In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from the Latin stella, "star".Stellation is the reciprocal or dual process to faceting.
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👉 Stellation in the context of Facet (geometry)
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically:
- In three-dimensional geometry, some authors call a facet of a polyhedron any polygon whose corners are vertices of the polyhedron, including polygons that are not faces. To facet a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to stellation and may also be applied to higher-dimensional polytopes.
- In polyhedral combinatorics and in the general theory of polytopes, a face that has dimension n − 1 (an (n − 1)-face or hyperface) is called a facet. In this terminology, every facet is a face. A facet of a facet, that is a (n − 2)-face, may be called a ridge.
- A facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex. For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics.
In this Dossier
- ⭐ Core Definition: Stellation
- 👉 Stellation in the context of Facet (geometry)
- Stellation in the context of Icosahedron
- Stellation in the context of Kepler–Poinsot polyhedra
- Stellation in the context of Regular icosahedron
- Stellation in the context of Dodecahedron
- Stellation in the context of Geometric figures
- Stellation in the context of 59 (number)
- Stellation in the context of Escher's solid
Stellation in the context of Icosahedron
In geometry, an icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" (/-drə/) or "icosahedrons".
There are infinitely many non-similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non-stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles.
View the full Wikipedia page for IcosahedronStellation in the context of Kepler–Poinsot polyhedra
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.
They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures. They can all be seen as three-dimensional analogues of the pentagram in one way or another.
View the full Wikipedia page for Kepler–Poinsot polyhedraStellation in the context of Regular icosahedron
The regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.
Many polyhedra and other related figures are constructed from the regular icosahedron, including its 59 stellations. The great dodecahedron, one of the Kepler–Poinsot polyhedra, is constructed by either stellation of the regular dodecahedron or faceting of the icosahedron. Some of the Johnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron's dual polyhedron is the regular dodecahedron, and their relation has a historical background in the comparison mensuration. It is analogous to a four-dimensional polytope, the 600-cell.
View the full Wikipedia page for Regular icosahedronStellation in the context of Dodecahedron
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular:The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry.
View the full Wikipedia page for DodecahedronStellation in the context of Geometric figures
Lists of shapes cover different types of geometric shape and related topics. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools.
View the full Wikipedia page for Geometric figuresStellation in the context of 59 (number)
59 (fifty-nine) is the natural number following 58 and preceding 60.
View the full Wikipedia page for 59 (number)Stellation in the context of Escher's solid
In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex Hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual appearance as two other shapes: a solid, Escher's solid, with 48 triangular faces, and a polyhedral compound of three flattened octahedra with 24 overlapping triangular faces.
Escher's solid can tessellate space to form the stellated rhombic dodecahedral honeycomb.
View the full Wikipedia page for Escher's solid