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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.
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.
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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
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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
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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
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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
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3D f0 = 12 f1 = 18 f2 = 8 |
truncated tetrahedron
x3x3o . . . | 12 | 1 2 | 2 1 ------+----+------+---- x . . | 2 | 6 * | 2 0 . x . | 2 | * 12 | 1 1 ------+----+------+---- x3x . | 6 | 3 3 | 4 * . x3o | 3 | 0 3 | * 4 |
hexagonal prism
x x6o . . . | 12 | 1 2 | 2 1 ------+----+------+---- x . . | 2 | 6 * | 2 0 . x . | 2 | * 12 | 1 1 ------+----+------+---- x x . | 4 | 2 2 | 6 * . x6o | 6 | 0 6 | * 2 |
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