1 1 | 0 1 1 1 1 | * 2N tetrahedra
10Y4-8T-para Description
N | 2 4 2 4 | 4 3 3 6 12 | 10 8
--+-----------+-------------+------
2 | N * * * | 2 1 1 2 0 | 4 2 plane,
square-inc.
2 | * 2N * * | 0 1 1 0 2 | 2 2 plane, not sq.-inc.
2 | * * N * | 2 0 0 0 4 | 4 2 obl.,
square-inc.
2 | * * * 2N | 0 0 0 2 2 | 2 2 obl., not sq.-inc.
--+-----------+-------------+------
4 | 2 0 2 0 | N * * * * | 2 0
3 | 1 2 0 0 | * N * * * | 2 0 plane,
4pyr-4pyr
3 | 1 2 0 0 | * * N * * | 0 2 plane, tet-tet
3 | 1 0 0 2 | * * * 2N * | 1 1 based on sq.-edge
3 | 0 1 1 1 | * * * * 2N | 1 1 else
--+-----------+-------------+------
5 | 2 2 2 2 | 1 1 0 1 2 | 2N * 4-pyramids
4 | 1 2 1 2 | 0 0 1 1 2 | * 2N tetrahedra
5Y4-4T-6P3-tri-n Description
2N | 2 4 1 1 2 | 2 2 4 3
3 3 6 | 5 6 4
---+--------------+--------------------+---------
2 | 2N * * * * | 1 1 0 1 1 1 0 |
2 2 1 plane, square-inc.
2 | * 4N * * * | 0 0 1 1 1 0 1 |
1 2 1 plane, not sq.-inc.
2 | * * N * * | 0 2 4 0 0 0
0 | 0 6 0 orthogonals
2 | * * * N * | 2 0 0 0 0 0
4 | 4 0 2 obl., square-inc.
2 | * * * * 2N | 0 0 0 0 0 2 2 |
2 0 2 obl., not sq.-inc.
---+--------------+--------------------+---------
4 | 2 0 0 2 0 | N * * * * *
* | 2 0 0 obl. squares
4 | 2 0 2 0 0 | * N * * * *
* | 0 2 0 ortho. squares
4 | 0 2 2 0 0 | * * 2N * * * * |
0 2 0 ortho. squares
3 | 1 2 0 0 0 | * * * 2N * * * |
1 1 0 plane, 4pyr-3pr
3 | 1 2 0 0 0 | * * * * 2N * * |
0 1 1 plane, 4pyr-tet
3 | 1 0 0 0 2 | * * * * * 2N * |
1 0 1 obl., square-inc.*)
3 | 0 1 0 1 1 | * * * * * * 4N |
1 0 1 obl., not sq.-inc.*)
---+--------------+--------------------+---------
5 | 2 2 0 2 2 | 1 0 0 1 0 1
2 | 2N * * 4-pyramids
6 | 2 4 3 0 0 | 0 1 2 1 1 0
0 | * 2N * 3-prisms
4 | 1 2 0 1 2 | 0 0 0 0 1 1
2 | * * 2N tetrahedra
*) only the oblique squares are referred to.
5Y4-4T-6P3-tri-ortho Description
2N | 2 2 2 1 1 2 | 2 4 2
3 3 3 3 3 | 5 3 3 4
---+-----------------+-----------------------+----------
2 | 2N * * * * * | 1 1 0 1 1 0
1 0 | 2 1 1 1 plane, square-inc.*)
2 | * 2N * * * * | 0 1 0 1 1 1
0 0 | 1 1 1 1 plane, not sq.-inc.*)
2 | * * 2N * * * | 0 0 1 1 1 0
0 1 | 1 1 1 1 plane, not sq.-inc.*)
2 | * * * N * * | 0 4 2 0 0
0 0 0 | 0 3 3 0 parallels
2 | * * * * N * | 2 0 0 0 0
2 0 2 | 4 0 0 2 obl., square-inc.*)
2 | * * * * * 2N | 0 0 0 0 0 1
2 1 | 2 0 0 2 obl., not sq.-inc.*)
---+-----------------+-----------------------+----------
4 | 2 0 0 0 2 0 | N * * * *
* * * | 2 0 0 0 obl. squares
4 | 1 1 0 2 0 0 | * 2N * * * *
* * | 0 1 1 0 para. squares
4 | 0 0 2 2 0 0 | * * N * *
* * * | 0 1 1 0 para. squares
3 | 1 1 1 0 0 0 | * * * 2N * *
* * | 1 1 0 0 4pyr.-3pr.
3 | 1 1 1 0 0 0 | * * * * 2N *
* * | 0 0 1 1 tet.-3pr.
3 | 0 1 0 0 1 1 | * * * * * 2N
* * | 1 0 0 1 oblique
3 | 1 0 0 0 0 2 | * * * * *
* 2N * | 1 0 0 1 oblique
3 | 0 0 1 0 1 1 | * * * * *
* * 2N | 1 0 0 1 oblique
---+-----------------+-----------------------+----------
5 | 2 1 1 0 2 2 | 1 0 0 1 0
1 1 1 | 2N * * * 4-pyramids
6 | 2 2 2 3 0 0 | 0 2 1 2 0
0 0 0 | * N * * 3pr., 4pyr.-inc.
6 | 2 2 2 3 0 0 | 0 2 1 0 2
0 0 0 | * * N * 3pr., tet.-inc.
4 | 1 1 1 0 1 2 | 0 0 0 0 1
1 1 1 | * * * 2N tetrahedra
*) oblique squares referred only
5Y4-4T-6P3-tri-meta
Description
2N | 2 2 2 1 1 1 1 | 2 4 2 3
3 3 3 3 | 5 6 4
---+------------------+-----------------------+---------
2 | 2N * * * * * * | 1 1 0 1 1 1
0 0 | 2 2 1 plane, square-inc.*)
2 | * 2N * * * * * | 0 1 0 1 1 0
1 0 | 1 2 1 plane, not sq.-inc.*)
2 | * * 2N * * * * | 0 0 1 1 1 0
0 1 | 1 2 1 plane, not sq.-inc.*)
2 | * * * N * * * | 0 4 2 0 0 0
0 0 | 0 6 0 paralells
2 | * * * * N * * | 0 0 0 0 0 2
2 0 | 2 0 2 obl., not sq.-inc.*)
2 | * * * * * N * | 0 0 0 0 0 2
0 2 | 2 0 2 obl., not sq.-inc.*)
2 | * * * * * * N | 2 0 0 0 0 0
2 2 | 4 0 2 obl., square-inc.*)
---+------------------+-----------------------+---------
4 | 2 0 0 0 0 0 2 | N * * * * *
* * | 2 0 0 oblique square
4 | 1 1 0 2 0 0 0 | * 2N * * * *
* * | 0 2 0 para., square-inc.*)
4 | 0 0 2 2 0 0 0 | * * N * * *
* * | 0 2 0 para., not sq.-inc.*)
3 | 1 1 1 0 0 0 0 | * * * 2N * *
* * | 1 1 0 4pyr.-3pr.
3 | 1 1 1 0 0 0 0 | * * * * 2N *
* * | 0 1 1 tet.-3pr.
3 | 1 0 0 0 1 1 0 | * * * * * 2N
* * | 1 0 1 oblique
3 | 0 1 0 0 1 0 1 | * * * * * *
2N * | 1 0 1 oblique
3 | 0 0 1 0 0 1 1 | * * * * * *
* 2N | 1 0 1 oblique
---+------------------+-----------------------+---------
5 | 2 1 1 0 1 1 2 | 1 0 0 1 0 1
1 1 | 2N * * 4-pyramids
6 | 2 2 2 3 0 0 0 | 0 2 1 1 1 0
0 0 | * 2N * 3-prisms
4 | 1 1 1 0 1 1 1 | 0 0 0 0 1 1
1 1 | * * 2N tetrahedra
*) oblique squares referred to only
5Y4-4T-6P3-tri-para Description
2N | 2 4 1 1 2 | 2 2 4 3
3 3 6 | 5 3 3 4
---+--------------+--------------------+----------
2 | 2N * * * * | 1 1 0 1 1 1 0 |
2 1 1 1 plane, square-inc.*)
2 | * 4N * * * | 0 0 1 1 1 0 1 |
1 1 1 1 plane, not sq.-inc.*)
2 | * * N * * | 0 2 4 0 0 0
0 | 0 3 3 0 paralells
2 | * * * N * | 2 0 0 0 0 0
4 | 4 0 0 2 obl., square-inc.*)
2 | * * * * 2N | 0 0 0 0 0 2 2 |
2 0 0 2 obl., not sq.-inc.*)
---+--------------+--------------------+----------
4 | 2 0 0 2 0 | N * * * * *
* | 2 0 0 0 oblique squares
4 | 2 0 2 0 0 | * N * * * *
* | 0 1 1 0 para., square-inc.*)
4 | 0 2 2 0 0 | * * 2N * * * * |
0 1 1 0 para., not sq.-inc.*)
3 | 1 2 0 0 0 | * * * 2N * * * |
1 1 0 0 4pyr.-3pr.
3 | 1 2 0 0 0 | * * * * 2N * * |
0 0 1 1 tet.-3pr.
3 | 1 0 0 0 2 | * * * * * 2N * |
1 0 0 1 oblique
3 | 0 1 0 1 1 | * * * * * * 4N |
1 0 0 1 oblique
---+--------------+--------------------+----------
5 | 2 2 0 2 2 | 1 0 0 1 0 1
2 | 2N * * * 4-pyramids
6 | 2 4 3 0 0 | 0 1 2 2 0 0
0 | * N * * 3pr., 4pyr.-inc.
6 | 2 4 3 0 0 | 0 1 2 0 2 0
0 | * * N * 3pr., tet.-inc.
4 | 1 2 0 1 2 | 0 0 0 0 1 1
2 | * * * 2N tetrahedra
*) oblique squares referred to only
3Q4-T-2P8-P4
Description
8N | 2 1 1 2 | 1 2
2 1 3 2 | 3 1 1 2
---+-------------+-------------------+----------
2 | 8N * * * | 1 1 1 0 0
1 | 2 0 1 1 plane, 4-8
2 | * 4N * * | 0 2 0 1 2
0 | 2 1 0 2 plane, 8-8
2 | * * 4N * | 0 0 2 1 0
0 | 0 0 1 2 paralells
2 | * * * 8N | 0 0 0 0 2
1 | 2 1 0 0 obliques
---+-------------+-------------------+----------
4 | 4 0 0 0 | 2N * * * *
* | 1 0 1 0 plane, squares
8 | 4 4 0 0 | * 2N * * *
* | 1 0 0 1 plane, octagons
4 | 2 0 2 0 | * * 4N * *
* | 0 0 1 1 para., square-inc.*)
4 | 0 2 2 0 | * * * 2N *
* | 0 0 0 2 para., not sq.-inc.*)
3 | 0 1 0 2 | * * * * 8N
* | 1 1 0 0 obl. triangles
4 | 2 0 0 2 | * * * *
* 4N | 2 0 0 0 obl. squares
---+-------------+-------------------+----------
12 | 8 4 0 8 | 1 1 0 0 4
4 | 2N * * * 4-cupolae
4 | 0 2 0 4 | 0 0 0 0
4 0 | * 2N * * tetrahedra
8 | 8 0 4 0 | 2 0 4 0
0 0 | * * N * cubes
16 | 8 8 8 0 | 0 2 4 4 0
0 | * * * N 8-prisms
*) plane squares are referred only
6Q4-2T Description
4N | 2 1 4 | 1 2 6 4 |
6 2
---+----------+-----------+------
2 | 4N * * | 1 1 0 2 | 4 0 plane, 4-8
2 | * 2N * | 0 2 4 0 | 4 2 plane, 8-8
2 | * * 8N | 0 0 2 1 | 2 1 oblique
---+----------+-----------+------
4 | 4 0 0 | N * * * | 2 0 plane,
squares
8 | 4 4 0 | * N * * | 2 0 plane,
octagons
3 | 0 1 2 | * * 8N * | 1 1 obl.
triangles
4 | 2 0 2 | * * * 4N | 2 0 obl.
squares
---+----------+-----------+------
12 | 8 4 8 | 1 1 4 4 | 2N * 4-cupolae
4 | 0 2 4 | 0 0 4 0 | * 2N tetrahedra
6Q3-2S3-gyro
Description
3N | 4 4 | 2 2 6 4 | 6
2
---+-------+------------+-----
2 | 6N * | 1 1 1 1 | 3 1 planars
2 | * 6N | 0 0 2 1 | 2 1 obliques
---+-------+------------+-----
3 | 3 0 | 2N * * * | 1 1 planar triangles
6 | 6 0 | * N * * | 2 0 hexagons
3 | 1 2 | * * 6N * | 1 1 obl. triangles
4 | 2 2 | * * * 3N | 2 0 obl. squares
---+-------+------------+-----
9 | 9 6 | 1 1 3 3 | 2N * 3-cupolas
6 | 6 6 | 2 0 6 0 | * N octahedra
6Q3-2S3-ortho
Description
3N | 2 2 4 | 1 1 2 6 4 |
6 2
---+----------+-------------+-----
2 | 3N * * | 1 0 1 2 0 | 4 0 plane,
3cup.-inc.*)
2 | * 3N * | 0 1 1 0 2 | 2 2 plane, oct.-inc.
2 | * * 6N | 0 0 0 2 1 | 2 1 obliques
---+----------+-------------+-----
3 | 3 0 0 | N * * * * | 2 0 plane,
3cup.-3cup.
3 | 0 3 0 | * N * * * | 0 2 plane,
oct.-oct.
6 | 3 3 0 | * * N * * | 2 0 planar
hexagons
3 | 1 0 2 | * * * 6N * | 1 1 obl. triangles
4 | 0 2 2 | * * * * 3N | 2 0 obl. squares
---+----------+-------------+-----
9 | 6 3 6 | 1 0 1 3 3 | 2N * 3-cupolas
6 | 0 6 6 | 0 2 0 6 0 | * N octahedra
3Q3-S3-2P6-2P3-gyro
Description
6N | 2 2 1 2 | 1 1
2 4 3 2 | 3 1 2 2
---+-------------+-------------------+----------
2 | 6N * * * | 1 0 1 1 0
1 | 2 0 1 1 plane, 3cup.-inc.*)
2 | * 6N * * | 0 1 1 1 1
0 | 1 1 1 1 plane, oct.-inc.
2 | * * 3N * | 0 0 0 4 0
0 | 0 0 2 2 parallels
2 | * * * 6N | 0 0 0 0 2
1 | 2 1 0 0 obliques
---+-------------+-------------------+----------
3 | 3 0 0 0 | 2N * * * *
* | 1 0 1 0 plane, 3cup.-3pr.
3 | 0 3 0 0 | * 2N * * *
* | 0 1 1 0 plane, oct.-3pr.
6 | 3 3 0 0 | * * 2N * *
* | 1 0 0 1 planar hexagons
4 | 1 1 2 0 | * * * 6N *
* | 0 0 1 1 parallel squares
3 | 0 1 0 2 | * * * * 6N
* | 1 1 0 0 obl. triangles
4 | 2 0 0 2 | * * * *
* 3N | 2 0 0 0 obl. squares
---+-------------+-------------------+----------
9 | 6 3 0 6 | 1 0 1 0
3 3 | 2N * * * 3-cupolas
6 | 0 6 0 6 | 0 2 0 0
6 0 | * N * * octahedra
6 | 3 3 3 0 | 1 1 0 3
0 0 | * * 2N * 3-prisms
12 | 6 6 6 0 | 0 0 2 6 0
0 | * * * N 6-prisms
3Q3-S3-2P6-2P3-ortho Description
6N | 2 2 1 2 | 1 1
2 2 2 3 2 | 3 1 1 1 2
---+-------------+----------------------+-----------
2 | 6N * * * | 1 0 1 1 0
0 1 | 2 0 1 0 1 plane, 3cup.-inc.*)
2 | * 6N * * | 0 1 1 0 1
1 0 | 1 1 0 1 1 plane, oct.-inc.
2 | * * 3N * | 0 0 0 2 2
0 0 | 0 0 1 1 2 parallels
2 | * * * 6N | 0 0 0 0 0
2 1 | 2 1 0 0 0 obliques
---+-------------+----------------------+-----------
3 | 3 0 0 0 | 2N * * * *
* * | 1 0 1 0 0 plane, 3cup.-3pr.
3 | 0 3 0 0 | * 2N * * *
* * | 0 1 0 1 0 plane, oct.-3pr.
6 | 3 3 0 0 | * * 2N * *
* * | 1 0 0 0 1 planar hexagons
4 | 2 0 2 0 | * * * 3N *
* * | 0 0 1 0 1 para. sq., 3cup.-inc.*)
4 | 0 2 2 0 | * * * * 3N
* * | 0 0 0 1 1 para. sq., oct.-inc.
3 | 0 1 0 2 | * * * *
* 6N * | 1 1 0 0 0 obl. triangles
4 | 2 0 0 2 | * * * *
* * 3N | 2 0 0 0 0 obl. squares
---+-------------+----------------------+-----------
9 | 6 3 0 6 | 1 0 1 0
0 3 3 | 2N * * * * 3-cupolas
6 | 0 6 0 6 | 0 2 0 0
0 6 0 | * N * * * octahedra
6 | 6 0 3 0 | 2 0 0 3
0 0 0 | * * N * * 3pr., 3cup.-inc.*)
6 | 0 6 3 0 | 0 2 0 0
3 0 0 | * * * N * 3pr., oct.-inc.
12 | 6 6 6 0 | 0 0 2 3 3
0 0 | * * * * N 6-prisms
*) top base is referred to only