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Hanner polytopes

The swedish mathematician O. Hanner in 1956 defined a subset of convex polytopes, accordingly called the "Hanner polytopes",inductively by

It becomes clear that the first statement is just the induction start, and that, because thetegum product is just the dual of the prism product of its dual components, either the second or the third statement could well be replaced by this other product without any change.On the other hand it should be stated that although the very start looks just likeprismaticextrusions ("|") and bipyramidalbi-taperings ("♢"), when applied to a starting edge line (as was already outlined as an addendum to the|,>,O devices),the Hanner polytopes still have a somewhat richer varity. This is simply because the later operations would restrictthe former products to such ones, where one of the factors always would be bound to be a mere line segment,instead of the liberty to be any other Hanner polytope as well.For instance neither the tegum product of 2cubes|||⊕ ||| nor its dual prism product of 2octs♢♢♢× ♢♢♢ (octdip) could be described by those operants solely.Note furthermore that the continued prism product just defines the unit balls of ℓ∞ norm (aka max norm) whilethe 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 forall 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 iscentrally symmetric. Further it becomes clear by induction that the total number of elements always is3D+1. E.g. thecube has1 (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 total vertex countand that there will be a disjoint parallel facet, which then uses the other subset. I.e. Hanner polytopes always are monostraticwhen 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, thatthere will be a diametral vertex at the opposite side.

It 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 theMahler 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'sarea 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 of the corresponding polytope, rather in the form which describes best its construction from the subdimensional polytopes, i.e. most often by means of extrusion| or bi-tapering♢,if those apply.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 2Dwe have|| ≡ ♢|, which in the here chosen normalization is the (±1, ±1)-squareu u = o4u, and|♢ ≡ ♢♢, which in this normalization is the dual {(±1, 0), (0, ±1)}-squareuo 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 continuethat doubling of combinatorical types all the way on. This multiplication of alike combinatorical types even gets worse in higher dimensions, as both sized squares could not only be used as a seed to build upon, but as that of each further factor again.

Volumes also can be derived hierarchically bottom up. This is because for extrusions it is obvious that within this normalizationVol(...|) = Vol(...) Vol(|) = 2 Vol(...)and for bi-taperingsD! Vol(...♢) = (D-1)! Vol(...) 1! Vol(♢), i.e.Vol(...♢) = 2 Vol(...)/D respectively,since obviouslyVol(|) = Vol(♢) = 2. But even more general it already derives from the respective products directly, as for these there are the formulaeVol(P×Q) = Vol(P) Vol(Q) anddim(P⊕Q) Vol(P⊕Q) = dim(P) Vol(P) dim(Q) Vol(Q) respectively.Esp. for the mere extrusional hypercubes this normalization then results inVol(||...|) = 2D,while for the mere bi-tapered orthoplexes it results accordingly inVol(♢♢...♢) = 2D/D!. Thence, as the Mahler volume was said to be constant for each dimension wrt. the Hanner polytopes, it well can be obtainedfrom those directly asVolMahler(D) = Vol(||...|) Vol(♢♢...♢) = 22D/D!.

External
links		wikipedia  polytopewiki  

NumberSymbolCoordinatesIncidence MatrixRemarks
1DMahler volume = 4
1.1
| =♢ - edge
  • (±1)
u. | 2
regular

single u-edge

Volume = 2

self-dual
2DMahler volume = 8
2.1
|× | =|| =♢| -{4}
  • (±1, ±1)
u u. .    | 4 | 2-------+---+--u .  & | 2 | 4
regular

all u-edges

Volume = 4

dual =2.2
2.2
|⊕ | =|♢ =♢♢ -{4}
  • (±1, 0)   & all permutations
uo ou&#zqo. o.    & | 4 | 2-----------+---+--oo oo&#q   | 2 | 4
regular

all q-edges

Volume = 2

dual =2.1
3DMahler volume = 32/3 = 10.666667
3.1
||× | =||| =♢|| -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

dual =3.4
3.2
||⊕ | =||♢ =♢|♢ -oct(metrically:squit)
  • (0, 0; ±1)
  • (±1, ±1; 0)
uo oo4ou&#zho. 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

dual =3.3
3.3
|♢× | =|♢| =♢♢| -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

dual =3.2
3.4
|♢⊕ | =|♢♢ =♢♢♢ -oct
  • (±1, 0, 0)   & all permutations
uo oq4oo&#zqo. 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

dual =3.1
4DMahler volume = 32/3 = 10.666667
4.01
|||× | =||× || =|||| =♢||| -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

dual =4.08
4.02
|||⊕ | =|||♢ =♢||♢ -cute
  • (0, 0, 0; ±1)
  • (±1, ±1, ±1; 0)
uo oo3oo4ou&#zuo. 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

dual =4.07
4.03
||♢× | =||♢| =♢|♢| -ope
  • (0, 0; ±1; ±1)
  • (±1, ±1; 0; ±1)
uu uo oo4ou&#zho. 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 0oo 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 | * 2oct var.
lacings: h-edges
1st + last factor: u-edges

Volume = 16/3 = 5.333333

dual =4.06
4.04
||♢⊕ | =||⊕ |♢ =||♢♢ =♢|♢♢ -hex
  • (0, 0; ±1, 0)   & all perms within last 2 coords
  • (±1, ±1; 0, 0)
qo4oo oo4ou&#zho.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 |  4oo4oo 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
lacings: h-edges
1st factor: q-edges
2nd factor: u-edges

Volume = 4/3 = 1.333333

dual =4.05
4.05
|♢|× | =|♢× || =|♢|| =♢♢|| -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 0q . . u |  4 |  2  2 | * 16 * | 1 1. . o4u |  4 |  0  4 | *  * 4 | 0 2--------+----+-------+--------+----q4o . u♦  8 |  8  4 | 2  4 0 | 4 *  tall 4pq . o4u♦  8 |  4  8 | 0  4 2 | * 4  narrow 4p
isogonal

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

Volume = 8

dual =4.04
4.06
|♢|⊕ | =|♢|♢ =♢♢|♢ -cute
  • (0, 0; 0; ±1)
  • (±1, 0; ±1; 0)   & all perms within first 2 coords
uo ou oq4oo&#zho. 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

dual =4.03
4.07
|♢♢× | =|♢♢| =♢♢♢| -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

dual =4.02
4.08
|♢♢⊕ | =|♢⊕ |♢ =|♢♢♢ =♢♢♢♢ -hex
  • (±1, 0, 0, 0)   & all permutations
uo oq3oo4oo&#zqo. 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

dual =4.01
4.09
||⊕ || -hex
  • (±1, ±1; 0, 0)
  • (0, 0; ±1, ±1)
uo4oo ou4oo&#zuo.4o. o.4o.    | 4 *♦ 2  4 * |  8  4 |  8.o4.o .o4.o    | * 4♦ 0  4 2 |  4  8 |  8---------------+-----+--------+-------+---u. .. .. ..    | 2 0 | 4  * * |  4  0 |  4oo4oo oo4oo&#u | 1 1 | * 16 * |  2  2 |  4.. .. .u ..    | 0 2 | *  * 4 |  0  4 |  4---------------+-----+--------+-------+---uo .. .. ..&#u | 2 1 | 1  2 0 | 16  * |  2.. .. ou ..&#u | 1 2 | 0  2 1 |  * 16 |  2---------------+-----+--------+-------+---uo .. ou ..&#u♦ 2 2 | 1  4 1 |  2  2 | 16
regular

all u-edges

Volume = 8/3 = 2.666667

dual =4.10
4.10
|♢× |♢ -tes
  • (±1, 0; ±1, 0)   & all permutations in each block
q4o q4o. . . . | 16♦  2  2 | 1  4 1 | 2 2--------+----+-------+--------+----q . . . |  2 | 16  * | 1  2 0 | 2 1. . q . |  2 |  * 16 | 0  2 1 | 1 2--------+----+-------+--------+----q4o . . |  4 |  4  0 | 4  * * | 2 0q . q . |  4 |  2  2 | * 16 * | 1 1. . q4o |  4 |  0  4 | *  * 4 | 0 2--------+----+-------+--------+----q4o q .♦  8 |  8  4 | 2  4 0 | 4 *q . q4o♦  8 |  4  8 | 0  4 2 | * 4
regular

all q-edges

Volume = 4

dual =4.09
5DMahler volume = 128/15 = 8.533333
5.01
||||× | =|||× || =||||| =♢|||| -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

dual =5.16
5.02
||||⊕ | =||||♢ =♢|||♢ - tessit
  • (0, 0, 0, 0; ±1)
  • (±1, ±1, ±1, ±1; 0)
uo oo3oo3oo4ou&#za   where: a = sqrt(5) = 2.236068o. 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 * |  2squippy var... .. .o3.o4.u    | 0  8 |  0 12 |  0  6 |  * 8 |  2cube------------------+------+-------+-------+------+---.. .. oo3oo4ou&#a | 1  8 |  8 12 | 12  6 |  6 1 | 16cubpy var.
lacings: a-edges
base: q-edges

Volume = 32/5 = 6.4

dual =5.15

all unit-edged variant is impossible
5.03
|||♢× | =|||♢| =♢||♢| -cutep
  • (0, 0, 0; ±1; ±1)
  • (±1, ±1, ±1; 0; ±1)
uu uo oo3oo4ou&#zuo. 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 0oo 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

dual =5.14
5.04
|||♢⊕ | =|||⊕ |♢ =|||♢♢ =♢||♢♢ - squacubdit
  • (0, 0, 0; ±1, 0)   & all perms within last 2 coords
  • (±1, ±1, ±1; 0, 0)
qo4oo oo3oo4ou&#zuo.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:cubeoo4oo 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  * |  2tet var... .. .. oo4ou&#u | 1 4 | 0  4  4 |  0  4 1 |  * 24 |  2squippy var.------------------+-----+---------+---------+-------+---qo .. .. oo4ou&#u | 2 4 | 1  8  4 |  4  8 1 |  4  2 | 24squasc var.
1st factor: q-edges
lacings + 2nd factor: u-edges

Volume = 8/5 = 1.6

dual =5.13

all unit-edged variant is impossible
5.05
||♢|× | =||♢× || =||♢|| =♢|♢|| -squoct
  • (0, 0; ±1; ±1, ±1)
  • (±1, ±1; 0; ±1, ±1)
oo4uu uo oo4ou&#zho.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 1oo4oo 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

dual =5.12
5.06
||♢|⊕ | =||♢|♢ =♢|♢|♢ -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 1o.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 0ooo 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 0squippy var.... ... ... ... oou&#(h,u,h) | 1 1 2 | 1  2 0  2 0 1 | 0  2 0  1 0  1 0 |  * 32 * * * |  1 1tet 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 0trip var.... ... .uo .oo4.ou&#zh      | 0 2 4 | 0  0 0  8 0 4 | 0  0 0  0 0  8 0 |  *  * * * 2 |  0 2oct 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 |  * 4octpy var.
u-edges
h-edges
q-edges

Volume = 32/15 = 2.133333

dual =5.11
5.07
||♢♢× | =||♢♢| =♢|♢♢| -hexip
  • (0, 0; ±1, 0; ±1)   & all perms of 3rd and 4th coords
  • (±1, ±1; 0, 0; ±1)
uu qo4oo oo4ou&#zho. 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 1oo 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 0uu .. .. .. ..&#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 0trip var.uu .. .. .. ou&#h  | 2 4 | 1 0  4 2 2 | 0  2  0  2 1 |  * 16  * |  2 0trip var... qo .. .. ou&#h  | 2 2 | 0 1  4 0 1 | 0  0  2  2 0 |  *  * 32 |  1 1tet var.-------------------+-----+------------+--------------+----------+-----uu qo .. .. ou&#h  | 4 4 | 2 2  8 2 2 | 1  4  4  4 1 |  2  2  2 | 16 *tepe var... qo4oo oo4ou&#zh | 4 4 | 0 4 16 0 4 | 0  0 16 16 0 |  0  0 16 |  * 2hex var.
lacings: h-edges
1st+3rd factor: u-edges
2nd factor: q-edges

Volume = 8/3 = 2.666667

dual =5.10
5.08
||♢♢⊕ | =||♢⊕ |♢ =||⊕ |♢♢ =||♢♢♢ =♢|♢♢♢ -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) = 1.870829o.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 |  4qo .. .. .. ..&#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  * |  2tet var.qo .. .. .. ou&#a | 2 2 |  1  4 1 | 0  2  2 |  * 48 |  2tet var.------------------+-----+---------+---------+-------+---qo3oo .. .. ou&#a | 3 2 |  3  6 1 | 1  6  3 |  2  3 | 32pen var.
lacings: a-edges
1st factor: q-edges
2nd factor: u-edges

Volume = 8/15 = 0.533333

dual =5.09
5.09
|♢||× | =|♢|× || =|♢× ||| =|♢||| =♢♢||| -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 0q . . . 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  tall4p var.q . . o4u |  8 |  4  8 | 0  4  2 |  * 24 * | 1 1  narrow4p 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 | * 4cube-pr. var.
1st factor: q-edges
2nd factor: u-edges

Volume = 16

dual =5.08
5.10
|♢||⊕ | =|♢||♢ =♢♢||♢ - tessit
  • (0, 0; 0, 0; ±1)
  • (±1, 0; ±1, ±1; 0)   & all perms within first 2 coords
uo oq4oo oo4ou&#zuo. 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  tallsquippy 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 2squippy.. .q4.o .. .u    | 0  8 |  0  8  4 |  0  0 2  4 0 | *  * * 4 * | 2 0  tall4p var... .q .. .o4.u    | 0  8 |  0  4  8 |  0  0 0  4 2 | *  * * * 4 | 0 2  narrow4p var.------------------+------+----------+--------------+------------+----.. oq4oo .. ou&#u | 1  8 |  8  8  4 |  8  4 2  4 0 | 2  4 0 1 0 | 8 *  tallcubpy var... oq .. oo4ou&#u | 1  8 |  8  4  8 |  4  8 0  4 2 | 0  4 2 0 1 | * 8  narrowcubpy var.
2nd factor: q-edges
others: u-edges

Volume = 16/5 = 3.2

dual =5.07

all unit-edged variant is impossible
5.11
|♢|♢× | =|♢|♢| =♢♢|♢| -cutep
  • (0, 0; 0; ±1; ±1)
  • (±1, 0; ±1; 0; ±1)   & all perms within first 2 coords
uu uo ou oq4oo&#zho. 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 0oo 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 0trip var.uu .. .. oq ..&#h  | 2  4 | 1  4 2 0  2 |  2  0  2 0 1 0 0 | * 16  * * * * | 1 1 0trip var... .. ou oq ..&#h  | 1  4 | 0  4 0 2  2 |  0  2  2 0 0 1 0 | *  * 16 * * * | 1 0 1squippy var... .. .. oq4oo&#h  | 1  4 | 0  4 0 0  4 |  0  0  4 0 0 0 1 | *  *  * 8 * * | 0 1 1squippy var..u .. .u .q ..     | 0  8 | 0  0 4 4  4 |  0  0  0 2 2 2 0 | *  *  * * 4 * | 2 0 0  narrow4p var..u .. .. .q4.o     | 0  8 | 0  0 4 0  8 |  0  0  0 0 4 0 2 | *  *  * * * 2 | 0 2 0  tall4p 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 | * * 2cute var.
early factors: u-edges
last factor: q-edges
lacings: h-edges

Volume = 4

dual =5.06
5.12
|♢|♢⊕ | =|♢|⊕ |♢ =|♢|♢♢ =♢♢|♢♢ - 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&#zho.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 2oo4oo 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 0tet var.qo .. .. oq ..&#h | 2 2 | 1  4 0 1 |  2  0  2 0 0 |  * 32  * * |  1 1tet var... .. ou oq ..&#h | 1 4 | 0  4 2 2 |  0  2  2 1 0 |  *  * 16 * |  2 0squippy var... .. .. oq4oo&#h | 1 4 | 0  4 0 4 |  0  0  4 0 1 |  *  *  * 8 |  0 2squippy 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 |  * 8squasc var.
q-edges
u-edges
h-edges

Volume = 4/5 = 0.8

dual =5.05

all unit-edged variant is impossible
5.13
|♢♢|× | =|♢♢× || =|♢♢|| =♢♢♢|| -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

dual =5.04
5.14
|♢♢|⊕ | =|♢♢|♢ =♢♢♢|♢ -opet
  • (0, 0, 0; 0; ±1)
  • (±1, 0, 0; ±1; 0)   & all perms within first 3 coords
uo ou oq3oo4oo&#zho. 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

dual =5.03
5.15
|♢♢♢× | =|♢♢♢| =♢♢♢♢| -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

dual =5.02
5.16
|♢♢♢⊕ | =|♢♢⊕ |♢ =|♢♢♢♢ =♢♢♢♢♢ -tac
  • (±1, 0, 0, 0, 0)   & all permutations
uo oq3oo3oo4oo&#zqo. 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

dual =5.01
5.17
|||× |♢ -pent
  • (±1, ±1, ±1; ±1, 0)   & all perms within last 2 coords
o3o4u q4o. . . . . | 32 |  3  2 |  3  6 1 | 1  6  3 | 2 3----------+----+-------+---------+---------+----. . u . . |  2 | 48  * |  2  2 0 | 1  4  1 | 2 2. . . q . |  2 |  * 32 |  0  3 1 | 0  3  3 | 1 3----------+----+-------+---------+---------+----. o4u . . |  4 |  4  0 | 24  * * | 1  2  0 | 2 1. . u q . |  4 |  2  2 |  * 48 * | 0  2  1 | 1 2. . . q4o |  4 |  0  4 |  *  * 8 | 0  0  3 | 0 3----------+----+-------+---------+---------+----o3o4u . .♦  8 | 12  0 |  6  0 0 | 4  *  * | 2 0. o4u q .♦  8 |  8  4 |  2  4 0 | * 24  * | 1 1. . u q4o♦  8 |  4  8 |  0  4 2 | *  * 12 | 0 2----------+----+-------+---------+---------+----o3o4u q . | 16 | 24  8 | 12 12 0 | 2  6  0 | 4 *  (q,u,u,u)-tes var.. o4u q4o | 16 | 16 16 |  4 16 4 | 0  4  4 | * 6  (q,q,u,u)-tes var.
q-edges
u-edges

Volume = 16

dual =5.24
5.18
||♢× |♢ -squoct
  • (0, 0; ±1; ±1, 0)   & all perms within last 2 coords
  • (±1, ±1; 0; ±1, 0)   & all perms within last 2 coords
uo oo4ou qq4oo&#zho. o.4o. o.4o.     | 8  * | 2  4  0  0 | 1  4  8  0 0 | 1  8 4 0 | 2 4.o .o4.o .o4.o     | * 16 | 0  2  2  2 | 0  4  4  4 1 | 1  8 2 2 | 2 4-------------------+------+------------+--------------+----------+----.. .. .. q. ..     | 2  0 | 8  *  *  * | 1  0  4  0 0 | 0  4 4 0 | 1 4oo oo4oo oo4oo&#h  | 1  1 | * 32  *  * | 0  2  2  0 0 | 1  4 1 0 | 2 2.. .. .u .. ..     | 0  2 | *  * 16  * | 0  2  0  2 0 | 1  4 0 1 | 2 2.. .. .. .q ..     | 0  2 | *  *  * 16 | 0  0  2  2 1 | 0  4 2 2 | 1 4-------------------+------+------------+--------------+----------+----.. .. .. q.4o.     | 4  0 | 4  0  0  0 | 2  *  *  * * | 0  0 4 0 | 0 4.. .. ou .. ..&#h  | 1  2 | 0  2  1  0 | * 32  *  * * | 1  2 0 0 | 2 1.. .. .. qq ..&#h  | 2  2 | 1  2  0  1 | *  * 32  * * | 0  2 1 0 | 1 2.. .. .u .q ..     | 0  4 | 0  0  2  2 | *  *  * 16 * | 0  2 0 1 | 1 2.. .. .. .q4.o     | 0  4 | 0  0  0  4 | *  *  *  * 4 | 0  0 2 2 | 0 4-------------------+------+------------+--------------+----------+----uo oo4ou .. ..&#zh | 2  4 | 0  8  4  0 | 0  8  0  0 0 | 4  * * * | 2 0  (h,u)-oct var... .. ou qq ..&#h  | 2  4 | 1  4  2  2 | 0  2  2  1 0 | * 32 * * | 1 1  (q,h,u)-trip var... .. .. qq4oo&#h  | 4  4 | 4  4  0  4 | 1  0  4  0 1 | *  * 8 * | 0 2  (h,q)-cube var... .. .u .q4.o     | 0  8 | 0  0  4  8 | 0  0  0  4 2 | *  * * 4 | 0 2  (u,q)-cube var.-------------------+------+------------+--------------+----------+----uo oo4ou qq ..&#zh | 4  8 | 2 16  8  4 | 0 16  8  4 0 | 2  8 0 0 | 4 *  (q,h,u)-ope var... .. ou qq4oo&#h  | 4  8 | 4  8  4  8 | 1  4  8  4 2 | 0  4 2 1 | * 8  (u,h,q)-tisdip var.
q-edges
h-edges
u-edges

Volume = 16/3 = 5.333333

dual =5.23
5.19
|♢|× |♢ -pent
  • (±1; ±1, 0; ±1, 0)   & all perms within each block
u q4o q4o. . . . . | 32 |  1  2  2 |  2  2 1  4 1 | 1  4 1 2 2 | 2 2 1----------+----+----------+--------------+------------+------u . . . . |  2 | 16  *  * |  2  2 0  0 0 | 1  4 1 0 0 | 2 2 0. q . . . |  2 |  * 32  * |  1  0 1  2 0 | 1  2 0 2 1 | 2 1 1. . . q . |  2 |  *  * 32 |  0  1 0  2 1 | 0  2 1 1 2 | 1 2 1----------+----+----------+--------------+------------+------u q . . . |  4 |  2  2  0 | 16  * *  * * | 1  2 0 0 0 | 2 1 0u . . q . |  4 |  2  0  2 |  * 16 *  * * | 0  2 1 0 0 | 1 2 0. q4o . . |  4 |  0  4  0 |  *  * 8  * * | 1  0 0 2 0 | 2 0 1. q . q . |  4 |  0  2  2 |  *  * * 32 * | 0  1 0 1 1 | 1 1 1. . . q4o |  4 |  0  0  4 |  *  * *  * 8 | 0  0 1 0 2 | 0 2 1----------+----+----------+--------------+------------+------u q4o . . |  8 |  4  8  0 |  4  0 2  0 0 | 4  * * * * | 2 0 0  (u,q,q)-cubeu q . q . |  8 |  4  4  4 |  2  2 0  2 0 | * 16 * * * | 1 1 0  (u,q,q)-cubeu . . q4o |  8 |  4  0  8 |  0  4 0  0 2 | *  * 4 * * | 0 2 0  (u,q,q)-cube. q4o q . |  8 |  0  8  4 |  0  0 2  4 0 | *  * * 8 * | 1 0 1  q-cube. q . q4o |  8 |  0  4  8 |  0  0 0  4 2 | *  * * * 8 | 0 1 1  q-cube----------+----+----------+--------------+------------+------u q4o q . | 16 |  8 16  8 |  8  4 4  8 0 | 2  4 0 2 0 | 4 * *  (u,q,q,q)-tesu q . q4o | 16 |  8  8 16 |  4  8 0  8 4 | 0  4 2 0 2 | * 4 *  (u,q,q,q)-tes. q4o q4o | 16 |  0 16 16 |  0  0 4 16 4 | 0  0 0 4 4 | * * 2  q-tes
q-edges

Volume = 8

dual =5.22
5.20
|♢♢× |♢ -squoct
  • (±1, 0; ±1, 0, 0)   & all perms within each block
q3o4o q4o. . . . . | 24 |  4  2 |  4  8 1 | 1  8  4 | 2 4----------+----+-------+---------+---------+----q . . . . |  2 | 48  * |  2  2 0 | 1  4  1 | 2 2. . . q . |  2 |  * 24 |  0  4 1 | 0  4  4 | 1 4----------+----+-------+---------+---------+----q3o . . . |  3 |  3  0 | 32  * * | 1  2  0 | 2 1q . . q . |  4 |  2  2 |  * 48 * | 0  2  1 | 1 2. . . q4o |  4 |  0  4 |  *  * 6 | 0  0  4 | 0 4----------+----+-------+---------+---------+----q3o4o . .♦  6 | 12  0 |  8  0 0 | 4  *  * | 2 0q3o . q .♦  6 |  6  3 |  2  3 0 | * 32  * | 1 1q . . q4o♦  8 |  4  8 |  0  4 2 | *  * 12 | 0 2----------+----+-------+---------+---------+----q3o4o q .♦ 12 | 24  6 | 16 12 0 | 2  8  0 | 4 *q3o . q4o♦ 12 | 12 12 |  4 12 3 | 0  4  3 | * 8
uniform

q-edges

Volume = 8/3 = 2.666667

dual =5.21
5.21
|||⊕ || - squacubdit
  • (±1, ±1; 0, 0, 0)
  • (0, 0; ±1, ±1, ±1)
uo4oo oo3oo4ou&#zy   where y = sqrt(5) = 2.236068o.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------------------+-----+---------+---------+-------+---u. .. .. .. ..    | 2 0 | 4  *  *♦  8  0 0 | 12  0 |  6oo4oo oo3oo4oo&#y | 1 1 | * 32  * |  2  3 0 |  6  3 |  6.. .. .. .. .u    | 0 2 | *  * 12 |  0  4 2 |  4  8 |  8------------------+-----+---------+---------+-------+---uo .. .. .. ..&#y | 2 1 | 1  2  0 | 32  * * |  3  0 |  3.. .. .. .. ou&#y | 1 2 | 0  2  1 |  * 48 * |  2  2 |  4.. .. .. .o4.u    | 0 4 | 0  0  4 |  *  * 6 |  0  4 |  4------------------+-----+---------+---------+-------+---uo .. .. .. ou&#y | 2 2 | 1  4  1 |  2  2 0 | 48  * |  2  taller 2ap (tet var.).. .. .. oo4ou&#y | 1 4 | 0  4  4 |  0  4 1 |  * 24 |  2  tallersquippy var.------------------+-----+---------+---------+-------+---uo .. .. oo4ou&#y | 2 4 | 1  8  4 |  4  8 1 |  4  2 | 24squasc var.
u-edges
y-edges

Volume = 16/5 = 3.2

dual =5.20

all unit-edged variant is impossible
5.22
||♢⊕ || -squoct
  • (±1; 0, 0; 0, 0)
  • (0; ±1, ±1; 0, 0)
  • (0; 0, 0; ±1, ±1)
uoo ouo4ooo oou4ooo&#z(h,u)o.. o..4o.. o..4o..        | 2 * *♦ 4 4 0  0 0 | 4 16 4  0  0 | 16 16  0 | 16.o. .o.4.o. .o.4.o.        | * 4 * | 2 0 2  4 0 | 4  8 0  8  4 | 16  8  8 | 16..o ..o4..o ..o4..o        | * * 4 | 0 2 0  4 2 | 0  8 4  4  8 |  8 16  8 | 16---------------------------+-------+------------+--------------+----------+---oo. oo.4oo. oo.4oo.&#h     | 1 1 0 | 8 * *  * * | 2  4 0  0  0 |  8  4  0 |  8o.o o.o4o.o o.o4o.o&#h     | 1 0 1 | * 8 *  * * | 0  4 2  0  0 |  4  8  0 |  8... .u. ... ... ...        | 0 2 0 | * * 4  * * | 2  0 0  4  0 |  8  0  4 |  8.oo .oo4.oo .oo4.oo&#u     | 0 1 1 | * * * 16 * | 0  2 0  2  2 |  4  4  4 |  8... ... ... ..u ...        | 0 0 2 | * * *  * 4 | 0  0 2  0  4 |  0  8  4 |  8---------------------------+-------+------------+--------------+----------+---... ou. ... ... ...&#h     | 1 2 0 | 2 0 1  0 0 | 8  * *  *  * |  4  0  0 |  4ooo ooo4ooo ooo4ooo&#(h,u) | 1 1 1 | 1 1 0  1 0 | * 32 *  *  * |  2  2  0 |  4... ... ... o.u ...&#h     | 1 0 2 | 0 2 0  0 1 | *  * 8  *  * |  0  4  0 |  4... .uo ... ... ...&#u     | 0 2 1 | 0 0 1  2 0 | *  * * 16  * |  2  0  2 |  4... ... ... .ou ...&#u     | 0 1 2 | 0 0 0  2 1 | *  * *  * 16 |  0  2  2 |  4---------------------------+-------+------------+--------------+----------+---... ouo ... ... ...&#(h,u) | 1 2 1 | 2 1 1  2 0 | 1  2 0  1  0 | 32  *  * |  2  shallower 3pyr (tet var.)... ... ... oou ...&#(h,u) | 1 1 2 | 1 2 0  2 1 | 0  2 1  0  1 |  * 32  * |  2  shallower 3pyr (tet var.)... .uo ... .ou ...&#u     | 0 2 2 | 0 0 1  4 1 | 0  0 0  2  2 |  *  * 16 |  2  u-tet---------------------------+-------+------------+--------------+----------+---... ouo ... oou ...&#(h,u) | 1 2 2 | 2 2 1  4 1 | 1  4 1  2  2 |  2  2  1 | 32  shallower tet-pyr (pen var.)
h-edges
u-edges

Volume = 16/15 = 1.066667

dual =5.19
5.23
|♢|⊕ || - squacubdit
  • (0, 0; 0; ±1, ±1)
  • (±1, 0; ±1; 0, 0)   & all perms within first 2 coords
oq4oo ou uo4oo&#zuo.4o. o. o.4o.    | 4 * | 2  8 0 0 |  8  4 16 0 0 | 2  4 16  8 | 4  8.o4.o .o .o4.o    | * 8 | 0  4 2 1 |  8  4  4 1 2 | 4  8  8  4 | 4  8------------------+-----+----------+--------------+------------+-----.. .. .. u. ..    | 2 0 | 4  * * * |  0  0  8 0 0 | 0  0  8  4 | 2  4oo4oo oo oo4oo&#u | 1 1 | * 32 * * |  2  1  2 0 0 | 1  2  4  2 | 2  4.q .. .. .. ..    | 0 2 | *  * 8 * |  4  0  0 1 1 | 4  4  4  0 | 4  4.. .. .u .. ..    | 0 2 | *  * * 4 |  0  4  0 0 2 | 0  8  0  4 | 0  8------------------+-----+----------+--------------+------------+-----oq .. .. .. ..&#u | 1 2 | 0  2 1 0 | 32  *  * * * | 1  1  2  0 | 2  2.. .. ou .. ..&#u | 1 2 | 0  2 0 1 |  * 16  * * * | 0  2  0  2 | 0  4.. .. .. uo ..&#u | 2 1 | 1  2 0 0 |  *  * 32 * * | 0  0  2  1 | 1  2.q4.o .. .. ..    | 0 4 | 0  0 4 0 |  *  *  * 2 * | 4  0  0  0 | 4  0.q .. .u .. ..    | 0 4 | 0  0 2 2 |  *  *  * * 4 | 0  4  0  0 | 0  4------------------+-----+----------+--------------+------------+-----oq4oo .. .. ..&#u | 1 4 | 0  4 4 0 |  4  0  0 1 0 | 8  *  *  * | 2  0  tallsquippy var.oq .. ou .. ..&#u | 1 4 | 0  4 2 2 |  2  2  0 0 1 | * 16  *  * | 0  2  rect.squippy var.oq .. .. uo ..&#u | 2 2 | 1  4 1 0 |  2  0  2 0 0 | *  * 32  * | 1  1tet var... .. ou uo ..&#u | 2 2 | 1  4 0 1 |  0  2  2 0 0 | *  *  * 16 | 0  2  u-tet------------------+-----+----------+--------------+------------+-----oq4oo .. uo ..&#u | 2 4 | 1  8 4 0 |  8  0  4 1 0 | 2  0  4  0 | 8  *squasc var.oq .. ou uo ..&#u | 2 4 | 1  8 2 2 |  4  4  4 0 1 | 0  2  2  2 | * 16squasc var.
q-edges
u-edges

Volume = 8/5 = 1.6

dual =5.18

all unit-edged variant is impossible
5.24
|♢♢⊕ || -tac
  • (±1, 0, 0; 0, 0)   & all perms within first 3 coords
  • (0, 0, 0; ±1, ±1)
uo4oo oq3oo4oo&#zho.4o. o.3o.4o.    | 4 * | 2  6  0 | 12 12 0 | 24  8 | 16.o4.o .o3.o4.o    | * 6 | 0  4  4 |  4 16 4 | 16 16 | 16------------------+-----+---------+---------+-------+---u. .. .. .. ..    | 2 0 | 4  *  *♦  6  0 0 | 12  0 |  8oo4oo oo3oo4oo&#h | 1 1 | * 24  * |  2  4 0 |  8  4 |  8.. .. .q .. ..    | 0 2 | *  * 12 |  0  4 2 |  4  8 |  8------------------+-----+---------+---------+-------+---uo .. .. .. ..&#h | 2 1 | 1  2  0 | 24  * * |  4  0 |  4.. .. oq .. ..&#h | 1 2 | 0  2  1 |  * 48 * |  2  2 |  4.. .. .q3.o ..    | 0 3 | 0  0  3 |  *  * 8 |  0  4 |  4------------------+-----+---------+---------+-------+---uo .. oq .. ..&#h | 2 2 | 1  4  1 |  2  2 0 | 48  * |  2  2ap var. (tet).. .. oq3oo ..&#h | 1 3 | 0  3  3 |  0  3 1 |  * 32 |  2  3pyr var. (tet)------------------+-----+---------+---------+-------+---uo .. oq3oo ..&#h | 2 3 | 1  6  3 |  3  6 1 |  3  2 | 32pen var.
q-edges
h-edges
u-edges

Volume = 8/15 = 0.533333

dual =5.17

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