External pieces of starry regulars

The starry regulars are the 14 nonconvex nonskew finite regular polytopes, being the four Kepler–Poinsot polyhedra and the ten Schläfli–Hess polychora, not counting the infinite number of nonconvex regular polygons. Even though they are derivations of the already existing Platonic solids (more specifically of the H_n symmetries), they are by far the hardest of the nonskew regular polytopes to visualize due to their external structural complexity, and of course due to being derivations of the largest regular polytopes. This page lists the external pieces of each starry regular as well as how they are positioned with respect to each cell.

To obtain the external pieces, I used one of three methods: for the Kepler–Poinsot polyhedra, I simply looked at the polyhedron (...) and for some of the Schläfli–Hess polychora I took the information from Bowers' How to Make Polychoron Models. As for the rest, I used Miratope to take two cell-first slices of the polychoron, being one just before the facet and another just afterwards, and took the difference between them by eye.

Kepler–Poinsot polyhedra

Small stellated dodecahedron

The small stellated dodecahedron has 12 pentagrams in a pentagonal vertex figure. Its external pieces are 60 isosceles triangles, edge lengths base 3−√54 and laterals 1+√58.

Its external hull can be made by augmenting pentagonal pyramids on the faces of the dodecahedron, or as a non-Catalan pentakis dodecahedron.

Great dodecahedron

The great dodecahedron has 12 pentagons in a pentagrammic vertex figure. Its external pieces are 60 isosceles triangles, edge lengths base 1 and laterals 3+√58.

Its external hull can be made by excavating triangular pyramids off of the faces of the icosahedron, or as a non-Catalan triakis icosahedron.

Great stellated dodecahedron

The great stellated dodecahedron has 12 pentagrams in a triangular vertex figure. Its external pieces are 60 isosceles triangles, edge lengths base 3−√54 and laterals 1+√58.

Its external hull can be made by augmenting tall triangular pyramids on the faces of the icosahedron, or as a non-Catalan triakis icosahedron.

Great icosahedron

The great icosahedron has 12 triangles in a pentagrammic vertex figure. Its external pieces are 60 isosceles triangles, edge lengths base $\sqrt{5}-2$ and laterals $\frac{5\sqrt{3}-2\sqrt{15}}{10}$, and 120 scalene triangles, edge lengths $\sqrt{2}-\frac{2\sqrt{10}}{5}$, $\frac{3-\sqrt{5}}{2}$ and $\sqrt{\frac{3-\sqrt{5}}{5}}$.

Its external hull can be made by excavating isosceles triangular pyramids off of the faces of the small stellated dodecahedron's external hull, or as a triakis pentakis dodecahedron.

Schläfli–Hess polychora

Small stellated hecatonicosachoron

The small stellated hecatonicosachoron has 120 small stellated dodecahedra in a dodecahedral vertex figure. Its external pieces are 1440 pentagonal pyramids, with the same edge lengths as the small stellated dodecahedron's external hull.

Its external hull can be made by augmenting dodecahedral pyramids on the cells of the hecatonicosachoron, or as a non-Catalan dodecakis hecatonicosachoron.

Great hecatonicosachoron

The great hecatonicosachoron has 120 great dodecahedra in a small stellated dodecahedral vertex figure. Its external pieces are 3600 digonal disphenoids, with edge lengths short base 3−√54, linking edges 3+√58 and long base 1.

Looking at the shape vertex/cell-first, the pieces are positioned with their long bases as the edges of its great dodecahedral cell with the short bases within the inner edges of the small stellated dodecahedral vertex figure, only making contact by their lateral edges and long base vertices.

Great stellated hecatonicosachoron

The great stellated hecatonicosachoron has 120 great stellated dodecahedra in an icosahedral vertex figure. Its external pieces are 2400 triangular antimilvi. The exact edge lengths are unknown.

Its triangular antimilvar pieces are the apices of the great stellated dodecahedral cell, with the sharper apex pointed outward.

Grand hecatonicosachoron

The grand hecatonicosachoron has 120 dodecahedra in a great icosahedral vertex figure. It has two types of isosceles triangular pyramidal external pieces, named "spike" and "dip", each numbering 7200, along with 7200 cut isosceles pyramids "gliders". Their edge lengths are:
Spikes: base base √5−14 and base laterals 3+√58, unknown laterals;
Dips: base base √5−14 and base laterals 3+√58, unknown laterals;
Gliders: base base 1 and base laterals 18, all else unknown. The cut is a tall kite pyramid, with its apex at the base apex its base at the base base's lateral face apex.

Looking at the shape vertex/cell-first, the gliders are placed askew the edges of the dodecahedral cell, with two meeting at an edge, without the piece faces meeting. The spikes are placed with their base apex at the cell's vertex. The remaining dips are placed with their base apex pointed at the face center. It is worth noting that all the pyramid's bases are within the plane of the cell faces, with a small pentagrammic "hole" to allow for a peak crossing.

Grand stellated hecatonicosachoron

The grand stellated hecatonicosachoron has 120 small stellated dodecahedra in a great dodecahedral vertex figure. Its external pieces are 7200 kite pyramids of unknown exact edge lengths.

Its kite pyramidal pieces are positioned with five at each small stellated dodecahedral cell apex, sharing edges and kite apex vertex, but not the faces, with the pyramid apex outside the base from its plane. The pyramid base is in the plane of the cell's faces. Their placement can be thought of as somewhat akin to the filling of the great icosahedron by a small stellated dodecahedron, but only the tips of the shape are considered.

Great grand hecatonicosachoron

The great grand hecatonicosachoron has 120 great dodecahedra in a great stellated dodecahedral vertex figure. Its external pieces are 3600 notches and 7200 cut kite pyramids, as seen on the animated image to the right, both of unknown exact edge lengths. The cut kite pyramids are cut by a kite pyramid on their shallow end.

The notches are placed with their base base edge along the middle section of the great dodecahedral cell's edges. The cut kite pyramids are positioned with five at each great dodecahedral cell vertex, sharing edges and oblique kite apex vertex, similarly to the placement of the grand stellated dodecahedron's pieces.

More information: Bowers' How to Make Gaghi

Great grand stellated hecatonicosachoron

The great grand stellated hecatonicosachoron has 120 great stellated dodecahedra in a tetrahedral vertex figure. Its external pieces are 2400 cut triangular antimilvi and 7200 kite pyramids, as seen on the animated image on the right, both of unknown exact edge lengths. The cut triangular antimilval piece, which is very close in shape to or even aequal to a triangular antitegum, is cut by a small triangular antimilvus/antitegum on one of its apices. The base of the kite pyramid is very close or aequal to in shape to a rhombus.

The cut triangular antimilvar pieces are positioned with their sharp apex at the vertices of the great stellated dodecahedral cell, while the kite pyramidal pieces are positioned with the longest lateral edge at two points on each edge of the cell, closer to the apex of the edge than the center, with five of them meeting at the center of another pentagrammic face.

More information: Bowers' How to Make Gogishi

Facetted hexacosichoron

The facetted hexacosichoron has 120 icosahedra in a great dodecahedral vertex figure. Its external pieces are 2400 triangular pyramids, with edge lengths base 1 and laterals $\sqrt{\frac{53+9\sqrt{5}-4\sqrt{58+30\sqrt{5}}}{96}}$.

Its external hull can be made by excavating tetrahedral pyramids on the cells of the hexacosichoron, or as a non-Catalan tetrakis hexacosichoron.

Great facetted hexacosichoron

The great facetted hexacosichoron has 120 great icosahedra in a small stellated dodecahedral vertex figure. It has 7200 identical external pieces. The Miratope method shows it has isosceles triangular pyramidal pieces, although due to the way the program works, the exact shape might be inaccurate. They will be referred to as isosceles triangular pyramids for now until the exact piece shape is discovered.

The isosceles triangular pyramids are positioned with five meeting at each cell vertex by their edges, with the base placed against an earlier pentagonal pyramid formation within the cell slice.

Grand hexacosichoron

The grand hexacosichoron has 600 tetrahedra in a great icosahedral vertex figure. Its external pieces are 7200 "spikes", 7200 "manta rays", 14400 "shards" and 7200 "fish". Spikes, manta rays are both cut isosceles triangular pyramids, cut by the intersection of two planes, which for the spike only cuts its apex but for the manta ray it also cuts part of its base. The shard is a cut asymmetrical tetrahedron (scalene triangular pyramid) with its apex and lateral edge corresponding to its second most acute angle cut. I really do not know how to describe the fish other than that it looks like a manta ray. All four are depicted in the animated image to the right.

The spikes are positioned with three at each tetrahedral cell vertex, the large base within the cell face plane, meeting only at the vertex. Six shards are placed alongside them, meeting the spikes and each other only at an edge. The manta rays is placed behind (before the base base of) the spikes. The fishes are placed with their "base" fully within the cell's triangular face, being at the center of the edges but further inwards.

More information: Bowers' How to Make Gax