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Hanner polytopes

The swedish mathematician O. Hanner in 1956 defined a subset of convex polytopes, accordingly the "Hanner polytopes", which could be constructed either by prismatic extrusion ("|") and bipyramidal bi-tapering (""), when applied to a starting edge line (as was already outlined as an addendum to the |,>,O devices). Those polytopes thus are fully described by any sequence of application of the prism product and the tegum product. And, as the tegum product is just the dual of the prism product of its dual components, these well could be defined equivalently by an according sequence of the prism product and the dualisation operation instead. Note that the continued prism product just defines the unit balls of ℓ norm (aka max norm) while the continued tegum product defines the unit balls of ℓ1, so there is a connection to the according Banach spaces too.

Because the general concern of this website is associated to at least orbiform polytopes mainly, it happens that just some of those Hanner polytopes can be given as such. It is esp. the all unit-edge requirement, which provides problems here. However there is a different normalization possible for all Hanner polytopes: Coordinates could be taken to be chosen from {+1, 0, -1} instead. This different representation will be the main concern of this very page: Often already elsewhere described polytopes, then metrically given with unit-edges, will be provided here once again, but now using that different coordinate representation and the thereby derived differently sized edges. And conversely, the ones where an all unit-edged variant is impossible, would be given in here solely.

Every Hanner polytope via construction obviously is centrally symmetric. Further it becomes clear by induction that the total number of elements always is 3D+1. E.g. the cube has 1 (nullitope) + 8 (vertices) + 12 (edges) + 6 (faces) + 1 (body) = 28 = 33+1. It also becomes clear thereby that every facet of a Hanner polytope summarizes exactly half of its vertex count and that there will be a disjoint parallel facet, which then uses the other subset. I.e. Hanner polytopes always are monostratic when oriented facet-first. Therefrom it follows also, as Hanner polytopes need not have a single facet type in general, that those various facet types all are bound to have the same vertex counts. Because duals of Hanner polytopes are Hanner polytopes again, it thus can be derived once more, for any vertex-first orientation, that there will be a diametral vertex at the opposite side.

It is immediate that not only extrusion | and bi-tapering are dual operations, but that quite generally any Hanner polytope's dual would be obtained when those operation signs just would be mutually swapped. Here it is obvious that the starting |, the seed edge line, without any effects well could have been represented as as well. It further was an observation of K. Mahler that the volume products of dual pairs of Hanner polytopes (when truely being given as metrical duals, which eg. is ensured by the below provided coordinates) would come out to be always the same for any such pair of the same dimension! Furthermore it is a still open conjecture that this product, thus being addressed as the Mahler volume, when extended for any dual pair of centrally symmetric convex bodies in general, would have its minimal value for these Hanner polytopes. (Just in order to provide an according example of inequality: The (±1, ±1)-square's area is 4, the dual {(±1, 0), (0, ±1)}-square's area is 2. So the Mahler volume here is 8. But the unit disc, which surely is a convex and centrally symmetric shape too, is selfdual and it has an area of π, so that product would become π2 = 9.869604 then instead.)

Within the below table the incidence matrix usually is not being given in the most symmetrical form, rather in the form which describes best its construction from the subdimensional polytopes. Further it should be noted that each combinatorical type of any Hanner polytope as such will get listed twice, i.e. by different metrics: this is simply because already within 2D we have || = ♢|, which in the here chosen normalization is the (±1, ±1)-square u u = o4u, and |♢ = ♢♢, which in this normalization is the dual {(±1, 0), (0, ±1)}-square uo ou&#zq = q4o. That is, both are similar regular squares, which are not only oriented but also sized differently. Accordingly the thereon based construction each would just continue that doubling of combinatorical types all the way on.

Volumes also can be derived hierarchically bottom up. This is because it is obvious that within this normalization Vol(...|) = Vol(...) Vol(|) = 2 Vol(...) and D! Vol(...♢) = (D-1)! Vol(...) 1! Vol(♢), i.e. Vol(...♢) = 2 Vol(...)/D respectively. Esp. for the mere extrusional hypercubes this normalization then results in Vol(||...|) = 2D, while for the mere bi-tapered orthoplexes it results accordingly in Vol(♢♢...♢) = 2D/D!. Thence, as the Mahler volume was said to be constant for each dimension wrt. the Hanner polytopes, it well can be obtained from those directly as VolMahler(D) = Vol(||...|) Vol(♢♢...♢) = 22D/D!.

Symbol Coordinates Incidence Matrix Remarks
1D   –   Mahler volume = 4
| =
♢ - edge
  • (±1)
u

. | 2
regular

single u-edge

Volume = 2
2D   –   Mahler volume = 8
|| =
♢| - {4}
  • (±1, ±1)
u u

. .    | 4 | 2
-------+---+--
u .  & | 2 | 4
regular

all u-edges

Volume = 4
|♢ =
♢♢ - {4}
  • (±1, 0)   & all permutations
uo ou&#zq

o. o.    & | 4 | 2
-----------+---+--
oo oo&#q   | 2 | 4
regular

all q-edges

Volume = 2
3D   –   Mahler volume = 32/3 = 10.666667
||| =
♢|| - cube
  • (±1, ±1, ±1)
u o4u

. . . | 8 | 1 2 | 2 1
------+---+-----+----
u . . | 2 | 4 * | 2 0
. . u | 2 | * 8 | 1 1
------+---+-----+----
u . u | 4 | 2 2 | 4 *
. o4u | 4 | 0 4 | * 2
regular

all u-edges

Volume = 8
||♢ =
♢|♢ - oct
  • (0, 0; ±1)
  • (±1, ±1; 0)
uo oo4ou&#zh

o. o.4o.    | 2 * | 4 0 | 4
.o .o4.o    | * 4 | 2 2 | 4
------------+-----+-----+--
oo oo4oo&#h | 1 1 | 8 * | 2
.. .. .u    | 0 2 | * 4 | 2
------------+-----+-----+--
.. .. ou&#h | 1 2 | 2 1 | 8
lacings: h-edges
bases: u-edges

Volume = 8/3 = 2.666667
|♢| =
♢♢| - cube
  • (±1, 0; ±1)   & all perms within first 2 coords
u q4o

. . . | 8 | 1 2 | 2 1
------+---+-----+----
u . . | 2 | 4 * | 2 0
. q . | 2 | * 8 | 1 1
------+---+-----+----
u q . | 4 | 2 2 | 4 *
. q4o | 4 | 0 4 | * 2
lacings: u-edges
bases: q-edges

Volume = 4
|♢♢ =
♢♢♢ - oct
  • (±1, 0, 0)   & all permutations
uo oq4oo&#zq

o. o.4o.    | 2 * | 4 0 | 4
.o .o4.o    | * 4 | 2 2 | 4
------------+-----+-----+--
oo oo4oo&#q | 1 1 | 8 * | 2
.. .q ..    | 0 2 | * 4 | 2
------------+-----+-----+--
.. oq ..&#q | 1 2 | 2 1 | 8
regular

all q-edges

Volume = 4/3 = 1.333333
4D   –   Mahler volume = 32/3 = 10.666667
|||| =
♢||| - tes
  • (±1, ±1, ±1, ±1)
u o3o4u

. . . . | 16  1  3 |  3  3 | 3 1
--------+----+------+-------+----
u . . . |  2 | 8  * |  3  0 | 3 0
. . . u |  2 | * 24 |  1  2 | 2 1
--------+----+------+-------+----
u . . u |  4 | 2  2 | 12  * | 2 0
. . o4u |  4 | 0  4 |  * 12 | 1 1
--------+----+------+-------+----
u . o4u   8 | 4  8 |  4  2 | 6 *
. o3o4u   8 | 0 12 |  0  6 | * 2
regular

all u-edges

Volume = 16
|||♢ =
♢||♢ - cute
  • (0, 0, 0; ±1)
  • (±1, ±1, ±1; 0)
uo oo3oo4ou&#zu

o. o.3o.4o.    | 2 *   8  0 | 12 0 |  6
.o .o3.o4.o    | * 8 |  2  3 |  6 3 |  6
---------------+-----+-------+------+---
oo oo3oo4oo&#u | 1 1 | 16  * |  3 0 |  3
.. .. .. .u    | 0 2 |  * 12 |  2 2 |  4
---------------+-----+-------+------+---
.. .. .. ox&#u | 1 2 |  2  1 | 24 * |  2
.. .. .o4.u    | 0 4 |  0  4 |  * 6 |  2
---------------+-----+-------+------+---
.. .. oo4ou&#u  1 4 |  4  4 |  4 1 | 12
all u-edges

Volume = 4
||♢| =
♢|♢| - ope
  • (0, 0; ±1; ±1)
  • (±1, ±1; 0; ±1)
uu uo oo4ou&#zh

o. o. o.4o.     | 4 * | 1  4 0 0 | 4  4 0 | 4 1
.o .o .o4.o     | * 8 | 0  2 1 2 | 2  4 2 | 4 1
----------------+-----+----------+--------+----
u. .. .. ..     | 2 0 | 2  * * * | 4  0 0 | 4 0
oo oo oo4oo&#h  | 1 1 | * 16 * * | 1  2 0 | 2 1
.u .. .. ..     | 0 2 | *  * 4 * | 2  0 2 | 4 0
.. .. .. .u     | 0 2 | *  * * 8 | 0  2 1 | 2 1
----------------+-----+----------+--------+----
uu .. .. ..&#h  | 2 2 | 1  2 1 0 | 8  * * | 2 0
.. .. .. ou&#h  | 1 2 | 0  2 0 1 | * 16 * | 1 1
.u .. .. .u     | 0 4 | 0  0 2 2 | *  * 4 | 2 0
----------------+-----+----------+--------+----
uu .. .. ou&#h  | 2 4 | 1  4 2 2 | 2  2 1 | 8 *  trip var.
.. uo oo4ou&#zh | 2 4 | 0  8 0 4 | 0  8 0 | * 2  oct var.
lacings: h-edges
1st + last factor: u-edges

Volume = 16/3 = 5.333333
||♢♢ =
♢|♢♢ - hex
  • (0, 0; ±1, 0)   & all perms within last 2 coords
  • (±1, ±1; 0, 0)
qo4oo oo4ou&#zh

o.4o. o.4o.    | 4 * | 2  4 0 |  8  4 |  8
.o4.o .o4.o    | * 4 | 0  4 2 |  4  8 |  8
---------------+-----+--------+-------+---
q. .. .. ..    | 2 0 | 4  * * |  4  0 |  4
oo4oo oo4oo&#h | 1 1 | * 16 * |  2  2 |  4
.. .. .. .u    | 0 2 | *  * 4 |  0  4 |  4
---------------+-----+--------+-------+---
qo .. .. ..&#h | 2 1 | 1  2 0 | 16  * |  2
.. .. .. ou&#h | 1 2 | 0  2 1 |  * 16 |  2
---------------+-----+--------+-------+---
qo .. .. ou&#h | 2 2 | 1  4 1 |  2  2 | 16  tet var.
lacings: h-edges
1st factor: q-edges
2nd factor: u-edges

Volume = 4/3 = 1.333333
|♢|| =
♢♢|| - tes
  • (±1, 0; ±1, ±1)   & all perms within first 2 coords
q4o o4u

. . . . | 16   2  2 | 1  4 1 | 2 2
--------+----+-------+--------+----
q . . . |  2 | 16  * | 1  2 0 | 2 1
. . . u |  2 |  * 16 | 0  2 1 | 1 2
--------+----+-------+--------+----
q4o . . |  4 |  4  0 | 4  * * | 2 0
q . . u |  4 |  2  2 | * 16 * | 1 1
. . o4u |  4 |  0  4 | *  * 4 | 0 2
--------+----+-------+--------+----
q4o . u   8 |  8  4 | 2  4 0 | 4 *  tall 4p
q . o4u   8 |  4  8 | 0  4 2 | * 4  narrow 4p
isogonal

1st factor: q-edges
2nd factor: u-edges

Volume = 8
|♢|♢ =
♢♢|♢ - cute
  • (0, 0; 0; ±1)
  • (±1, 0; ±1; 0)   & all perms within first 2 coords
uo ou oq4oo&#zh

o. o. o.4o.    | 2 *   8 0 0 | 4  8 0 0 | 4 2
.o .o .o4.o    | * 8 |  2 1 2 | 2  4 2 1 | 4 2
---------------+-----+--------+----------+----
oo oo oo4oo&#h | 1 1 | 16 * * | 1  2 0 0 | 2 1
.. .u .. ..    | 0 2 |  * 4 * | 2  0 2 0 | 4 0
.. .. .q ..    | 0 2 |  * * 8 | 0  2 1 1 | 2 2
---------------+-----+--------+----------+----
.. ou .. ..&#h | 1 2 |  2 1 0 | 8  * * * | 2 0
.. .. oq ..&#h | 1 2 |  2 0 1 | * 16 * * | 1 1
.. .u .q ..    | 0 4 |  0 2 2 | *  * 4 * | 2 0
.. .. .q4.o    | 0 4 |  0 0 4 | *  * * 2 | 0 2
---------------+-----+--------+----------+----
.. ou oq ..&#h  1 4 |  4 2 2 | 2  2 1 0 | 8 *
.. .. oq4oo&#h  1 4 |  4 0 4 | 0  4 0 1 | * 4
u-edges
q-edges
h-edges

Volume = 2
|♢♢| =
♢♢♢| - ope
  • (±1, 0, 0; ±1)   & all perms within first 3 coords
u q3o4o

. . . . | 12  1  4 |  4  4 | 4 1
--------+----+------+-------+----
u . . . |  2 | 6  * |  4  0 | 4 0
. q . . |  2 | * 24 |  1  2 | 2 1
--------+----+------+-------+----
u q . . |  4 | 2  2 | 12  * | 2 0
. q3o . |  3 | 0  3 |  * 16 | 1 1
--------+----+------+-------+----
u q3o .   6 | 3  6 |  3  2 | 8 *
. q3o4o   6 | 0 12 |  0  8 | * 2
lacing u-edges
base q-edges

Volume = 8/3 = 2.666667
|♢♢♢ =
♢♢♢♢ - hex
  • (±1, 0, 0, 0)   & all permutations
uo oq3oo4oo&#zq

o. o.3o.4o.    | 2 *   6  0 | 12 0 |  8
.o .o3.o4.o    | * 6   2  4 |  8 4 |  8
---------------+-----+-------+------+---
oo oo3oo4oo&#q | 1 1 | 12  * |  4 0 |  4
.. .q .. ..    | 0 2 |  * 12 |  2 2 |  4
---------------+-----+-------+------+---
.. oq .. ..&#q | 1 2 |  2  1 | 24 * |  2
.. .q3.o ..    | 0 3 |  0  3 |  * 8 |  2
---------------+-----+-------+------+---
.. oq3oo ..&#q  1 3 |  3  3 |  3 1 | 16
regular

all q-edges

Volume = 2/3 = 0.666667
5D   –   Mahler volume = 128/15 = 8.533333
||||| =
♢|||| - pent
  • (±1, ±1, ±1, ±1, ±1)
u o3o3o4u

. . . . . | 32   1  4 |  4  6 |  6  4 | 4 1
----------+----+-------+-------+-------+----
u . . . . |  2 | 16  *   4  0 |  6  0 | 4 0
. . . . u |  2 |  * 64   1  3 |  3  3 | 3 1
----------+----+-------+-------+-------+----
u . . . u |  4 |  2  2 | 32  * |  3  0 | 3 0
. . . o4u |  4 |  0  4 |  * 48 |  1  2 | 2 1
----------+----+-------+-------+-------+----
u . . o4u   8 |  4  8 |  4  2 | 24  * | 2 0
. . o3o4u   8 |  0 12 |  0  6 |  * 16 | 1 1
----------+----+-------+-------+-------+----
u . o3o4u  16 |  8 24 | 12 12 |  6  2 | 8 *
. o3o3o4u  16 |  0 32 |  0 24 |  0  8 | * 2
regular

all u-edges

Volume = 32
||||♢ =
♢|||♢ - tessit
  • (0, 0, 0, 0; ±1)
  • (±1, ±1, ±1, ±1; 0)
uo oo3oo3oo4ou&#za   where: a = sqrt(5) = 2.236068

o. o.3o.3o.4o.    | 2  * | 16  0 | 32  0 | 24 0 |  8  verf: tes
.o .o3.o3.o4.o    | * 16 |  2  4 |  8  6 | 12 4 |  8
------------------+------+-------+-------+------+---
oo oo3oo3oo4oo&#a | 1  1 | 32  * |  4  0 |  6 0 |  4  ef: tet
.. .. .. .. .u    | 0  2 |  * 32 |  2  3 |  6 3 |  6
------------------+------+-------+-------+------+---
.. .. .. .. ou&#a | 1  2 |  2  1 | 64  * |  3 0 |  3
.. .. .. .o4.u    | 0  4 |  0  4 |  * 24 |  2 2 |  4
------------------+------+-------+-------+------+---
.. .. .. oo4ou&#a | 1  4 |  4  4 |  4  1 | 48 * |  2  squippy var.
.. .. .o3.o4.u    | 0  8 |  0 12 |  0  6 |  * 8 |  2  cube
------------------+------+-------+-------+------+---
.. .. oo3oo4ou&#a | 1  8 |  8 12 | 12  6 |  6 1 | 16  cubpy var.
lacings: a-edges
base: q-edges

Volume = 32/5 = 6.4

all unit-edged variant is impossible
|||♢| =
♢||♢| - cutep
  • (0, 0, 0; ±1; ±1)
  • (±1, ±1, ±1; 0; ±1)
uu uo oo3oo4ou&#zu

o. o. o.3o.4o.     | 4  * | 1  8 0  0 |  8 12  0  0 | 12  6 0 |  6 1
.o .o .o3.o4.o     | * 16 | 0  2 1  3 |  2  6  3  3 |  6  6 3 |  6 1
-------------------+------+-----------+-------------+---------+-----
u. .. .. .. ..     | 2  0 | 2  * *  *   8  0  0  0 | 12  0 0 |  6 0
oo oo oo3oo4oo&#u  | 1  1 | * 32 *  * |  1  3  0  0 |  3  3 0 |  3 1
.u .. .. .. ..     | 0  2 | *  * 8  * |  2  0  3  0 |  6  0 3 |  6 0
.. .. .. .. .u     | 0  2 | *  * * 24 |  0  2  1  2 |  2  4 2 |  4 1
-------------------+------+-----------+-------------+---------+-----
uu .. .. .. ..&#u  | 2  2 | 1  2 1  0 | 16  *  *  * |  3  0 0 |  3 0
.. .. .. .. ox&#u  | 1  2 | 0  2 0  1 |  * 48  *  * |  1  2 0 |  2 1
.u .. .. .. .u     | 0  4 | 0  0 2  2 |  *  * 12  * |  2  0 2 |  4 0
.. .. .. .o4.u     | 0  4 | 0  0 0  4 |  *  *  * 12 |  0  2 1 |  2 1
-------------------+------+-----------+-------------+---------+-----
uu .. .. .. ou&#u   2  4 | 1  4 2  2 |  2  2  1  0 | 24  * * |  2 0
.. .. .. oo4ou&#u   1  4 | 0  4 0  4 |  0  4  0  1 |  * 24 * |  1 1
.u .. .. .o4.u      0  8 | 0  0 4  8 |  0  0  4  2 |  *  * 6 |  2 0
-------------------+------+-----------+-------------+---------+-----
uu .. .. oo4ou&#u   2  8 | 1  8 4  8 |  4  8  4  2 |  4  2 1 | 12 *
.. uo oo3oo4ou&#zu  2  8 | 0 16 0 12 |  0 24  0  6 |  0 12 0 |  * 2
all u-edges

Volume = 8
|||♢♢ =
♢||♢♢ - squacubdit
  • (0, 0, 0; ±1, 0)   & all perms within last 2 coords
  • (±1, ±1, ±1; 0, 0)
qo4oo oo3oo4ou&#zu

o.4o. o.3o.4o.    | 4 * | 2  8  0 | 16 12 0 | 24  6 | 12
.o4.o .o3.o4.o    | * 8 | 0  4  3 |  4 12 3 | 12 12 | 12
------------------+-----+---------+---------+-------+---
q. .. .. .. ..    | 2 0 | 4  *  * |  8  0 0 | 12  0 |  6  ef: cube
oo4oo oo3oo4oo&#u | 1 1 | * 32  * |  2  3 0 |  6  3 |  6
.. .. .. .. .u    | 0 2 | *  * 12 |  0  4 2 |  4  8 |  8
------------------+-----+---------+---------+-------+---
qo .. .. .. ..&#u | 2 1 | 1  2  0 | 32  * * |  3  0 |  3
.. .. .. .. ou&#u | 1 2 | 0  2  1 |  * 48 * |  2  2 |  4
.. .. .. .o4.u    | 0 4 | 0  0  4 |  *  * 6 |  0  4 |  4
------------------+-----+---------+---------+-------+---
qo .. .. .. ou&#u | 2 2 | 1  4  1 |  2  2 0 | 48  * |  2  tet var.
.. .. .. oo4ou&#u | 1 4 | 0  4  4 |  0  4 1 |  * 24 |  2  squippy var.
------------------+-----+---------+---------+-------+---
qo .. .. oo4ou&#u | 2 4 | 1  8  4 |  4  8 1 |  4  2 | 24  squasc var.
1st factor: q-edges
lacings + 2nd factor: u-edges

Volume = 8/5 = 1.6

all unit-edged variant is impossible
||♢|| =
♢|♢|| - squoct
  • (0, 0; ±1; ±1, ±1)
  • (±1, ±1; 0; ±1, ±1)
oo4uu uo oo4ou&#zh

o.4o. o. o.4o.     | 8  * | 2  4  0  0 | 1  8  4 0  0 | 4  8 1 0 | 4 2
.o4.o .o .o4.o     | * 16 | 0  2  2  2 | 0  4  4 1  4 | 2  8 1 2 | 4 2
-------------------+------+------------+--------------+----------+----
.. u. .. .. ..     | 2  0 | 8  *  *  * | 1  4  0 0  0 | 4  4 0 0 | 4 1
oo4oo oo oo4oo&#h  | 1  1 | * 32  *  * | 0  2  2 0  0 | 1  4 1 0 | 2 2
.. .u .. .. ..     | 0  2 | *  * 16  * | 0  2  0 1  2 | 2  4 0 2 | 4 1
.. .. .. .. .u     | 0  2 | *  *  * 16 | 0  0  2 0  2 | 0  4 1 1 | 2 2
-------------------+------+------------+--------------+----------+----
o.4u. .. .. ..     | 4  0 | 4  0  0  0 | 2  *  * *  * | 4  0 0 0 | 4 0
.. uu .. .. ..&#h  | 2  2 | 1  2  1  0 | * 32  * *  * | 1  2 0 0 | 2 1
.. .. .. .. ou&#h  | 1  2 | 0  2  0  1 | *  * 32 *  * | 0  2 1 0 | 1 2
.o4.u .. .. ..     | 0  4 | 0  0  4  0 | *  *  * 4  * | 2  0 0 2 | 4 0
.. .u .. .. .u     | 0  4 | 0  0  2  2 | *  *  * * 16 | 0  2 0 1 | 2 1
-------------------+------+------------+--------------+----------+----
oo4uu .. .. ..&#h  | 4  4 | 4  4  4  0 | 1  4  0 1  0 | 8  * * * | 2 0  (h,u)-cube var.
.. uu .. .. ou&#h  | 2  4 | 1  4  2  2 | 0  2  2 0  1 | * 32 * * | 1 1  (h,u)-trip var.
.. .. uo oo4ou&#zh | 2  4 | 0  8  0  4 | 0  0  8 0  0 | *  * 4 * | 0 2  (h,u)-oct var.
.o4.u .. .. .u     | 0  8 | 0  0  8  4 | 0  0  0 2  4 | *  * * 4 | 2 0  u-cube
-------------------+------+------------+--------------+----------+----
oo4uu .. .. ou&#h  | 4  8 | 4  8  8  4 | 1  8  4 2  4 | 2  4 0 1 | 8 *  (h,u)-tisdip var.
.. uu uo oo4ou&#zh | 4  8 | 2 16  4  8 | 0  8 16 0  4 | 0  8 2 0 | * 4  (h,u)-ope var.
1st factor: u-edges
2nd factor: q-edges

Volume = 32/3 = 10.666667
||♢|♢ =
♢|♢|♢ - opet
  • (0, 0; 0; 0; ±1
  • (0, 0; ±1; ±1; 0)
  • (±1, ±1; 0; ±1; 0)
uoo ouu ouo ooo4oou&#z(h,u,h)

o.. o.. o.. o..4o..          | 2 * * | 4  8 0  0 0 0 | 2 16 4  8 0  0 0 |  8 16 4 0 0 |  8 2
.o. .o. .o. .o.4.o.          | * 4 * | 2  0 1  4 0 0 | 2  8 0  0 4  4 0 |  8  8 0 4 1 |  8 2
..o ..o ..o ..o4..o          | * * 8 | 0  2 0  2 1 2 | 0  4 2  4 2  4 2 |  4  8 4 4 1 |  8 2
-----------------------------+-------+---------------+------------------+-------------+-----
oo. oo. oo. oo.4oo.&#h       | 1 1 0 | 8  * *  * * * | 1  4 0  0 0  0 0 |  4  4 0 0 0 |  4 1
o.o o.o o.o o.o4o.o&#u       | 1 0 1 | * 16 *  * * * | 0  2 1  2 0  0 0 |  2  4 2 0 0 |  4 1
... .u. ... ... ...          | 0 2 0 | *  * 2  * * * | 2  0 0  0 4  0 0 |  8  0 0 4 0 |  8 0
.oo .oo .oo .oo4.oo&#h       | 0 1 1 | *  * * 16 * * | 0  2 0  0 1  2 0 |  2  4 0 2 1 |  4 2
... ..u ... ... ...          | 0 0 2 | *  * *  * 4 * | 0  0 2  0 2  0 2 |  4  0 4 4 0 |  8 0
... ... ... ... ..u          | 0 0 2 | *  * *  * * 8 | 0  0 0  2 0  2 1 |  0  4 2 2 1 |  4 2
-----------------------------+-------+---------------+------------------+-------------+-----
... ou. ... ... ...&#h       | 1 2 0 | 2  0 1  0 0 0 | 4  * *  * *  * * |  4  0 0 0 0 |  4 0
ooo ooo ooo ooo4ooo&#(h,u,h) | 1 1 1 | 1  1 0  1 0 0 | * 32 *  * *  * * |  1  2 0 0 0 |  2 1
... o.u ... ... ...&#h       | 1 0 2 | 0  2 0  0 1 0 | *  * 8  * *  * * |  2  0 2 0 0 |  4 0
... ... ... ... o.u&#h       | 1 0 2 | 0  2 0  0 0 1 | *  * * 16 *  * * |  0  2 1 0 0 |  2 1
... .uu ... ... ...&#h       | 0 2 2 | 0  0 1  2 1 0 | *  * *  * 8  * * |  2  0 0 2 0 |  4 0
... ... ... ... .ou&#h       | 0 1 2 | 0  0 0  2 0 1 | *  * *  * * 16 * |  0  2 0 1 1 |  2 2
... ..u ... ... ..u          | 0 0 4 | 0  0 0  0 2 2 | *  * *  * *  * 4 |  0  0 2 2 0 |  4 0
-----------------------------+-------+---------------+------------------+-------------+-----
... ouu ... ... ...&#(h,u,h) | 1 2 2 | 2  2 1  2 1 0 | 1  2 1  0 1  0 0 | 16  * * * * |  2 0  squippy var.
... ... ... ... oou&#(h,u,h) | 1 1 2 | 1  2 0  2 0 1 | 0  2 0  1 0  1 0 |  * 32 * * * |  1 1  tet var.
... o.u ... ... o.u&#u       | 1 0 4 | 0  4 0  0 2 2 | 0  0 2  2 0  0 1 |  *  * 8 * * |  2 0  u-squippy
... .uu ... ... .ou&#h       | 0 2 4 | 0  0 1  4 2 2 | 0  0 0  0 2  2 1 |  *  * * 8 * |  2 0  trip var.
... ... .uo .oo4.ou&#zh      | 0 2 4 | 0  0 0  8 0 4 | 0  0 0  0 0  8 0 |  *  * * * 2 |  0 2  oct var.
-----------------------------+-------+---------------+------------------+-------------+-----
... ouu ... ... oou&#(h,u,h) | 1 2 4 | 2  4 1  4 2 2 | 1  4 2  2 2  2 1 |  2  2 1 1 0 | 16 *  trippy var.
... ... ouo ooo4oou&#(h,u,h) | 1 2 4 | 2  4 0  8 0 4 | 0  8 0  4 0  8 0 |  0  8 0 0 1 |  * 4  octpy var.
u-edges
h-edges
q-edges

Volume = 32/15 = 2.133333
||♢♢| =
♢|♢♢| - hexip
  • (0, 0; ±1, 0; ±1)   & all perms of 3rd and 4th coords
  • (±1, ±1; 0, 0; ±1)
uu qo4oo oo4ou&#zh

o. o.4o. o.4o.     | 8 * | 1 2  4 0 0 | 2  4  8  4 0 |  8  4  8 |  8 1
.o .o4.o .o4.o     | * 8 | 0 0  4 1 2 | 0  4  4  8 2 |  4  8  8 |  8 1
-------------------+-----+------------+--------------+----------+-----
u. .. .. .. ..     | 2 0 | 4 *  * * * | 2  4  0  0 0 |  8  4  0 |  8 0
.. q. .. .. ..     | 2 0 | * 8  * * * | 1  0  4  0 0 |  4  0  4 |  4 1
oo oo4oo oo4oo&#h  | 1 1 | * * 32 * * | 0  1  2  2 0 |  2  2  4 |  4 1
.u .. .. .. ..     | 0 2 | * *  * 4 * | 0  4  0  0 2 |  4  8  0 |  8 0
.. .. .. .. .u     | 0 2 | * *  * * 8 | 0  0  0  4 1 |  0  4  4 |  4 1
-------------------+-----+------------+--------------+----------+-----
u. q. .. .. ..     | 4 0 | 2 2  0 0 0 | 4  *  *  * * |  4  0  0 |  4 0
uu .. .. .. ..&#h  | 2 2 | 1 0  2 1 0 | * 16  *  * * |  2  2  0 |  4 0
.. qo .. .. ..&#h  | 2 1 | 0 1  2 0 0 | *  * 32  * * |  1  0  2 |  2 1
.. .. .. .. ou&#h  | 1 2 | 0 0  2 0 1 | *  *  * 32 * |  0  1  2 |  2 1
.u .. .. .. .u     | 0 4 | 0 0  0 2 2 | *  *  *  * 4 |  0  4  0 |  4 0
-------------------+-----+------------+--------------+----------+-----
uu qo .. .. ..&#h  | 4 2 | 2 2  4 1 0 | 1  2  2  0 0 | 16  *  * |  2 0   var.
uu .. .. .. ou&#h  | 2 4 | 1 0  4 2 2 | 0  2  0  2 1 |  * 16  * |  2 0   var.
.. qo .. .. ou&#h  | 2 2 | 0 1  4 0 1 | 0  0  2  2 0 |  *  * 32 |  1 1   var.
-------------------+-----+------------+--------------+----------+-----
uu qo .. .. ou&#h  | 4 4 | 2 2  8 2 2 | 1  4  4  4 1 |  2  2  2 | 16 *   var.
.. qo4oo oo4ou&#zh | 4 4 | 0 4 16 0 4 | 0  0 16 16 0 |  0  0 16 |  * 2   var.
lacings: h-edges
1st+3rd factor: u-edges
2nd factor: q-edges

Volume = 8/3 = 2.666667
||♢♢♢ =
♢|♢♢♢ - tac
  • (0, 0; ±1, 0, 0)   & all perms within last 3 coords
  • (±1, ±1; 0, 0, 0)
qo3oo4oo oo4ou&#za   where: a = sqrt(7/2) = 1.870829

o.3o.4o. o.4o.    | 6 * |  4  4 0 | 4 16  4 | 16 16 | 16  verf: hex var.
.o3.o4.o .o4.o    | * 4 |  0  6 2 | 0 12 12 |  8 24 | 16  verf: hex var.
------------------+-----+---------+---------+-------+---
q. .. .. .. ..    | 2 0 | 12  * * | 2  4  0 |  8  4 |  8  ef: oct var.
oo3oo4oo oo4oo&#a | 1 1 |  * 24 * | 0  4  2 |  4  8 |  8  ef: oct var.
.. .. .. .. .u    | 0 2 |  *  * 4 | 0  0  6 |  0 12 |  8  ef: oct
------------------+-----+---------+---------+-------+---
q.3o. .. .. ..    | 3 0 |  3  0 0 | 8  *  * |  4  0 |  4
qo .. .. .. ..&#a | 2 1 |  1  2 0 | * 48  * |  2  2 |  4
.. .. .. .. ou&#a | 1 2 |  0  2 1 | *  * 24 |  0  4 |  4
------------------+-----+---------+---------+-------+---
qo3oo .. .. ..&#a | 3 1 |  3  3 0 | 1  3  0 | 32  * |  2  tet var.
qo .. .. .. ou&#a | 2 2 |  1  4 1 | 0  2  2 |  * 48 |  2  tet var.
------------------+-----+---------+---------+-------+---
qo3oo .. .. ou&#a | 3 2 |  3  6 1 | 1  6  3 |  2  3 | 32  pen var.
lacings: a-edges
1st factor: q-edges
2nd factor: u-edges

Volume = 8/15 = 0.533333
|♢||| =
♢♢||| - pent
  • (±1, 0; ±1, ±1, ±1)   & all perms within first 2 coords
q4o o3o4u

. . . . . | 32 |  2  3 | 1  6  3 |  3  6 1 | 3 2
----------+----+-------+---------+---------+----
q . . . . |  2 | 32  * | 1  3  0 |  3  3 0 | 3 1
. . . . u |  2 |  * 48 | 0  2  2 |  1  4 1 | 2 2
----------+----+-------+---------+---------+----
q4o . . . |  4 |  4  0 | 8  *  * |  3  0 0 | 3 0
q . . . u |  4 |  2  2 | * 48  * |  1  2 0 | 2 1
. . . o4u |  4 |  0  4 | *  * 24 |  0  2 1 | 1 2
----------+----+-------+---------+---------+----
q4o . . u |  8 |  8  4 | 2  4  0 | 12  * * | 2 0  tall 4p var.
q . . o4u |  8 |  4  8 | 0  4  2 |  * 24 * | 1 1  narrow 4p var.
. . o3o4u |  8 |  0 12 | 0  0  6 |  *  * 4 | 0 2  u-cube
----------+----+-------+---------+---------+----
q4o . o4u | 16 | 16 16 | 4 16  4 |  4  4 0 | 6 *  (4,4)-dip var.
q . o3o4u | 16 |  8 24 | 0 12 12 |  0  6 2 | * 4  cube-pr. var.
1st factor: q-edges
2nd factor: u-edges

Volume = 16
|♢||♢ =
♢♢||♢ - tessit
  • (0, 0; 0, 0; ±1)
  • (±1, 0; ±1, ±1; 0)   & all perms within first 2 coords
uo oq4oo oo4ou&#zu

o. o.4o. o.4o.    | 2  * | 16  0  0 | 16 16 0  0 0 | 4 16 4 0 0 | 4 4
.o .o4.o .o4.o    | * 16 |  2  2  2 |  4  4 1  4 1 | 2  8 2 2 2 | 4 4
------------------+------+----------+--------------+------------+----
oo oo4oo oo4oo&#u | 1  1 | 32  *  * |  2  2 0  0 0 | 1  4 1 0 0 | 2 2
.. .q .. .. ..    | 0  2 |  * 16  * |  2  0 1  2 0 | 2  4 0 2 1 | 4 2
.. .. .. .. .u    | 0  2 |  *  * 16 |  0  2 0  2 1 | 0  4 2 1 2 | 2 4
------------------+------+----------+--------------+------------+----
.. oq .. .. ..&#u | 1  2 |  2  1  0 | 32  * *  * * | 1  2 0 0 0 | 2 1
.. .. .. .. ou&#u | 1  2 |  2  0  1 |  * 32 *  * * | 0  2 1 0 0 | 1 2
.. .q4.o .. ..    | 0  4 |  0  4  0 |  *  * 4  * * | 2  0 0 2 0 | 4 0
.. .q .. .. .u    | 0  4 |  0  2  2 |  *  * * 16 * | 0  2 0 1 1 | 2 2
.. .. .. .o4.u    | 0  4 |  0  0  4 |  *  * *  * 4 | 0  0 2 0 2 | 0 4
------------------+------+----------+--------------+------------+----
.. oq4oo .. ..&#u | 1  4 |  4  4  0 |  4  0 1  0 0 | 8  * * * * | 2 0  tall squippy var.
.. oq .. .. ou&#u | 1  4 |  4  2  2 |  2  2 0  1 0 | * 32 * * * | 1 1  rect. squippy var.
.. .. .. oo4ou&#u | 1  4 |  4  0  4 |  0  4 0  0 1 | *  * 8 * * | 0 2  squippy
.. .q4.o .. .u    | 0  8 |  0  8  4 |  0  0 2  4 0 | *  * * 4 * | 2 0  tall 4p var.
.. .q .. .o4.u    | 0  8 |  0  4  8 |  0  0 0  4 2 | *  * * * 4 | 0 2  narrow 4p var.
------------------+------+----------+--------------+------------+----
.. oq4oo .. ou&#u | 1  8 |  8  8  4 |  8  4 2  4 0 | 2  4 0 1 0 | 8 *  tall cubpy var.
.. oq .. oo4ou&#u | 1  8 |  8  4  8 |  4  8 0  4 2 | 0  4 2 0 1 | * 8  narrow cubpy var.
2nd factor: q-edges
others: u-edges

Volume = 16/5 = 3.2

all unit-edged variant is impossible
|♢|♢| =
♢♢|♢| - cutep
  • (0, 0; 0; ±1; ±1)
  • (±1, 0; ±1; 0; ±1)   & all perms within first 2 coords
uu uo ou oq4oo&#zh

o. o. o. o.4o.     | 4  * | 1  8 0 0  0 |  8  4  8 0 0 0 0 | 4  8  4 2 0 0 | 4 2 1
.o .o .o .o4.o     | * 16 | 0  2 1 1  2 |  2  2  4 1 2 2 1 | 2  4  4 2 2 1 | 4 2 1
-------------------+------+-------------+------------------+---------------+------
u. .. .. .. ..     | 2  0 | 2  * * *  * |  8  0  0 0 0 0 0 | 4  8  0 0 0 0 | 4 2 0
oo oo oo oo4oo&#h  | 1  1 | * 32 * *  * |  1  1  2 0 0 0 0 | 1  2  2 1 0 0 | 2 1 1
.u .. .. .. ..     | 0  2 | *  * 8 *  * |  2  0  0 1 2 0 0 | 2  4  0 0 2 1 | 4 2 0
.. .. .u .. ..     | 0  2 | *  * * 8  * |  0  2  0 1 0 2 0 | 2  0  4 0 2 0 | 4 0 1
.. .. .. .q ..     | 0  2 | *  * * * 16 |  0  0  2 0 1 1 1 | 0  2  2 2 1 2 | 2 2 1
-------------------+------+-------------+------------------+---------------+------
uu .. .. .. ..&#h  | 2  2 | 1  2 1 0  0 | 16  *  * * * * * | 1  2  0 0 0 0 | 2 1 0
.. .. ou .. ..&#h  | 1  2 | 0  2 0 1  0 |  * 16  * * * * * | 1  0  2 0 0 0 | 2 0 1
.. .. .. oq ..&#h  | 1  2 | 0  2 0 0  1 |  *  * 32 * * * * | 0  1  1 1 0 0 | 1 1 1
.u .. .u .. ..     | 0  4 | 0  0 2 2  0 |  *  *  * 4 * * * | 2  0  0 0 2 0 | 4 0 0
.u .. .. .q ..     | 0  4 | 0  0 2 0  2 |  *  *  * * 8 * * | 0  2  0 0 1 1 | 2 2 0
.. .. .u .q ..     | 0  4 | 0  0 0 2  2 |  *  *  * * * 8 * | 0  0  2 0 1 0 | 2 0 1
.. .. .. .q4.o     | 0  4 | 0  0 0 0  4 |  *  *  * * * * 4 | 0  0  0 2 0 1 | 0 2 1
-------------------+------+-------------+------------------+---------------+------
uu .. ou .. ..&#h  | 2  4 | 1  4 2 2  0 |  2  2  0 1 0 0 0 | 8  *  * * * * | 2 0 0  trip var.
uu .. .. oq ..&#h  | 2  4 | 1  4 2 0  2 |  2  0  2 0 1 0 0 | * 16  * * * * | 1 1 0  trip var.
.. .. ou oq ..&#h  | 1  4 | 0  4 0 2  2 |  0  2  2 0 0 1 0 | *  * 16 * * * | 1 0 1  squippy var.
.. .. .. oq4oo&#h  | 1  4 | 0  4 0 0  4 |  0  0  4 0 0 0 1 | *  *  * 8 * * | 0 1 1  squippy var.
.u .. .u .q ..     | 0  8 | 0  0 4 4  4 |  0  0  0 2 2 2 0 | *  *  * * 4 * | 2 0 0  narrow 4p var.
.u .. .. .q4.o     | 0  8 | 0  0 4 0  8 |  0  0  0 0 4 0 2 | *  *  * * * 2 | 0 2 0  tall 4p var.
-------------------+------+-------------+------------------+---------------+------
uu .. ou oq ..&#h  | 2  8 | 1  8 4 4  4 |  4  4  4 2 2 2 0 | 2  2  2 0 1 0 | 8 * *  squippyp var.
uu .. .. oq4oo&#h  | 2  8 | 1  8 4 0  8 |  4  0  8 0 4 0 2 | 0  4  0 2 0 1 | * 4 *  squippyp var.
.. uo ou oq4oo&#zh | 2  8 | 0 16 0 4  8 |  0  8 16 0 0 4 2 | 0  0  8 4 0 0 | * * 2  cute var.
early factors: u-edges
last factor: q-edges
lacings: h-edges

Volume = 4
|♢|♢♢ =
♢♢|♢♢ - squacubdit
  • (0, 0; 0; ±1, 0)   & all perms within last 2 coords
  • (±1, 0; ±1; 0, 0)   & all perms within first 2 coords
qo4oo ou oq4oo&#zh

o.4o. o. o.4o.    | 4 * | 2  8 0 0 | 16  4  8 0 0 |  8 16  4 2 |  8 4
.o4.o .o .o4.o    | * 8 | 0  4 1 2 |  4  4  8 2 1 |  4  8  8 4 |  8 4
------------------+-----+----------+--------------+------------+-----
q. .. .. .. ..    | 2 0 | 4  * * * |  8  0  0 0 0 |  4  8  0 0 |  4 2
oo4oo oo oo4oo&#h | 1 1 | * 32 * * |  2  1  2 0 0 |  2  4  2 1 |  4 2
.. .. .u .. ..    | 0 2 | *  * 4 * |  0  4  0 2 0 |  4  0  8 0 |  8 0
.. .. .. .q ..    | 0 2 | *  * * 8 |  0  0  4 1 1 |  0  4  4 4 |  4 4
------------------+-----+----------+--------------+------------+-----
qo .. .. .. ..&#h | 2 1 | 1  2 0 0 | 32  *  * * * |  1  2  0 0 |  2 1
.. .. ou .. ..&#h | 1 2 | 0  2 1 0 |  * 16  * * * |  2  0  2 0 |  4 0
.. .. .. oq ..&#h | 1 2 | 0  2 0 1 |  *  * 32 * * |  0  2  1 1 |  2 2
.. .. .u .q ..    | 0 4 | 0  0 2 2 |  *  *  * 4 * |  0  0  4 0 |  4 0
.. .. .. .q4.o    | 0 4 | 0  0 0 4 |  *  *  * * 2 |  0  0  0 4 |  0 4
------------------+-----+----------+--------------+------------+-----
qo .. ou .. ..&#h | 2 2 | 1  4 1 0 |  2  2  0 0 0 | 16  *  * * |  2 0  tet var.
qo .. .. oq ..&#h | 2 2 | 1  4 0 1 |  2  0  2 0 0 |  * 32  * * |  1 1  tet var.
.. .. ou oq ..&#h | 1 4 | 0  4 2 2 |  0  2  2 1 0 |  *  * 16 * |  2 0  squippy var.
.. .. .. oq4oo&#h | 1 4 | 0  4 0 4 |  0  0  4 0 1 |  *  *  * 8 |  0 2  squippy var.
------------------+-----+----------+--------------+------------+-----
qo .. ou oq ..&#h | 2 4 | 1  8 2 2 |  4  4  4 1 0 |  2  2  2 0 | 16 *  squasc var.
qo .. .. oq4oo&#h | 2 4 | 1  8 0 4 |  4  0  8 0 1 |  0  4  0 2 |  * 8  squasc var.
q-edges
u-edges
h-edges

Volume = 4/5 = 0.8

all unit-edged variant is impossible
|♢♢|| =
♢♢♢|| - squoct
  • (±1, 0, 0; ±1, ±1)   & all perms within first 3 coords
o4u q3o4o

. . . . . | 24 |  2  4 | 1  8  4 |  4  8 1 | 4 2
----------+----+-------+---------+---------+----
. u . . . |  2 | 24  * | 1  4  0 |  4  4 0 | 4 1
. . q . . |  2 |  * 48 | 0  2  2 |  1  4 1 | 2 2
----------+----+-------+---------+---------+----
o4u . . . |  4 |  4  0 | 6  *  * |  4  0 0 | 4 0
. u q . . |  4 |  2  2 | * 48  * |  1  2 0 | 2 1
. . q3o . |  3 |  0  3 | *  * 32 |  0  2 1 | 1 2
----------+----+-------+---------+---------+----
o4u q . . |  8 |  8  4 | 2  4  0 | 12  * * | 2 0  (q,u)-cube var.
. u q3o . |  6 |  3  6 | 0  3  2 |  * 32 * | 1 1  (u,q)-trip var.
. . q3o4o |  6 |  0 12 | 0  0  8 |  *  * 4 | 0 2  q-oct
----------+----+-------+---------+---------+----
o4u q3o . | 12 | 12 12 | 3 12  4 |  3  4 0 | 8 *  (q,u)-tisdip var.
. u q3o4o | 12 |  6 24 | 0 12 16 |  0  8 2 | * 4  (u,q)-ope var.
1st factor: u-edges
2nd factor: q-edges

Volume = 16/3 = 5.333333
|♢♢|♢ =
♢♢♢|♢ - opet
  • (0, 0, 0; 0; ±1)
  • (±1, 0, 0; ±1; 0)   & all perms within first 3 coords
uo ou oq3oo4oo&#zh

o. o. o.3o.4o.    | 2  * | 12 0  0 |  6 24  0  0 | 12 16 0 0 |  8 2
.o .o .o3.o4.o    | * 12 |  2 1  4 |  2  8  4  4 |  8  8 4 1 |  8 2
------------------+------+---------+-------------+-----------+-----
oo oo oo3oo4oo&#h | 1  1 | 24 *  * |  1  4  0  0 |  4  4 0 0 |  4 1
.. .u .. .. ..    | 0  2 |  * 6  * |  2  0  4  0 |  8  0 4 0 |  8 0
.. .. .q .. ..    | 0  2 |  * * 24 |  0  2  1  2 |  2  4 2 1 |  4 2
------------------+------+---------+-------------+-----------+-----
.. ou .. .. ..&#h | 1  2 |  2 1  0 | 12  *  *  * |  4  0 0 0 |  4 0
.. .. oq .. ..&#h | 1  2 |  2 0  1 |  * 48  *  * |  1  2 0 0 |  2 1
.. .u .q .. ..    | 0  4 |  0 2  2 |  *  * 12  * |  2  0 2 0 |  4 0
.. .. .q3.o ..    | 0  3 |  0 0  3 |  *  *  * 16 |  0  2 1 1 |  2 2
------------------+------+---------+-------------+-----------+-----
.. ou oq .. ..&#h | 1  4 |  4 2  2 |  2  2  1  0 | 24  * * * |  2 0  (h,q,u)-squippy var.
.. .. oq3oo ..&#h | 1  3 |  3 0  3 |  0  3  0  1 |  * 32 * * |  1 1  (h,q)-tet var.
.. .u .q3.o ..    | 0  6 |  0 3  6 |  0  0  3  2 |  *  * 8 * |  2 0  (u,q)-trip var.
.. .. .q3.o4.o    | 0  6 |  0 0 12 |  0  0  0  8 |  *  * * 2 |  0 2  q-oct
------------------+------+---------+-------------+-----------+-----
.. ou oq3oo ..&#h | 1  6 |  6 3  6 |  3  6  3  2 |  3  2 1 0 | 16 *  (h,u,q)-trippy var.
.. .. oq3oo4oo&#h | 1  6 |  6 0 12 |  0 12  0  8 |  0  8 0 1 |  * 4  (h,q)-octpy var.
u-edges
h-edges
q-edges

Volume = 16/15 = 1.066667
|♢♢♢| =
♢♢♢♢| - hexip
  • (±1, 0, 0, 0; ±1)   & all perms within first 4 coords
u q3o3o4o

. . . . . | 16 | 1  6 |  6 12 | 12  8 |  8 1
----------+----+------+-------+-------+-----
u . . . . |  2 | 8  *   6  0 | 12  0 |  8 0
. q . . . |  2 | * 48 |  1  4 |  4  4 |  4 1
----------+----+------+-------+-------+-----
u q . . . |  4 | 2  2 | 24  * |  4  0 |  4 0
. q3o . . |  3 | 0  3 |  * 64 |  1  2 |  2 1
----------+----+------+-------+-------+-----
u q3o . .   6 | 3  6 |  3  2 | 32  * |  2 0
. q3o3o .   4 | 0  6 |  0  4 |  * 32 |  1 1
----------+----+------+-------+-------+-----
u q3o3o .   8 | 4 12 |  6  8 |  4  2 | 16 *
. q3o3o4o   8 | 0 24 |  0 32 |  0 16 |  * 2
lacings: u-edges
base: q-edges

Volume = 4/3 = 1.333333
|♢♢♢♢ =
♢♢♢♢♢ - tac
  • (±1, 0, 0, 0, 0)   & all permutations
uo oq3oo3oo4oo&#zq

o. o.3o.3o.4o.    | 2 *   8  0 | 24  0 | 32  0 | 16
.o .o3.o3.o4.o    | * 8   2  6 | 12 12 | 24  8 | 16
------------------+-----+-------+-------+-------+---
oo oo3oo3oo4oo&#q | 1 1 | 16  *   6  0 | 12  0 |  8
.. .q .. .. ..    | 0 2 |  * 24   2  4 |  8  4 |  8
------------------+-----+-------+-------+-------+---
.. oq .. .. ..&#q | 1 2 |  2  1 | 48  * |  4  0 |  4
.. .q3.o .. ..    | 0 3 |  0  3 |  * 32 |  2  2 |  4
------------------+-----+-------+-------+-------+---
.. oq3oo .. ..&#q  1 3 |  3  3 |  3  1 | 64  * |  2
.. .q3.o3.o ..     0 4 |  0  6 |  0  4 |  * 16 |  2
------------------+-----+-------+-------+-------+---
.. oq3oo3oo ..&#q  1 4 |  4  6 |  6  4 |  4  1 | 32
regular

q-edges

Volume = 4/15 = 0.266667


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