x5o           x5o                 
                        x5x                        
                                                   
            o5f                     o5f            
     x5o                                   x5o     
                        F5o                        
                 x5f           x5f                 
                                                   
     x5x                                   x5x     
            F5o                     F5o            
                                                   
x5o                                             x5o
                                                   
            x5f                     x5f            
     o5f                                   o5f     
                                                   
                 F5o           F5o                 
                        x5f                        
     x5o                                   x5o     
            x5x                     x5x            
                                                   
                        o5f                        
                 x5o           x5o                 	where F = ff = f+x

This polychoron occurs as a triacontadiminishing of oFxxox5Fofxfo oxofox5ooxofx&#zx. In fact, the 1st layer of that tegum sum representation is dropped, representing the vertex set of an inscribed F-{5}. The relation of that un-diminished one to ex runs as follows: ex in pentagonal duoprismatic subsymmetry can be given as xfooxo5xofxoo oxofox5ooxofx&#zx. That will be transformed into (-x)fooxo5Fofxoo oxofox5ooxofx&#zx. And finally into (-x)foo(-x)o5Fofxfo oxofox5ooxofx&#zx. Then a Stott expansion wrt. the first node produces that un-diminished polychoron.

The relation of this polychoron to cydadex can be seen when displaying that one in this symmetry as fooxo5ofxoo xofox5oxofx&#zx, then rewriting the same vertex set as foo(-x)o5ofxfo xofox5oxofx&#zx and thereafter applying a Stott expansion to the first nodes each. – In fact, the diminishing, which obtains cydadex from ex or which obtains this polychoron from oFxxox5Fofxfo oxofox5ooxofx&#zx do commutate with the here resp. above described expanded kaleido-facetings.

Incidence matrix according to Dynkin symbol

Fxxox5ofxfo xofox5oxofx&#zx   → all existing heights = 0

o....5o.... o....5o....     | 25  *  *  *  * |  2   4  2  0  0   0  0  0  0   0   0   0  0  0  0 | 1  4  2  1   4  0  0  0  0  0   0  0  0  0  0   0  0  0  0  0  0 |  2  2  2 0  0  0  0  0  0  0 0 0
.o...5.o... .o...5.o...     |  * 50  *  *  * |  0   2  0  1  2   2  1  0  0   0   0   0  0  0  0 | 0  1  2  0   2  2  1  2  2  1   2  0  0  0  0   0  0  0  0  0  0 |  1  1  2 1  2  2  0  0  0  0 0 0
..o..5..o.. ..o..5..o..     |  *  * 50  *  * |  0   0  1  0  0   2  0  1  1   2   2   0  0  0  0 | 0  0  0  1   2  0  0  2  1  0   2  2  2  2  1   2  0  0  0  0  0 |  0  2  1 0  1  2  2  1  1  0 0 0
...o.5...o. ...o.5...o.     |  *  *  * 25  * |  0   0  0  0  0   0  2  0  0   4   0   4  0  0  0 | 0  0  0  0   0  0  0  0  0  1   4  2  0  0  0   4  2  2  0  0  0 |  0  2  0 0  0  2  2  0  0  1 0 0
....o5....o ....o5....o     |  *  *  *  * 50 |  0   0  0  0  0   0  0  0  0   0   2   2  2  1  1 | 0  0  0  0   0  0  0  0  0  0   0  0  2  1  2   2  2  2  1  2  2 |  0  1  0 0  0  0  2  2  1  2 1 1
----------------------------+----------------+---------------------------------------------------+------------------------------------------------------------------+---------------------------------
..... ..... x.... .....     |  2  0  0  0  0 | 25   *  *  *  *   *  *  *  *   *   *   *  *  *  * | 1  2  0  0   0  0  0  0  0  0   0  0  0  0  0   0  0  0  0  0  0 |  2  1  0 0  0  0  0  0  0  0 0 0
oo...5oo... oo...5oo...&#x  |  1  1  0  0  0 |  * 100  *  *  *   *  *  *  *   *   *   *  *  *  * | 0  1  1  0   1  0  0  0  0  0   0  0  0  0  0   0  0  0  0  0  0 |  1  1  1 0  0  0  0  0  0  0 0 0
o.o..5o.o.. o.o..5o.o..&#x  |  1  0  1  0  0 |  *   * 50  *  *   *  *  *  *   *   *   *  *  *  * | 0  0  0  1   2  0  0  0  0  0   0  0  0  0  0   0  0  0  0  0  0 |  0  2  1 0  0  0  0  0  0  0 0 0
.x... ..... ..... .....     |  0  2  0  0  0 |  *   *  * 25  *   *  *  *  *   *   *   *  *  *  * | 0  0  0  0   0  2  0  2  0  1   0  0  0  0  0   0  0  0  0  0  0 |  0  0  0 1  2  2  0  0  0  0 0 0
..... ..... ..... .x...     |  0  2  0  0  0 |  *   *  *  * 50   *  *  *  *   *   *   *  *  *  * | 0  0  1  0   0  1  1  0  1  0   0  0  0  0  0   0  0  0  0  0  0 |  1  0  1 1  1  0  0  0  0  0 0 0
.oo..5.oo.. .oo..5.oo..&#x  |  0  1  1  0  0 |  *   *  *  *  * 100  *  *  *   *   *   *  *  *  * | 0  0  0  0   1  0  0  1  1  0   1  0  0  0  0   0  0  0  0  0  0 |  0  1  1 0  1  1  0  0  0  0 0 0
.o.o.5.o.o. .o.o.5.o.o.&#x  |  0  1  0  1  0 |  *   *  *  *  *   * 50  *  *   *   *   *  *  *  * | 0  0  0  0   0  0  0  0  0  1   2  0  0  0  0   0  0  0  0  0  0 |  0  1  0 0  0  2  0  0  0  0 0 0
..x.. ..... ..... .....     |  0  0  2  0  0 |  *   *  *  *  *   *  * 25  *   *   *   *  *  *  * | 0  0  0  0   0  0  0  2  0  0   0  2  2  0  0   0  0  0  0  0  0 |  0  0  0 0  1  2  2  1  0  0 0 0
..... ..x.. ..... .....     |  0  0  2  0  0 |  *   *  *  *  *   *  *  * 25   *   *   *  *  *  * | 0  0  0  1   0  0  0  0  0  0   0  0  0  2  0   0  0  0  0  0  0 |  0  2  0 0  0  0  0  0  1  0 0 0
..oo.5..oo. ..oo.5..oo.&#x  |  0  0  1  1  0 |  *   *  *  *  *   *  *  *  * 100   *   *  *  *  * | 0  0  0  0   0  0  0  0  0  0   1  1  0  0  0   1  0  0  0  0  0 |  0  1  0 0  0  1  1  0  0  0 0 0
..o.o5..o.o ..o.o5..o.o&#x  |  0  0  1  0  1 |  *   *  *  *  *   *  *  *  *   * 100   *  *  *  * | 0  0  0  0   0  0  0  0  0  0   0  0  1  1  1   1  0  0  0  0  0 |  0  1  0 0  0  0  1  1  1  0 0 0
...oo5...oo ...oo5...oo&#x  |  0  0  0  1  1 |  *   *  *  *  *   *  *  *  *   *   * 100  *  *  * | 0  0  0  0   0  0  0  0  0  0   0  0  0  0  0   1  1  1  0  0  0 |  0  1  0 0  0  0  1  0  0  1 0 0
....x ..... ..... .....     |  0  0  0  0  2 |  *   *  *  *  *   *  *  *  *   *   *   * 50  *  * | 0  0  0  0   0  0  0  0  0  0   0  0  1  0  0   0  1  0  1  1  1 |  0  0  0 0  0  0  1  1  0  1 1 1
..... ..... ....x .....     |  0  0  0  0  2 |  *   *  *  *  *   *  *  *  *   *   *   *  * 25  * | 0  0  0  0   0  0  0  0  0  0   0  0  0  0  0   0  0  2  0  2  0 |  0  1  0 0  0  0  0  0  0  2 1 0
..... ..... ..... ....x     |  0  0  0  0  2 |  *   *  *  *  *   *  *  *  *   *   *   *  *  * 25 | 0  0  0  0   0  0  0  0  0  0   0  0  0  0  2   0  0  0  0  0  2 |  0  0  0 0  0  0  0  2  1  0 0 1
----------------------------+----------------+---------------------------------------------------+------------------------------------------------------------------+---------------------------------
..... ..... x....5o....     |  5  0  0  0  0 |  5   0  0  0  0   0  0  0  0   0   0   0  0  0  0 | 5  *  *  *   *  *  *  *  *  *   *  *  *  *  *   *  *  *  *  *  * |  2  0  0 0  0  0  0  0  0  0 0 0
..... ..... xo... .....&#x  |  2  1  0  0  0 |  1   2  0  0  0   0  0  0  0   0   0   0  0  0  0 | * 50  *  *   *  *  *  *  *  *   *  *  *  *  *   *  *  *  *  *  * |  1  1  0 0  0  0  0  0  0  0 0 0
..... ..... ..... ox...&#x  |  1  2  0  0  0 |  0   2  0  0  1   0  0  0  0   0   0   0  0  0  0 | *  * 50  *   *  *  *  *  *  *   *  *  *  *  *   *  *  *  *  *  * |  1  0  1 0  0  0  0  0  0  0 0 0
..... o.x.. ..... .....&#x  |  1  0  2  0  0 |  0   0  2  0  0   0  0  0  1   0   0   0  0  0  0 | *  *  * 25   *  *  *  *  *  *   *  *  *  *  *   *  *  *  *  *  * |  0  2  0 0  0  0  0  0  0  0 0 0
ooo..5ooo.. ooo..5ooo..&#x  |  1  1  1  0  0 |  0   1  1  0  0   1  0  0  0   0   0   0  0  0  0 | *  *  *  * 100  *  *  *  *  *   *  *  *  *  *   *  *  *  *  *  * |  0  1  1 0  0  0  0  0  0  0 0 0
.x... ..... ..... .x...     |  0  4  0  0  0 |  0   0  0  2  2   0  0  0  0   0   0   0  0  0  0 | *  *  *  *   * 25  *  *  *  *   *  *  *  *  *   *  *  *  *  *  * |  0  0  0 1  1  0  0  0  0  0 0 0
..... ..... .o...5.x...     |  0  5  0  0  0 |  0   0  0  0  5   0  0  0  0   0   0   0  0  0  0 | *  *  *  *   *  * 10  *  *  *   *  *  *  *  *   *  *  *  *  *  * |  1  0  0 1  0  0  0  0  0  0 0 0
.xx.. ..... ..... .....&#x  |  0  2  2  0  0 |  0   0  0  1  0   2  0  1  0   0   0   0  0  0  0 | *  *  *  *   *  *  * 50  *  *   *  *  *  *  *   *  *  *  *  *  * |  0  0  0 0  1  1  0  0  0  0 0 0
..... ..... ..... .xo..&#x  |  0  2  1  0  0 |  0   0  0  0  1   2  0  0  0   0   0   0  0  0  0 | *  *  *  *   *  *  *  * 50  *   *  *  *  *  *   *  *  *  *  *  * |  0  0  1 0  1  0  0  0  0  0 0 0
.x.o. ..... ..... .....&#x  |  0  2  0  1  0 |  0   0  0  1  0   0  2  0  0   0   0   0  0  0  0 | *  *  *  *   *  *  *  *  * 25   *  *  *  *  *   *  *  *  *  *  * |  0  0  0 0  0  2  0  0  0  0 0 0
.ooo.5.ooo. .ooo.5.ooo.&#x  |  0  1  1  1  0 |  0   0  0  0  0   1  1  0  0   1   0   0  0  0  0 | *  *  *  *   *  *  *  *  *  * 100  *  *  *  *   *  *  *  *  *  * |  0  1  0 0  0  1  0  0  0  0 0 0
..xo. ..... ..... .....&#x  |  0  0  2  1  0 |  0   0  0  0  0   0  0  1  0   2   0   0  0  0  0 | *  *  *  *   *  *  *  *  *  *   * 50  *  *  *   *  *  *  *  *  * |  0  0  0 0  0  1  1  0  0  0 0 0
..x.x ..... ..... .....&#x  |  0  0  2  0  2 |  0   0  0  0  0   0  0  1  0   0   2   0  1  0  0 | *  *  *  *   *  *  *  *  *  *   *  * 50  *  *   *  *  *  *  *  * |  0  0  0 0  0  0  1  1  0  0 0 0
..... ..x.o ..... .....&#x  |  0  0  2  0  1 |  0   0  0  0  0   0  0  0  1   0   2   0  0  0  0 | *  *  *  *   *  *  *  *  *  *   *  *  * 50  *   *  *  *  *  *  * |  0  1  0 0  0  0  0  0  1  0 0 0
..... ..... ..... ..o.x&#x  |  0  0  1  0  2 |  0   0  0  0  0   0  0  0  0   0   2   0  0  0  1 | *  *  *  *   *  *  *  *  *  *   *  *  *  * 50   *  *  *  *  *  * |  0  0  0 0  0  0  0  1  1  0 0 0
..ooo5..ooo ..ooo5..ooo&#x  |  0  0  1  1  1 |  0   0  0  0  0   0  0  0  0   1   1   1  0  0  0 | *  *  *  *   *  *  *  *  *  *   *  *  *  *  * 100  *  *  *  *  * |  0  1  0 0  0  0  1  0  0  0 0 0
...ox ..... ..... .....&#x  |  0  0  0  1  2 |  0   0  0  0  0   0  0  0  0   0   0   2  1  0  0 | *  *  *  *   *  *  *  *  *  *   *  *  *  *  *   * 50  *  *  *  * |  0  0  0 0  0  0  1  0  0  1 0 0
..... ..... ...ox .....&#x  |  0  0  0  1  2 |  0   0  0  0  0   0  0  0  0   0   0   2  0  1  0 | *  *  *  *   *  *  *  *  *  *   *  *  *  *  *   *  * 50  *  *  * |  0  1  0 0  0  0  0  0  0  1 0 0
....x5....o ..... .....     |  0  0  0  0  5 |  0   0  0  0  0   0  0  0  0   0   0   0  5  0  0 | *  *  *  *   *  *  *  *  *  *   *  *  *  *  *   *  *  * 10  *  * |  0  0  0 0  0  0  0  0  0  0 1 1
....x ..... ....x .....     |  0  0  0  0  4 |  0   0  0  0  0   0  0  0  0   0   0   0  2  2  0 | *  *  *  *   *  *  *  *  *  *   *  *  *  *  *   *  *  *  * 25  * |  0  0  0 0  0  0  0  0  0  1 1 0
....x ..... ..... ....x     |  0  0  0  0  4 |  0   0  0  0  0   0  0  0  0   0   0   0  2  0  2 | *  *  *  *   *  *  *  *  *  *   *  *  *  *  *   *  *  *  *  * 25 |  0  0  0 0  0  0  0  1  0  0 0 1
----------------------------+----------------+---------------------------------------------------+------------------------------------------------------------------+---------------------------------
..... ..... xo...5ox...&#x  ♦  5  5  0  0  0 |  5  10  0  0  5   0  0  0  0   0   0   0  0  0  0 | 1  5  5  0   0  0  1  0  0  0   0  0  0  0  0   0  0  0  0  0  0 | 10  *  * *  *  *  *  *  *  * * *
..... ofxfo xofox .....&#xt ♦  2  2  4  2  2 |  1   4  4  0  0   4  2  0  2   4   4   4  0  1  0 | 0  2  0  2   4  0  0  0  0  0   4  0  0  2  0   4  0  2  0  0  0 |  * 25  * *  *  *  *  *  *  * * *
..... ..... ..... oxo..&#x  ♦  1  2  1  0  0 |  0   2  1  0  1   2  0  0  0   0   0   0  0  0  0 | 0  0  1  0   2  0  0  0  1  0   0  0  0  0  0   0  0  0  0  0  0 |  *  * 50 *  *  *  *  *  *  * * *
.x... ..... .o...5.x...     ♦  0 10  0  0  0 |  0   0  0  5 10   0  0  0  0   0   0   0  0  0  0 | 0  0  0  0   0  5  2  0  0  0   0  0  0  0  0   0  0  0  0  0  0 |  *  *  * 5  *  *  *  *  *  * * *
.xx.. ..... ..... .xo..&#x  ♦  0  4  2  0  0 |  0   0  0  2  2   4  0  1  0   0   0   0  0  0  0 | 0  0  0  0   0  1  0  2  2  0   0  0  0  0  0   0  0  0  0  0  0 |  *  *  * * 25  *  *  *  *  * * *
.xxo. ..... ..... .....&#x  ♦  0  2  2  1  0 |  0   0  0  1  0   2  2  1  0   2   0   0  0  0  0 | 0  0  0  0   0  0  0  1  0  1   2  1  0  0  0   0  0  0  0  0  0 |  *  *  * *  * 50  *  *  *  * * *
..xox ..... ..... .....&#x  ♦  0  0  2  1  2 |  0   0  0  0  0   0  0  1  0   2   2   2  1  0  0 | 0  0  0  0   0  0  0  0  0  0   0  1  1  0  0   2  1  0  0  0  0 |  *  *  * *  *  * 50  *  *  * * *
..x.x ..... ..... ..o.x&#x  ♦  0  0  2  0  4 |  0   0  0  0  0   0  0  1  0   0   4   0  2  0  2 | 0  0  0  0   0  0  0  0  0  0   0  0  2  0  2   0  0  0  0  0  1 |  *  *  * *  *  *  * 25  *  * * *
..... ..x.o ..... ..o.x&#x  ♦  0  0  2  0  2 |  0   0  0  0  0   0  0  0  1   0   4   0  0  0  1 | 0  0  0  0   0  0  0  0  0  0   0  0  0  2  2   0  0  0  0  0  0 |  *  *  * *  *  *  *  * 25  * * *
...ox ..... ...ox .....&#x  ♦  0  0  0  1  4 |  0   0  0  0  0   0  0  0  0   0   0   4  2  2  0 | 0  0  0  0   0  0  0  0  0  0   0  0  0  0  0   0  2  2  0  1  0 |  *  *  * *  *  *  *  *  * 25 * *
....x5....o ....x .....     ♦  0  0  0  0 10 |  0   0  0  0  0   0  0  0  0   0   0   0 10  5  0 | 0  0  0  0   0  0  0  0  0  0   0  0  0  0  0   0  0  0  2  5  0 |  *  *  * *  *  *  *  *  *  * 5 *
....x5....o ..... ....x     ♦  0  0  0  0 10 |  0   0  0  0  0   0  0  0  0   0   0   0 10  0  5 | 0  0  0  0   0  0  0  0  0  0   0  0  0  0  0   0  0  0  2  0  5 |  *  *  * *  *  *  *  *  *  * * 5