Acronym spysp
Name small pyramidic swirlprism
Circumradius (1+sqrt(5))/2 = 1.618034
Vertex figure
Colonel of regiment sub-regimental to ex
Dihedral angles
uniform relative:
related scaliform:
sisp   gypasp  
general polytopal classes:
noble polytopes  

This polychoron is based on the ex and indeed all vertices do coincide with that. Even the edges and the triangles of this polychoron are subsets of those of ex. Infact it could be understood, when systematically all 5-tet-rosettes (considered alone as pescs) get excavated into concave pairs of peppies by dropping that central edge each and replacing it by the according pentagon underneath. That "systematic way" thereby is obtained as follows. Consider gap as an ex diminishing. That one alredy omits 2 decagonal edge cycles. 10 more such decagonal edge cycles of the same Hopf fibration of ex will be swirling around the there contained paps (5 for each of the 2 pap-cycles). – Infact, all those 12 decagonal edge cycles would be used as guides of according pap-cycles in sisp. – But in the case og spysp each edge of any of those 12 decagons gets replaced by its orthogonal pentagon underneath, keeping just the outer connecting triangles for lacing faces of the introduced peppies pairs. That is, each of those 12 decagons of that fibration becomes a necklace of adjoined peppy pairs. In total those would leave no tet any more.

The spysp vertex figure is similarily derived from the pap, when all the descending lacing edges get excavated to the orthogonal f-edge underneath.

As abstract polytope spysp is isomorph to gypasp, thereby replacing pentagons by pentagrams, resp. replacing peppy by stappy.

By construction the pentagons provide non-convex dihedral angles. Infact the complements of those of pesc. Nonetheless, this polychoron has the strange property that it's entire surface is exposed, even though it is not convex, that is, it's a non-self-intersecting concave polychoron.

Incidence matrix

120 |  10 |  15   5 |  12
  2 | 600 |   3   1 |   4
  3 |   3 | 600   * |   2
  5 |   5 |   * 120 |   2  : {5}
  6 |  10 |   5   1 | 240  : peppy

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