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The heptagons {(H,h,H,h,H,h,H)} only show up axial symmetry. Their corner angles are h = 108° resp. H = 144°. The edges of this polyhedron are of size x (those, which are incident to triangles) resp. of size v = 1/f, i.e. in the golden ratio smaller.

Incidence matrix according to Dynkin symbol

VooFxfu oVofFxu ooVxfFu&#z(x,v) (F=ff=x+f, V=2f, u=2x, v=1/f=f-x)   → all existing heights = 0
(tegum sum of 3 orthogonal V-edges, 3 each parallely oriented (x,f,F)-cubes, and a likewise parallely oriented u-cube)

o...... o...... o......         | 2 * * * * * * | 4 0 0 0 0 0 0 0 0 | 2 2 0 0 0 0  [(3,h)2]
.o..... .o..... .o.....         | * 2 * * * * * | 0 4 0 0 0 0 0 0 0 | 0 0 2 2 0 0  [(3,h)2]
..o.... ..o.... ..o....         | * * 2 * * * * | 0 0 4 0 0 0 0 0 0 | 0 0 0 0 2 2  [(3,h)2]
...o... ...o... ...o...         | * * * 8 * * * | 1 0 0 1 1 0 0 0 0 | 1 1 0 1 0 0  [3,H2]
....o.. ....o.. ....o..         | * * * * 8 * * | 0 1 0 0 0 1 1 0 0 | 0 0 1 1 1 0  [3,H2]
.....o. .....o. .....o.         | * * * * * 8 * | 0 0 1 0 0 0 0 1 1 | 1 0 0 0 1 1  [3,H2]
......o ......o ......o         | * * * * * * 8 | 0 0 0 0 1 0 1 0 1 | 1 0 0 1 1 0  [h3]
--------------------------------+---------------+-------------------+------------
o..o... o..o... o..o...&#x      | 1 0 0 1 0 0 0 | 8 * * * * * * * * | 1 1 0 0 0 0
.o..o.. .o..o.. .o..o..&#x      | 0 1 0 0 1 0 0 | * 8 * * * * * * * | 0 0 1 1 0 0
..o..o. ..o..o. ..o..o.&#x      | 0 0 1 0 0 1 0 | * * 8 * * * * * * | 0 0 0 0 1 1
....... ....... ...x...         | 0 0 0 2 0 0 0 | * * * 4 * * * * * | 0 1 0 1 0 0
...o..o ...o..o ...o..o&#v      | 0 0 0 1 0 0 1 | * * * * 8 * * * * | 1 0 0 1 0 0
....x.. ....... .......         | 0 0 0 0 2 0 0 | * * * * * 4 * * * | 0 0 1 0 1 0
....o.o ....o.o ....o.o&#v      | 0 0 0 0 1 0 1 | * * * * * * 8 * * | 0 0 0 1 1 0
....... .....x. .......         | 0 0 0 0 0 2 0 | * * * * * * * 4 * | 1 0 0 0 0 1
.....oo .....oo .....oo&#v      | 0 0 0 0 0 1 1 | * * * * * * * * 8 | 1 0 0 0 1 0
--------------------------------+---------------+-------------------+------------
....... o..f.xu .......&#(x,v)t | 1 0 0 2 0 2 2 | 2 0 0 0 2 0 0 1 2 | 4 * * * * *  {(H,h,H,h,H,h,H)} tower: A-D-G-F
....... ....... o..x...&#x      | 1 0 0 2 0 0 0 | 2 0 0 1 0 0 0 0 0 | * 4 * * * *
.o..x.. ....... .......&#x      | 0 1 0 0 2 0 0 | 0 2 0 0 0 1 0 0 0 | * * 4 * * *
....... ....... .o.xf.u&#(x,v)t | 0 1 0 2 2 0 2 | 0 2 0 1 2 0 2 0 0 | * * * 4 * *  {(H,h,H,h,H,h,H)} tower: B-E-G-D
..o.xfu ....... .......&#(x,v)t | 0 0 1 0 2 2 2 | 0 0 2 0 0 1 2 0 2 | * * * * 4 *  {(H,h,H,h,H,h,H)} tower: C-F-G-E
....... ..o..x. .......&#x      | 0 0 1 0 0 2 0 | 0 0 2 0 0 0 0 1 0 | * * * * * 4

or
o...... o...... o......         & | 6  * * |  4  0  0 |  2  2  [(3,h)2]
...o... ...o... ...o...         & | * 24 * |  1  1  1 |  2  1  [3,H2]
......o ......o ......o           | *  * 8 |  0  0  3 |  3  0  [h3]
----------------------------------+--------+----------+------
o..o... o..o... o..o...&#x      & | 1  1 0 | 24  *  * |  1  1
....... ....... ...x...         & | 0  2 0 |  * 12  * |  1  1
...o..o ...o..o ...o..o&#v      & | 0  1 1 |  *  * 24 |  2  0
----------------------------------+--------+----------+------
....... o..f.xu .......&#(x,v)t & | 1  4 2 |  2  1  4 | 12  *  {(H,h,H,h,H,h,H)} tower: A-D-G-F
....... ....... o..x...&#x      & | 1  2 0 |  2  1  0 |  * 12