In three-dimensional space, applying the operation of stellation to a polyhedron extends its faces (or edges and planes) until they generate new vertices that bound a newly formed figure. Stellation represents the dual action to faceting a polyhedron.
Originating from studies of star polyhedra in 14th century Europe, a proper mathematical account of polyhedral stellations was given by Johannes Kepler in his 1619 classic work, Harmonices Mundi. Progress later ensued on detailing and enumerating stellations of prominent stars, such as the regular Kepler-Poinsot polyhedra, with developments on different stellation methods occurring in the 1900s, principally from and soon afterward, .
A short generalized table of the most notable polyhedral stellations belonging to convex uniform polyhedra is provided, with complete sets of stellations for the Platonic solids (including the fifty-nine icosahedral stellations), as well as for select Catalan solids (e.g., the rhombic dodecahedron and the rhombic triacontahedron). Stellations of non-convex uniform polyhedra of structure with facial planes passing through their centers (i.e., hemipolyhedral), which render unbounded vertices, are also included; these are stellations to infinity — per — conforming to extensions on traditional definitions of polyhedra.
Background Star polytopes
Experimentation with star polygons and star polyhedra since the fourteenth century AD led the way to formal theories for stellating polyhedra:
It was in 1619 that the first geometric description of a stellation was given, by Johannes Kepler in his landmark book, Harmonices Mundi: the process of extending the edges (or faces) of a figure until new vertices are generated, which collectively form a new figure. Using this method, Kepler was able to discover the small stellated dodecahedron and the great stellated dodecahedron. In 1809, Louis Poinsot rediscovered Kepler's star figures and discovered a further two, the great icosahedron and great dodecahedron; he achieved this by experimenting assembling regular star polygons and convex regular polygons on vertices of the regular icosahedron and dodecahedron (i.e., pentagons, pentagrams and equilateral triangles). Three years later, Augustin-Louis Cauchy proved, using concepts of symmetry, that these four stellations are the only regular star-polyhedra, eventually termed the Kepler–Poinsot polyhedra. As with most non-convex polyhedra including stellations and other star polyhedra, the Kepler-Poinsot polyhedra, with regular self-intersecting faces, are now known to be inequivalent to the topological sphere as a simple connected surface (this is in contrast with the traditional convex uniform polyhedra and their corresponding homotopy invariance).
Stellation process
details, for the first time, all stellations of the regular icosahedron with specific rules proposed by J. C. P. Miller. Generalizing these (Miller's rules) for stellating any uniform polyhedron yields the following:
These rules are ideal for stellating smaller uniform solids, such as the regular polyhedra; however, when assessing stellations of other larger uniform polyhedra, this method can quickly become overwhelming. (For example, there are a total of 358,833,072 stellations to the rhombic triacontahedron using this set of rules.) To address this, proposed a set of rules that restrict the number of stellations to a more manageable set of fully supported stellations that are radially convex, such that an outward ray from the center of the original polyhedron (in any direction) crosses the stellation surface only once (that is to say, all visible parts of a face are seen from the same side).
In the 1948 first edition of Regular Polytopes, H. S. M. Coxeter describes the stellation process as the reciprocal action to faceting, identifying the four Kepler-Poinsot polyhedra as stellations and facetings of the regular dodecahedron and icosahedron. He specifies the construction of a star polyhedron as a stellation of its core (with congruent face-planes), or by faceting its case — the former requires the addition of solid pieces that generate new vertices, while the latter involves the removal of solid pieces, without forming any new vertices (the core of a star polyhedron or compound is the largest convex solid that can be drawn inside them, while their case is the smallest convex solid that contains them).
Lists Lists for polyhedral stellations contain non-convex polyhedra; some of the most notable examples include:
Stellations that topologically do not fit into standard definitions of uniform polyhedra are listed further down (i.e. stellations of hemipolyhedra).
+ Notable stellations of various polyhedra Image Name Stellation core Face diagram Refs. Notes Great dodecahedron Regular dodecahedron Great stellated dodecahedron Small stellated dodecahedron Great icosahedron Regular icosahedron Stella octangula (compound of two tetrahedra) Regular octahedron Compound of five tetrahedra Regular icosahedron Compound of ten tetrahedra Compound of five octahedra Compound of five cubes Rhombic triacontahedron Compound of cube and octahedronCuboctahedron Compound of dodecahedron and icosahedronIcosidodecahedron Compound of great icosahedron and great stellated dodecahedron Compound of great dodecahedron and small stellated dodecahedron Small triambic icosahedron Regular icosahedron Final stellation of the icosahedron First stellation of the rhombic dodecahedron Rhombic dodecahedron
KEY * Kepler-Poinsot polyhedron (star polyhedron with regular facets)
† Regular compound polyhedron (vertex, edge, and face-transitive compound)
‡ Compound of dual regular polyhedra (Platonic or Kepler-Poinsot duals)
¶ First/outermost stellation of stellation core
"Stellation core" describes a stellated regular (Platonic), semi-regular (Archimedean), or dual to a semi-regular (Catalan) figure."Face diagram" represents the lines of intersection from extended polyhedral edges that are used in the stellation process."Refs." (references) such as indexes found in using the Crennells' illustration notation (C), and (W).
Enumerations The table below is adapted from research by Robert Webb, using his program Stella. It enumerates fully supported stellations and stellations per Miller's process, of the regular Platonic solids as well as the semi-regular Archimedean solids and their Catalan duals. In this list, the elongated square gyrobicupola and its dual polyhedron are not included (these are sometimes considered a fourteenth Archimedean and Catalan solid, respectively). The base polyhedron stellation core is included as a zeroth convex stellation following the Crennells' indexing, with stellation totals the sum of chiral and reflexible stellations (a "chiral" stellation is enantiomorphous, while a "reflexible" stellation maintains the same group symmetry as its stellation core, yet remains achiral – for a count of these separately, visit the parent source).
Stellation totals of convex polyhedra by group symmetry and order (Td, Oh, Ih)▼PolyhedronCell typesFully supported stellationsMiller stellationsTetrahedron111Cube111Octahedron222Dodecahedron444Icosahedron111859Truncated tetrahedron4610Cuboctahedron81321Truncated octahedron91845Truncated cube91845Rhombicuboctahedron4818827? (128723453647 reflexible)Truncated cuboctahedron4922632? (317650001638 reflexible)Snub cube274299050957776?Icosidodecahedron4184770841855109Truncated icosahedron4511173082649548558 Truncated dodecahedron4511412645087084526Rhombicosidodecahedron273298832037395?Truncated icosidodecahedron2941016992138164?Snub dodecahedron1940? (579 reflexible)?Triakis tetrahedron921188Rhombic dodecahedron445Tetrakis hexahedron101762143383367876Triakis octahedron323083218044256331Deltoidal icositetrahedron321201253811894971Disdyakis dodecahedron292? (14728897413 reflexible)?Pentagonal icositetrahedron6972621?Rhombic triacontahedron29227358833098Pentakis dodecahedron25371112946668?Triakis icosahedron24113902332663?Deltoidal hexecontahedron2267146284014?Disdyakis triacontahedron2033? (~ 1012 reflexible)?Pentagonal hexecontahedron53630049378413796?
"Cell types" are sets of symmetrically equivalent stellation cells, where "stellation cells" are the minimal 3D spaces enclosed on all sides by the original polyhedron's extended facial planes."?" denotes an unknown total number of stellations; however, the number of reflexible stellations are sometimes known for these (where chiral stellations are excluded).
Stellations of Platonic solids Only three of the five Platonic solids produce stellations: the regular octahedron, regular dodecahedron, and regular icosahedron. The regular tetrahedron and cube are unable to generate stellations when extending their faces, since extending their vertices only form one possible convex hull.
Stellations of the octahedron
The stella octangula (or stellated octahedron) is the only stellation of the regular octahedron. This stellation is made of self-dual tetrahedra, as the simplest regular polyhedral compound:
Figure Stellation
Stellations of the dodecahedron All stellations of the regular dodecahedron are Kepler-Poinsot polyhedra:
Platonic solid Kepler–Poinsot solids+ Stellations of the regular dodecahedronRegular dodecahedronSmall stellated dodecahedronGreat dodecahedronGreat stellated dodecahedron
Stellations of the icosahedron
detailed the stellations of the regular icosahedron with rules proposed by J. C. P. Miller. As found in , the following table lists all stellations of the icosahedron per the Crennells' indexing (in it, the regular icosahedron (or snub octahedron) stellation core is indexed as "1"):
Stellations of the regular icosahedron Crennell Cells Faces Figure Face diagramA 0 B 1C 2D 3 4E 5 6 7F 8 9 10G 11 12H 13e1 3' 5f1 5' 6' 9 10g1 10' 12e1f1 3' 6' 9 10 e1f1g13' 6' 9 12f1g15' 6' 9 12 e24' 6 7 f27' 8g28' 9'11 e2f24' 6 8 e2f2g24' 6 9' 11 f2g27' 9' 11De14 5Ef17 9 10Fg18 9 12De1f14 6' 9 10 De1f1g14 6' 9 12 Ef1g17 9 12De23 6 7Ef25 6 8Fg210 11De2f23 6 8De2f2g23 6 9' 11 Ef2g25 6 9' 11 f15' 6 9 10e1f13 5 6 9 10De1f14 5 6 9 10 f1g15' 6 9 10 12 e1f1g13 5 6 9 10 12De1f1g14 5 6 9 10 12 f1g25' 6 8 9 10 11 e1f1g23 5 6 8 9 10 11 De1f1g24 5 6 8 9 10 11 f1f2g25' 6 7 9 10 11 e1f1f2g23 5 6 7 9 10 11 De1f1f2g24 5 6 7 9 10 11 e2f14 5 6 7 9 10De2f13 5 6 7 9 10 Ef15 6 7 9 10e2f1g14 5 6 7 9 10 12 De2f1g13 5 6 7 9 10 12 Ef1g15 6 7 9 10 12 e2f1f24 5 6 8 9 10De2f1f23 5 6 8 9 10 Ef1f25 6 8 9 10 e2f1f2g14 5 6 8 9 10 12 De2f1f2g13 5 6 8 9 10 12 Ef1f2g15 6 8 9 10 12 e2f1f2g24 5 6 9 10 11 De2f1f2g23 5 6 9 10 11 Ef1f2g25 6 9 10 11
"Cells" (du Val notation) correspond to the internal congruent spaces formed by extending face-planes of the regular icosahedron.
A subset of these are illustrated in , alongside constructions for physical models (W19–W66).
Stellations of Catalan solids Stellations of the rhombic dodecahedron
The rhombic dodecahedron produces three fully supported stellations, described in : + Rhombic dodecahedron stellationsStellationFigureFace diagram
An additional fourth stellation is possible under Miller's rules. The first stellation of the rhombic dodecahedron is notable for being able to form a honeycomb in three-dimensional space, using copies of itself.
Stellations of the rhombic triacontahedron shows the rhombic triacontahedron produces 227 fully supported stellations, including the rhombic triacontahedron itself. Some of these are shown in the table below:
+ Select stellations to the rhombic triacontahedron StellationFigureFace diagram
Of these, the compound of five cubes is notable for being a regular compound polyhedron. The medial rhombic triacontahedron and the great rhombic triacontahedron are also notable for being star (non-convex) isotoxal polyhedra.
Hemipolychrons