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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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