twins

Twin polytopes

A pair of polytopes is considered to be a twin, if its face vector vec(f) = (f₀, f₁, f₂, ...) = (V, E, F, ...), i.e. the total count of elements wrt. every single subdimension, would be exactly the same, but still the polytopes are neither somehow scaled or moved variants of each other, neither are conjugates – like {5, 5/2} (gad) and {5/2, 5} (sissid) – nor even are different realisations of the same abstract polytope.

This even could be driven to the outermost point when considering for instance the pair x3/2o5x5*a (saddid) and x5/3o3x5*a (sidditdid), which not only have all the same elements throughout but also have all the same respective incidences of types too. Therefore these well feature exactly the same incidence matrices – when considering symmetry equivalence classes. However when considering the mere 0-1-matrices instead, i.e. the true representation of the respective abstract polytopes, then it occurs that they well differ within the respective pseudo faces, which are squares in the former, hexagons in the latter case. Thus indeed, in a very limiting reading those indeed could be considered twins. But sure, the very purpose of considering twins, i.e. comparing qualitatively different polytopes with same facet vectors then would still not be met for obvious reasons: their incidence matrices (by equivalence classes of symmetry) still coincides.

Further trivial twins will be provided here only in a mere shortlist. Those then are sorts like multidiminishings at different locations and/or according gyrations.

3D f₀ = 10, f₁ = 20, f₂ = 12 → mibdi, pap
3D f₀ = 12, f₁ = 24, f₂ = 14 → co, tobcu
3D f₀ = 14, f₁ = 26, f₂ = 14 → pabauhip, mabauhip
3D f₀ = 16, f₁ = 32, f₂ = 18 → squobcu, squigybcu
3D f₀ = 18, f₁ = 36, f₂ = 20 → etobcu, etigybcu
3D f₀ = 20, f₁ = 40, f₂ = 22 → pobcu, pegybcu
3D f₀ = 22, f₁ = 40, f₂ = 20 → pabaud, mabaud
3D f₀ = 24, f₁ = 48, f₂ = 26 → sirco, esquigybcu
3D f₀ = 25, f₁ = 50, f₂ = 27 → pocuro, pegycuro
3D f₀ = 30, f₁ = 60, f₂ = 32 → epobcu, epigybcu
3D f₀ = 30, f₁ = 60, f₂ = 32 → id, pobro
3D f₀ = 35, f₁ = 70, f₂ = 37 → epocuro, epgycuro
3D f₀ = 50, f₁ = 90, f₂ = 42 → pabidrid, mabidrid, gybadrid
3D f₀ = 55, f₁ = 105, f₂ = 52 → dirid, pagydrid, magydrid
3D f₀ = 60, f₁ = 120, f₂ = 62 → srid, gyrid, pabgyrid, mabgyrid, tagyrid
3D f₀ = 70, f₁ = 120, f₂ = 52 → pabautid, mabautid

After sieving out all those various types of mutual closeness, the remainder of this page will be concerned within the following table of known "true" twin polytopes. Accordingly they then show up largely different incidence matrices – even when given by symmetry type. And for sure, this then not only occurs because of representation wrt. a different (sub)symmetry or orientation.

It shall be emphasized that whenever a pair of polytopes happens to be a twin (within any of the above readings), then clearly the pair of respective prisms or of duoprisms with any common further factor will take over the same property.

3D f₀ = 7 f₁ = 12 f₂ = 7	elongated triangular pyramid oxx3ooo&#xt → height(1,2) = sqrt(3/8) = 0.612372 height(2,3) = 1 o..3o.. \| 1 * * \| 3 0 0 0 \| 3 0 0 .o.3.o. \| * 3 * \| 1 2 1 0 \| 2 2 0 ..o3..o \| * * 3 \| 0 0 1 2 \| 0 2 1 -----------+-------+---------+------ oo.3oo.&#x \| 1 1 0 \| 3 * * * \| 2 0 0 .x. ... \| 0 2 0 \| * 3 * * \| 1 1 0 .oo3.oo&#x \| 0 1 1 \| * * 3 * \| 0 2 0 ..x ... \| 0 0 2 \| * * * 3 \| 0 1 1 -----------+-------+---------+------ ox. ...&#x \| 1 2 0 \| 2 1 0 0 \| 3 * * .xx ...&#x \| 0 2 2 \| 0 1 2 1 \| * 3 * ..x3..o \| 0 0 3 \| 0 0 0 3 \| * * 1	hexagonal hexagonal pyramid ox6oo&#y → height = sqrt(y²-1) where y > 1 tip o.6o. \| 1 * \| 6 0 \| 6 0 base .o6.o \| * 6 \| 1 2 \| 2 1 --------------+-----+-----+---- lace oo6oo&#y \| 1 1 \| 6 * \| 2 0 base .x .. \| 0 2 \| * 6 \| 1 1 --------------+-----+-----+---- coat ox ..&#y \| 1 2 \| 2 1 \| 6 * base .x6.o \| 0 6 \| 0 6 \| * 1
3D f₀ = 8 f₁ = 18 f₂ = 12	snub disphenoid xoBo oBox&#xt → outer heights = 0.578369 inner height = 0.411123 where B = 1.289169 (pseudo) o... o... & \| 4 * \| 1 2 1 0 \| 2 2 .o.. .o.. & \| * 4 \| 0 2 1 2 \| 1 4 ---------------+-----+---------+---- x... .... & \| 2 0 \| 2 * * * \| 2 0 oo.. oo..&#x & \| 1 1 \| * 8 * * \| 1 1 o.o. o.o.&#x & \| 1 1 \| * * 4 * \| 0 2 .oo. .oo.&#x \| 0 2 \| * * * 4 \| 0 2 ---------------+-----+---------+---- xo.. ....&#x & \| 2 1 \| 1 2 0 0 \| 4 * ooo. ooo.&#x & \| 1 2 \| 0 1 1 1 \| * 8	hexagonal bipyramid oxo6ooo&#yt → both heights = sqrt(y²-1) where y > 1 o..6o.. \| 1 * * \| 6 0 0 \| 6 0 .o.6.o. \| * 6 * \| 1 2 1 \| 2 2 ..o6..o \| * * 1 \| 0 0 6 \| 0 6 -----------+-------+-------+---- oo.6oo.&#y \| 1 1 0 \| 6 * * \| 2 0 .x. ... \| 0 2 0 \| * 6 * \| 1 1 .oo6.oo&#y \| 0 1 1 \| * * 6 \| 0 2 -----------+-------+-------+---- ox. ...&#y \| 1 2 0 \| 2 1 0 \| 6 * .xo ...&#y \| 0 2 1 \| 0 1 2 \| * 6
3D f₀ = 12 f₁ = 24 f₂ = 14	cuboctahedron o3x4o o3o4o \| 12 \| 4 \| 2 2 ------+----+----+---- . x . \| 2 \| 24 \| 1 1 ------+----+----+---- o3x . \| 3 \| 3 \| 8 * . x4o \| 4 \| 4 \| * 6 (or trigonal orthobicupola, cf. above)	hexagonal antiprism xo6ox&#x → height = sqrt[sqrt(3)-1] = 0.855600 o.6o. \| 6 * \| 2 2 0 \| 1 2 1 0 .o6.o \| * 6 \| 0 2 2 \| 0 1 2 1 ---------+-----+--------+-------- x. .. \| 2 0 \| 6 * * \| 1 1 0 0 oo6oo&#x \| 1 1 \| * 12 * \| 0 1 1 0 .. .x \| 0 2 \| * * 6 \| 0 0 1 1 ---------+-----+--------+-------- x.6o. \| 6 0 \| 6 0 0 \| 1 * * * xo ..&#x \| 2 1 \| 1 2 0 \| * 6 * * .. ox&#x \| 1 2 \| 0 2 1 \| * * 6 * .o6.x \| 0 6 \| 0 0 6 \| * * * 1
3D f₀ = 12 f₁ = 30 f₂ = 20	icosahedron x3o5o o3o5o \| 12 \| 5 \| 5 ------+----+----+--- x . . \| 2 \| 30 \| 2 ------+----+----+--- x3o . \| 3 \| 3 \| 20	decagonal bipyramid oxo10ooo&#yt → both heights = sqrt(y²-f²) where y > f = (1+sqrt(5))/2 = 1.618034 o..10o.. \| 1 * * \| 10 0 0 \| 10 0 .o.10.o. \| * 10 * \| 1 2 1 \| 2 2 ..o10..o \| * * 1 \| 0 0 10 \| 0 10 ------------+--------+----------+------ oo.10oo.&#y \| 1 1 0 \| 10 * * \| 2 0 .x. ... \| 0 2 0 \| * 10 * \| 1 1 .oo10.oo&#y \| 0 1 1 \| * * 10 \| 0 2 ------------+--------+----------+------ ox. ...&#y \| 1 2 0 \| 2 1 0 \| 10 * .xo ...&#y \| 0 2 1 \| 0 1 2 \| * 10
3D f₀ = 14 f₁ = 26 f₂ = 14	bilunabirotunda xfofx oxfxo&#xt → outer heights = (1+sqrt(5))/4 = 0.809017 inner heights = 1/2 o.... o.... & \| 4 * * \| 1 2 0 0 0 \| 1 2 0 0 .o... .o... & \| * 8 * \| 0 1 1 1 1 \| 1 1 1 1 ..o.. ..o.. \| * * 2 \| 0 0 0 4 0 \| 0 2 0 2 -------------------+-------+-----------+-------- x.... ..... & \| 2 0 0 \| 2 * * * * \| 0 2 0 0 oo... oo...&#x & \| 1 1 0 \| * 8 * * * \| 1 1 0 0 ..... .x... & \| 0 2 0 \| * * 4 * * \| 1 0 1 0 .oo.. .oo..&#x & \| 0 1 1 \| * * * 8 * \| 0 1 0 1 .o.o. .o.o.&#x \| 0 2 0 \| * * * * 4 \| 0 0 1 1 -------------------+-------+-----------+-------- ..... ox...&#x & \| 1 2 0 \| 0 2 1 0 0 \| 4 * * * {3} xfo.. .....&#xt & \| 2 2 1 \| 1 2 0 2 0 \| * 4 * * {5} ..... .x.x.&#x \| 0 4 0 \| 0 0 2 0 2 \| * * 2 * {4} .ooo. .ooo.&#xt \| 0 2 1 \| 0 0 0 2 1 \| * * * 4 {3}	parabiaugmented hexagonal prism oxxxo oxuxo&#xt → outer heights = 1/sqrt(2) = 0.707107 inner heights = sqrt(3)/2 = 0.866025 o.... o.... & \| 2 * * \| 4 0 0 0 0 \| 2 2 0 0 .o... .o... & \| * 8 * \| 1 1 1 1 0 \| 1 1 1 1 ..o.. ..o.. \| * * 4 \| 0 0 0 2 1 \| 0 0 2 1 -------------------+-------+-----------+-------- oo... oo...&#x & \| 1 1 0 \| 8 * * * * \| 1 1 0 0 .x... ..... & \| 0 2 0 \| * 4 * * * \| 1 0 1 0 ..... .x... & \| 0 2 0 \| * * 4 * * \| 0 1 0 1 .oo.. .oo..&#x & \| 0 1 1 \| * * * 8 * \| 0 0 1 1 ..x.. ..... \| 0 0 2 \| * * * * 2 \| 0 0 2 0 -------------------+-------+-----------+-------- ox... .....&#x & \| 1 2 0 \| 2 1 0 0 0 \| 4 * * * {3} ..... ox...&#x & \| 1 2 0 \| 2 0 1 0 0 \| * 4 * * {3} .xx.. .....&#x & \| 0 2 2 \| 0 1 0 2 1 \| * * 4 * {4} ..... .xux.&#xt \| 0 4 2 \| 0 0 2 4 0 \| * * * 2 {6} (or metabiaugmented ..., cf. above)
3D f₀ = 20 f₁ = 30 f₂ = 12	dodecahedron o3o5x o3o5o \| 20 \| 3 \| 3 ------+----+----+--- . . x \| 2 \| 30 \| 2 ------+----+----+--- . o5x \| 5 \| 5 \| 12	decagonal prism x x10o o o10o \| 20 \| 1 2 \| 2 1 -------+----+-------+----- x . . \| 2 \| 10 * \| 2 0 . x . \| 2 \| * 20 \| 1 1 -------+----+-------+----- x x . \| 4 \| 2 2 \| 10 * . x10o \| 10 \| 0 10 \| * 2
3D f₀ = 24 f₁ = 36 f₂ = 14	truncated cube o3x4x . . . \| 24 \| 2 1 \| 1 2 ------+----+-------+---- . x . \| 2 \| 24 * \| 1 1 . . x \| 2 \| * 12 \| 0 2 ------+----+-------+---- o3x . \| 3 \| 3 0 \| 8 * . x4x \| 8 \| 4 4 \| * 6	truncated octahedron x3x4o . . . \| 24 \| 1 2 \| 2 1 ------+----+-------+---- x . . \| 2 \| 12 * \| 2 0 . x . \| 2 \| * 24 \| 1 1 ------+----+-------+---- x3x . \| 6 \| 3 3 \| 8 * . x4o \| 4 \| 0 4 \| * 6
3D f₀ = 30 f₁ = 60 f₂ = 32	icosidodecahedron o3x5o o3o5o \| 30 \| 4 \| 2 2 ------+----+----+------ . x . \| 2 \| 60 \| 1 1 ------+----+----+------ o3x . \| 3 \| 3 \| 20 * . x5o \| 5 \| 5 \| * 12 (or pentagonal orthobirotunda, cf. above)	elongated pentagonal orthobicupola xxxx5oxxo&#xt → outer heights = sqrt((5-sqrt(5))/10) = 0.525731 inner height = 1 o...5o... & \| 10 * \| 2 2 0 0 0 \| 1 2 1 0 0 .o..5.o.. & \| * 20 \| 0 1 1 1 1 \| 0 1 1 1 1 ---------------+-------+----------------+------------ x... .... & \| 2 0 \| 10 * * * * \| 1 1 0 0 0 oo..5oo..&#x & \| 1 1 \| * 20 * * * \| 0 1 1 0 0 .x.. .... & \| 0 2 \| * * 10 * * \| 0 1 0 1 0 .... .x.. & \| 0 2 \| * * * 10 * \| 0 0 1 0 1 .oo.5.oo.&#x \| 0 2 \| * * * * 10 \| 0 0 0 1 1 ---------------+-------+----------------+------------ x...5o... & \| 5 0 \| 5 0 0 0 0 \| 2 * * * * xx.. ....&#x & \| 2 2 \| 1 2 1 0 0 \| * 10 * * * .... ox..&#x & \| 1 2 \| 0 2 0 1 0 \| * * 10 * * .xx. ....&#x \| 0 4 \| 0 0 2 0 2 \| * * * 5 * .... .xx.&#x \| 0 4 \| 0 0 0 2 2 \| * * * * 5 (or ... gyrobicupola, cf. above)
3D f₀ = 60 f₁ = 90 f₂ = 32	truncated icosahedron x3x5o o3o5o \| 60 \| 1 2 \| 2 1 ------+----+-------+------ x . . \| 2 \| 30 * \| 2 0 . x . \| 2 \| * 60 \| 1 1 ------+----+-------+------ x3x . \| 6 \| 3 3 \| 20 * . x5o \| 5 \| 0 5 \| * 12	truncated dodecahedron o3x5x . . . \| 60 \| 2 1 \| 1 2 ------+----+-------+------ . x . \| 2 \| 60 * \| 1 1 . . x \| 2 \| * 30 \| 0 2 ------+----+-------+------ o3x . \| 3 \| 3 0 \| 20 * . x5x \| 10 \| 5 5 \| * 12
4D f₀ = 2400 f₁ = 10800 f₂ = 7680 f₃ = 960	small tritrigonal trishecatonicosihexacosachoron x3x3o3o5/4a5/2c . . . . \| 2400 \| 6 3 \| 6 3 3 3 \| 3 3 1 1 ------------------+------+-----------+---------------------+---------------- x . . . \| 2 \| 7200 * \| 1 1 1 0 \| 1 1 1 0 . x . . \| 2 \| * 3600 \| 2 0 0 2 \| 2 1 0 1 ------------------+------+-----------+---------------------+---------------- x3x . . \| 6 \| 3 3 \| 2400 * * * \| 1 1 0 0 x . o . a5/2c \| 5 \| 5 0 \| * 1440 * * \| 1 0 1 0 x . . o5/4a \| 5 \| 5 0 \| * 1440 * \| 0 1 1 0 . x3o . \| 3 \| 0 3 \| * * * 2400 \| 1 0 0 1 ------------------+------+-----------+---------------------+---------------- x3x3o . a5/2c ♦ 60 \| 60 60 \| 20 12 0 20 \| 120 * * * x3x . o5/4a ♦ 60 \| 60 30 \| 20 0 12 0 \| 120 * * x . o3o5/4a5/2c ♦ 20 \| 60 0 \| 0 12 12 0 \| * * 120 * . x3o3o ♦ 4 \| 0 6 \| 0 0 0 4 \| * * * 600	small tritrigonal hexacositrishecatonicosachoron o3o3x5x3a5/4c . . . . \| 2400 \| 6 3 \| 3 3 3 6 \| 1 1 3 3 ----------------+------+-----------+---------------------+---------------- . . x . \| 2 \| 7200 * \| 1 0 1 1 \| 1 0 1 1 . . . x \| 2 \| * 3600 \| 0 2 0 2 \| 0 1 2 1 ----------------+------+-----------+---------------------+---------------- o . x . a5/4c \| 5 \| 5 0 \| 1440 * * * \| 1 0 1 0 o . . x3a \| 3 \| 0 3 \| 2400 * * \| 0 1 1 0 . o3x . \| 3 \| 3 0 \| * * 2400 * \| 1 0 0 1 . . x5x \| 10 \| 5 5 \| * * * 1440 \| 0 0 1 1 ----------------+------+-----------+---------------------+---------------- o3o3x . a5/4c ♦ 20 \| 60 0 \| 12 0 20 0 \| 120 * * * o3o . x3a ♦ 4 \| 0 6 \| 0 4 0 0 \| 600 * * o . x5x3a5/4c ♦ 60 \| 60 60 \| 12 20 0 12 \| * * 120 * . o3x5x ♦ 60 \| 60 30 \| 0 0 20 12 \| * * * 120

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3D f₀ = 7 f₁ = 12 f₂ = 7	elongated triangular pyramid oxx3ooo&#xt → height(1,2) = sqrt(3/8) = 0.612372 height(2,3) = 1 o..3o.. \| 1 * * \| 3 0 0 0 \| 3 0 0 .o.3.o. \| * 3 * \| 1 2 1 0 \| 2 2 0 ..o3..o \| * * 3 \| 0 0 1 2 \| 0 2 1 -----------+-------+---------+------ oo.3oo.&#x \| 1 1 0 \| 3 * * * \| 2 0 0 .x. ... \| 0 2 0 \| * 3 * * \| 1 1 0 .oo3.oo&#x \| 0 1 1 \| * * 3 * \| 0 2 0 ..x ... \| 0 0 2 \| * * * 3 \| 0 1 1 -----------+-------+---------+------ ox. ...&#x \| 1 2 0 \| 2 1 0 0 \| 3 * * .xx ...&#x \| 0 2 2 \| 0 1 2 1 \| * 3 * ..x3..o \| 0 0 3 \| 0 0 0 3 \| * * 1	hexagonal hexagonal pyramid ox6oo&#y → height = sqrt(y²-1) where y > 1 tip o.6o. \| 1 * \| 6 0 \| 6 0 base .o6.o \| * 6 \| 1 2 \| 2 1 --------------+-----+-----+---- lace oo6oo&#y \| 1 1 \| 6 * \| 2 0 base .x .. \| 0 2 \| * 6 \| 1 1 --------------+-----+-----+---- coat ox ..&#y \| 1 2 \| 2 1 \| 6 * base .x6.o \| 0 6 \| 0 6 \| * 1
3D f₀ = 8 f₁ = 18 f₂ = 12	snub disphenoid xoBo oBox&#xt → outer heights = 0.578369 inner height = 0.411123 where B = 1.289169 (pseudo) o... o... & \| 4 * \| 1 2 1 0 \| 2 2 .o.. .o.. & \| * 4 \| 0 2 1 2 \| 1 4 ---------------+-----+---------+---- x... .... & \| 2 0 \| 2 * * * \| 2 0 oo.. oo..&#x & \| 1 1 \| * 8 * * \| 1 1 o.o. o.o.&#x & \| 1 1 \| * * 4 * \| 0 2 .oo. .oo.&#x \| 0 2 \| * * * 4 \| 0 2 ---------------+-----+---------+---- xo.. ....&#x & \| 2 1 \| 1 2 0 0 \| 4 * ooo. ooo.&#x & \| 1 2 \| 0 1 1 1 \| * 8	hexagonal bipyramid oxo6ooo&#yt → both heights = sqrt(y²-1) where y > 1 o..6o.. \| 1 * * \| 6 0 0 \| 6 0 .o.6.o. \| * 6 * \| 1 2 1 \| 2 2 ..o6..o \| * * 1 \| 0 0 6 \| 0 6 -----------+-------+-------+---- oo.6oo.&#y \| 1 1 0 \| 6 * * \| 2 0 .x. ... \| 0 2 0 \| * 6 * \| 1 1 .oo6.oo&#y \| 0 1 1 \| * * 6 \| 0 2 -----------+-------+-------+---- ox. ...&#y \| 1 2 0 \| 2 1 0 \| 6 * .xo ...&#y \| 0 2 1 \| 0 1 2 \| * 6
3D f₀ = 12 f₁ = 24 f₂ = 14	cuboctahedron o3x4o o3o4o \| 12 \| 4 \| 2 2 ------+----+----+---- . x . \| 2 \| 24 \| 1 1 ------+----+----+---- o3x . \| 3 \| 3 \| 8 * . x4o \| 4 \| 4 \| * 6 (or trigonal orthobicupola, cf. above)	hexagonal antiprism xo6ox&#x → height = sqrt[sqrt(3)-1] = 0.855600 o.6o. \| 6 * \| 2 2 0 \| 1 2 1 0 .o6.o \| * 6 \| 0 2 2 \| 0 1 2 1 ---------+-----+--------+-------- x. .. \| 2 0 \| 6 * * \| 1 1 0 0 oo6oo&#x \| 1 1 \| * 12 * \| 0 1 1 0 .. .x \| 0 2 \| * * 6 \| 0 0 1 1 ---------+-----+--------+-------- x.6o. \| 6 0 \| 6 0 0 \| 1 * * * xo ..&#x \| 2 1 \| 1 2 0 \| * 6 * * .. ox&#x \| 1 2 \| 0 2 1 \| * * 6 * .o6.x \| 0 6 \| 0 0 6 \| * * * 1
3D f₀ = 12 f₁ = 30 f₂ = 20	icosahedron x3o5o o3o5o \| 12 \| 5 \| 5 ------+----+----+--- x . . \| 2 \| 30 \| 2 ------+----+----+--- x3o . \| 3 \| 3 \| 20	decagonal bipyramid oxo10ooo&#yt → both heights = sqrt(y²-f²) where y > f = (1+sqrt(5))/2 = 1.618034 o..10o.. \| 1 * * \| 10 0 0 \| 10 0 .o.10.o. \| * 10 * \| 1 2 1 \| 2 2 ..o10..o \| * * 1 \| 0 0 10 \| 0 10 ------------+--------+----------+------ oo.10oo.&#y \| 1 1 0 \| 10 * * \| 2 0 .x. ... \| 0 2 0 \| * 10 * \| 1 1 .oo10.oo&#y \| 0 1 1 \| * * 10 \| 0 2 ------------+--------+----------+------ ox. ...&#y \| 1 2 0 \| 2 1 0 \| 10 * .xo ...&#y \| 0 2 1 \| 0 1 2 \| * 10
3D f₀ = 14 f₁ = 26 f₂ = 14	bilunabirotunda xfofx oxfxo&#xt → outer heights = (1+sqrt(5))/4 = 0.809017 inner heights = 1/2 o.... o.... & \| 4 * * \| 1 2 0 0 0 \| 1 2 0 0 .o... .o... & \| * 8 * \| 0 1 1 1 1 \| 1 1 1 1 ..o.. ..o.. \| * * 2 \| 0 0 0 4 0 \| 0 2 0 2 -------------------+-------+-----------+-------- x.... ..... & \| 2 0 0 \| 2 * * * * \| 0 2 0 0 oo... oo...&#x & \| 1 1 0 \| * 8 * * * \| 1 1 0 0 ..... .x... & \| 0 2 0 \| * * 4 * * \| 1 0 1 0 .oo.. .oo..&#x & \| 0 1 1 \| * * * 8 * \| 0 1 0 1 .o.o. .o.o.&#x \| 0 2 0 \| * * * * 4 \| 0 0 1 1 -------------------+-------+-----------+-------- ..... ox...&#x & \| 1 2 0 \| 0 2 1 0 0 \| 4 * * * {3} xfo.. .....&#xt & \| 2 2 1 \| 1 2 0 2 0 \| * 4 * * {5} ..... .x.x.&#x \| 0 4 0 \| 0 0 2 0 2 \| * * 2 * {4} .ooo. .ooo.&#xt \| 0 2 1 \| 0 0 0 2 1 \| * * * 4 {3}	parabiaugmented hexagonal prism oxxxo oxuxo&#xt → outer heights = 1/sqrt(2) = 0.707107 inner heights = sqrt(3)/2 = 0.866025 o.... o.... & \| 2 * * \| 4 0 0 0 0 \| 2 2 0 0 .o... .o... & \| * 8 * \| 1 1 1 1 0 \| 1 1 1 1 ..o.. ..o.. \| * * 4 \| 0 0 0 2 1 \| 0 0 2 1 -------------------+-------+-----------+-------- oo... oo...&#x & \| 1 1 0 \| 8 * * * * \| 1 1 0 0 .x... ..... & \| 0 2 0 \| * 4 * * * \| 1 0 1 0 ..... .x... & \| 0 2 0 \| * * 4 * * \| 0 1 0 1 .oo.. .oo..&#x & \| 0 1 1 \| * * * 8 * \| 0 0 1 1 ..x.. ..... \| 0 0 2 \| * * * * 2 \| 0 0 2 0 -------------------+-------+-----------+-------- ox... .....&#x & \| 1 2 0 \| 2 1 0 0 0 \| 4 * * * {3} ..... ox...&#x & \| 1 2 0 \| 2 0 1 0 0 \| * 4 * * {3} .xx.. .....&#x & \| 0 2 2 \| 0 1 0 2 1 \| * * 4 * {4} ..... .xux.&#xt \| 0 4 2 \| 0 0 2 4 0 \| * * * 2 {6} (or metabiaugmented ..., cf. above)
3D f₀ = 20 f₁ = 30 f₂ = 12	dodecahedron o3o5x o3o5o \| 20 \| 3 \| 3 ------+----+----+--- . . x \| 2 \| 30 \| 2 ------+----+----+--- . o5x \| 5 \| 5 \| 12	decagonal prism x x10o o o10o \| 20 \| 1 2 \| 2 1 -------+----+-------+----- x . . \| 2 \| 10 * \| 2 0 . x . \| 2 \| * 20 \| 1 1 -------+----+-------+----- x x . \| 4 \| 2 2 \| 10 * . x10o \| 10 \| 0 10 \| * 2
3D f₀ = 24 f₁ = 36 f₂ = 14	truncated cube o3x4x . . . \| 24 \| 2 1 \| 1 2 ------+----+-------+---- . x . \| 2 \| 24 * \| 1 1 . . x \| 2 \| * 12 \| 0 2 ------+----+-------+---- o3x . \| 3 \| 3 0 \| 8 * . x4x \| 8 \| 4 4 \| * 6	truncated octahedron x3x4o . . . \| 24 \| 1 2 \| 2 1 ------+----+-------+---- x . . \| 2 \| 12 * \| 2 0 . x . \| 2 \| * 24 \| 1 1 ------+----+-------+---- x3x . \| 6 \| 3 3 \| 8 * . x4o \| 4 \| 0 4 \| * 6
3D f₀ = 30 f₁ = 60 f₂ = 32	icosidodecahedron o3x5o o3o5o \| 30 \| 4 \| 2 2 ------+----+----+------ . x . \| 2 \| 60 \| 1 1 ------+----+----+------ o3x . \| 3 \| 3 \| 20 * . x5o \| 5 \| 5 \| * 12 (or pentagonal orthobirotunda, cf. above)	elongated pentagonal orthobicupola xxxx5oxxo&#xt → outer heights = sqrt((5-sqrt(5))/10) = 0.525731 inner height = 1 o...5o... & \| 10 * \| 2 2 0 0 0 \| 1 2 1 0 0 .o..5.o.. & \| * 20 \| 0 1 1 1 1 \| 0 1 1 1 1 ---------------+-------+----------------+------------ x... .... & \| 2 0 \| 10 * * * * \| 1 1 0 0 0 oo..5oo..&#x & \| 1 1 \| * 20 * * * \| 0 1 1 0 0 .x.. .... & \| 0 2 \| * * 10 * * \| 0 1 0 1 0 .... .x.. & \| 0 2 \| * * * 10 * \| 0 0 1 0 1 .oo.5.oo.&#x \| 0 2 \| * * * * 10 \| 0 0 0 1 1 ---------------+-------+----------------+------------ x...5o... & \| 5 0 \| 5 0 0 0 0 \| 2 * * * * xx.. ....&#x & \| 2 2 \| 1 2 1 0 0 \| * 10 * * * .... ox..&#x & \| 1 2 \| 0 2 0 1 0 \| * * 10 * * .xx. ....&#x \| 0 4 \| 0 0 2 0 2 \| * * * 5 * .... .xx.&#x \| 0 4 \| 0 0 0 2 2 \| * * * * 5 (or ... gyrobicupola, cf. above)
3D f₀ = 60 f₁ = 90 f₂ = 32	truncated icosahedron x3x5o o3o5o \| 60 \| 1 2 \| 2 1 ------+----+-------+------ x . . \| 2 \| 30 * \| 2 0 . x . \| 2 \| * 60 \| 1 1 ------+----+-------+------ x3x . \| 6 \| 3 3 \| 20 * . x5o \| 5 \| 0 5 \| * 12	truncated dodecahedron o3x5x . . . \| 60 \| 2 1 \| 1 2 ------+----+-------+------ . x . \| 2 \| 60 * \| 1 1 . . x \| 2 \| * 30 \| 0 2 ------+----+-------+------ o3x . \| 3 \| 3 0 \| 20 * . x5x \| 10 \| 5 5 \| * 12
4D f₀ = 2400 f₁ = 10800 f₂ = 7680 f₃ = 960	small tritrigonal trishecatonicosihexacosachoron x3x3o3o5/4a5/2c . . . . \| 2400 \| 6 3 \| 6 3 3 3 \| 3 3 1 1 ------------------+------+-----------+---------------------+---------------- x . . . \| 2 \| 7200 * \| 1 1 1 0 \| 1 1 1 0 . x . . \| 2 \| * 3600 \| 2 0 0 2 \| 2 1 0 1 ------------------+------+-----------+---------------------+---------------- x3x . . \| 6 \| 3 3 \| 2400 * * * \| 1 1 0 0 x . o . a5/2c \| 5 \| 5 0 \| * 1440 * * \| 1 0 1 0 x . . o5/4a \| 5 \| 5 0 \| * 1440 * \| 0 1 1 0 . x3o . \| 3 \| 0 3 \| * * * 2400 \| 1 0 0 1 ------------------+------+-----------+---------------------+---------------- x3x3o . a5/2c ♦ 60 \| 60 60 \| 20 12 0 20 \| 120 * * * x3x . o5/4a ♦ 60 \| 60 30 \| 20 0 12 0 \| 120 * * x . o3o5/4a5/2c ♦ 20 \| 60 0 \| 0 12 12 0 \| * * 120 * . x3o3o ♦ 4 \| 0 6 \| 0 0 0 4 \| * * * 600	small tritrigonal hexacositrishecatonicosachoron o3o3x5x3a5/4c . . . . \| 2400 \| 6 3 \| 3 3 3 6 \| 1 1 3 3 ----------------+------+-----------+---------------------+---------------- . . x . \| 2 \| 7200 * \| 1 0 1 1 \| 1 0 1 1 . . . x \| 2 \| * 3600 \| 0 2 0 2 \| 0 1 2 1 ----------------+------+-----------+---------------------+---------------- o . x . a5/4c \| 5 \| 5 0 \| 1440 * * * \| 1 0 1 0 o . . x3a \| 3 \| 0 3 \| 2400 * * \| 0 1 1 0 . o3x . \| 3 \| 3 0 \| * * 2400 * \| 1 0 0 1 . . x5x \| 10 \| 5 5 \| * * * 1440 \| 0 0 1 1 ----------------+------+-----------+---------------------+---------------- o3o3x . a5/4c ♦ 20 \| 60 0 \| 12 0 20 0 \| 120 * * * o3o . x3a ♦ 4 \| 0 6 \| 0 4 0 0 \| 600 * * o . x5x3a5/4c ♦ 60 \| 60 60 \| 12 20 0 12 \| * * 120 * . o3x5x ♦ 60 \| 60 30 \| 0 0 20 12 \| * * * 120