### Twin polytopes

Two different realisations of the same abstract polytope are called to be isomorphic. Examples would be {5, 5/2} (gad) and {5/2, 5} (sissid). Accordingly their incidence matrices always will be alike.

In contrast to the former a pair of polytopes is considered to be a twin, if its facet 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 not isomorphic. Thence they will have different incidences, not isomorphic facet types, etc.; in short: they will require for qualitatively different incidence matrices. (The restriction "qualitatively" here is inserted in order to exclude potential differences which only would be due to representations wrt. different subsymmetries.)

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

The following table then is an explicite listing of some known true twin polytopes.

 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 ``` 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 = 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)

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