| Site Map | Polytopes | Dynkin Diagrams | Vertex Figures, etc. | Incidence Matrices | Index |
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 the tegum 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 like prismatic extrusions ("|") and bipyramidal bi-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 restrict the 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 2 cubes ||| ⊕ ||| nor its dual prism product of 2 octs ♢♢♢ × ♢♢♢ (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) 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 total 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 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 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 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. 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 normalization Vol(...|) = Vol(...) Vol(|) = 2 Vol(...) and for bi-taperings D! Vol(...♢) = (D-1)! Vol(...) 1! Vol(♢), i.e. Vol(...♢) = 2 Vol(...)/D respectively, since obviously Vol(|) = Vol(♢) = 2. But even more general it already derives from the respective products directly, as for these there are the formulae Vol(P×Q) = Vol(P) Vol(Q) and dim(P⊕Q) Vol(P⊕Q) = dim(P) Vol(P) dim(Q) Vol(Q) 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!.
| Number | Symbol | Coordinates | Incidence Matrix | Remarks |
| 1D | Mahler volume = 4 | |||
|---|---|---|---|---|
| 1.1 |
| = ♢ - edge |
|
u . | 2 |
regular
single u-edge Volume = 2 self-dual |
| 2D | Mahler volume = 8 | |||
| 2.1 |
| × | = || = ♢| - {4} |
|
u u . . | 4 | 2 -------+---+-- u . & | 2 | 4 |
regular
all u-edges Volume = 4 dual = 2.2 |
| 2.2 |
| ⊕ | = |♢ = ♢♢ - {4} |
|
uo ou&#zq o. o. & | 4 | 2 -----------+---+-- oo oo&#q | 2 | 4 |
regular
all q-edges Volume = 2 dual = 2.1 |
| 3D | Mahler volume = 32/3 = 10.666667 | |||
| 3.1 |
|| × | = ||| = ♢|| - cube |
|
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) |
|
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 dual = 3.3 |
| 3.3 |
|♢ × | = |♢| = ♢♢| - cube |
|
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 |
|
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 dual = 3.1 |
| 4D | Mahler volume = 32/3 = 10.666667 | |||
| 4.01 |
||| × | = || × || = |||| = ♢||| - tes |
|
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 |
|
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 dual = 4.07 |
| 4.03 |
||♢ × | = ||♢| = ♢|♢| - ope |
|
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 dual = 4.06 |
| 4.04 |
||♢ ⊕ | = || ⊕ |♢ = ||♢♢ = ♢|♢♢ - hex |
|
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 |
lacings: h-edges 1st factor: q-edges 2nd factor: u-edges Volume = 4/3 = 1.333333 dual = 4.05 |
| 4.05 |
|♢| × | = |♢ × || = |♢|| = ♢♢|| - tes |
|
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 dual = 4.04 |
| 4.06 |
|♢| ⊕ | = |♢|♢ = ♢♢|♢ - cute |
|
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 dual = 4.03 |
| 4.07 |
|♢♢ × | = |♢♢| = ♢♢♢| - ope |
|
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 |
|
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 dual = 4.01 |
| 4.09 |
|| ⊕ || - hex |
|
uo4oo ou4oo&#zu o.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 | 4 oo4oo 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 |
|
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 0 q . 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 |
| 5D | Mahler volume = 128/15 = 8.533333 | |||
| 5.01 |
|||| × | = ||| × || = ||||| = ♢|||| - pent |
|
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
|
|
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 dual = 5.15 all unit-edged variant is impossible |
| 5.03 |
|||♢ × | = |||♢| = ♢||♢| - cutep |
|
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 dual = 5.14 |
| 5.04 |
|||♢ ⊕ | = ||| ⊕ |♢ = |||♢♢ = ♢||♢♢ - squacubdit |
|
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 dual = 5.13 all unit-edged variant is impossible |
| 5.05 |
||♢| × | = ||♢ × || = ||♢|| = ♢|♢|| - squoct |
|
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 dual = 5.12 |
| 5.06 |
||♢| ⊕ | = ||♢|♢ = ♢|♢|♢ - opet |
|
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 dual = 5.11 |
| 5.07 |
||♢♢ × | = ||♢♢| = ♢|♢♢| - hexip |
|
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 trip var. uu .. .. .. ou&#h | 2 4 | 1 0 4 2 2 | 0 2 0 2 1 | * 16 * | 2 0 trip var. .. qo .. .. ou&#h | 2 2 | 0 1 4 0 1 | 0 0 2 2 0 | * * 32 | 1 1 tet 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 | * 2 hex var. |
lacings: h-edges 1st+3rd factor: u-edges 2nd factor: q-edges Volume = 8/3 = 2.666667 dual = 5.10 |
| 5.08 |
||♢♢ ⊕ | = ||♢ ⊕ |♢ = || ⊕ |♢♢ = ||♢♢♢ = ♢|♢♢♢ - tac |
|
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 dual = 5.09 |
| 5.09 |
|♢|| × | = |♢| × || = |♢ × ||| = |♢||| = ♢♢||| - pent |
|
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 dual = 5.08 |
| 5.10 |
|♢|| ⊕ | =
|♢||♢ =
♢♢||♢ - tessit
|
|
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 dual = 5.07 all unit-edged variant is impossible |
| 5.11 |
|♢|♢ × | = |♢|♢| = ♢♢|♢| - cutep |
|
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 dual = 5.06 |
| 5.12 |
|♢|♢ ⊕ | = |♢| ⊕ |♢ = |♢|♢♢ = ♢♢|♢♢ - squacubdit |
|
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 dual = 5.05 all unit-edged variant is impossible |
| 5.13 |
|♢♢| × | = |♢♢ × || = |♢♢|| = ♢♢♢|| - squoct |
|
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 |
|
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 dual = 5.03 |
| 5.15 |
|♢♢♢ × | = |♢♢♢| = ♢♢♢♢| - hexip |
|
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 |
|
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 dual = 5.01 |
| 5.17 |
||| × |♢ - pent |
|
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 |
|
uo oo4ou qq4oo&#zh o. 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 4 oo 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 |
|
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 0 u . . 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)-cube u q . q . | 8 | 4 4 4 | 2 2 0 2 0 | * 16 * * * | 1 1 0 (u,q,q)-cube u . . 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)-tes u 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 |
|
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 1 q . . 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 0 q3o . q . ♦ 6 | 6 3 | 2 3 0 | * 32 * | 1 1 q . . 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
|
|
uo4oo oo3oo4ou&#zy where y = sqrt(5) = 2.236068 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 ------------------+-----+---------+---------+-------+--- u. .. .. .. .. | 2 0 | 4 * * ♦ 8 0 0 | 12 0 | 6 oo4oo 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 taller squippy var. ------------------+-----+---------+---------+-------+--- uo .. .. oo4ou&#y | 2 4 | 1 8 4 | 4 8 1 | 4 2 | 24 squasc var. |
u-edges y-edges Volume = 16/5 = 3.2 dual = 5.20 all unit-edged variant is impossible |
| 5.22 |
||♢ ⊕ || - squoct |
|
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 | 8 o.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 | 4 ooo 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
|
|
oq4oo ou uo4oo&#zu o.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 4 oo4oo 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 tall squippy 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 1 tet 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 | * 16 squasc var. |
q-edges u-edges Volume = 8/5 = 1.6 dual = 5.18 all unit-edged variant is impossible |
| 5.24 |
|♢♢ ⊕ || - tac |
|
uo4oo oq3oo4oo&#zh o.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 | 8 oo4oo 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 | 32 pen var. |
q-edges h-edges u-edges Volume = 8/15 = 0.533333 dual = 5.17 |
© 2004-2024 | top of page |