In geometry, a
Contents 1 Characteristics 1.1 Non-convexity
1.2
2 Relationships among the regular polyhedra 3 History 4 Regular star polyhedra in art and culture 5 See also 6 References 7 External links Characteristics[edit]
Non-convexity[edit]
These figures have pentagrams (star pentagons) as faces or vertex
figures. The small and great stellated dodecahedron have nonconvex
regular pentagram faces. The great dodecahedron and great icosahedron
have convex polygonal faces, but pentagrammic vertex figures.
In all cases, two faces can intersect along a line that is not an edge
of either face, so that part of each face passes through the interior
of the figure. Such lines of intersection are not part of the
polyhedral structure and are sometimes called false edges. Likewise
where three such lines intersect at a point that is not a corner of
any face, these points are false vertices. The images below show
golden balls at the true vertices, and silver rods along the true
edges.
For example, the small stellated dodecahedron has 12 pentagram faces
with the central pentagonal part hidden inside the solid. The visible
parts of each face comprise five isosceles triangles which touch at
five points around the pentagon. We could treat these triangles as 60
separate faces to obtain a new, irregular polyhedron which looks
outwardly identical. Each edge would now be divided into three shorter
edges (of two different kinds), and the 20 false vertices would become
true ones, so that we have a total of 32 vertices (again of two
kinds). The hidden inner pentagons are no longer part of the
polyhedral surface, and can disappear. Now the Euler's formula holds:
60 − 90 + 32 = 2. However this
polyhedron is no longer the one described by the
χ = V − E + F = 2 displaystyle chi =V-E+F=2 does not always hold. Schläfli held that all polyhedra must have χ = 2, and he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never widely held. A modified form of Euler's formula, using density (D) of the vertex figures ( d v displaystyle d_ v ) and faces ( d f displaystyle d_ f ) was given by Arthur Cayley, and holds both for convex polyhedra (where the correction factors are all 1), and the Kepler–Poinsot polyhedra: d v V − E + d f F = 2 D . displaystyle d_ v V-E+d_ f F=2D. Duality[edit] The Kepler–Poinsot polyhedra exist in dual pairs:
Summary[edit] Name Picture Spherical tiling Stellation diagram Schläfli p,q and Coxeter-Dynkin Faces p Edges Vertices q Vertex figure (config.) χ Density Symmetry Dual Small stellated dodecahedron 5/2,5 12 5/2 30 12 5 (5/2)5 −6 3 Ih Great dodecahedron Great stellated dodecahedron 5/2,3 12 5/2 30 20 3 (5/2)3 2 7 Ih Great icosahedron Great dodecahedron 5,5/2 12 5 30 12 5/2 (55)/2 −6 3 Ih Small stellated dodecahedron Great icosahedron 3,5/2 20 3 30 12 5/2 (35)/2 2 7 Ih Great stellated dodecahedron Relationships among the regular polyhedra[edit] These share the same vertex arrangements: These share the same vertex and edge arrangements: The icosahedron, small stellated dodecahedron, great icosahedron, and great dodecahedron. The small stellated dodecahedron and great icosahedron. The dodecahedron and great stellated dodecahedron. The icosahedron and great dodecahedron. The small stellated dodecahedron and great icosahedron share the same vertices and edges. The icosahedron and great dodecahedron also share the same vertices and edges. The three dodecahedra are all stellations of the regular convex dodecahedron, and the great icosahedron is a stellation of the regular convex icosahedron. The small stellated dodecahedron and the great icosahedron are facettings of the convex dodecahedron, while the two great dodecahedra are facettings of the regular convex icosahedron. If the intersections are treated as new edges and vertices, the figures obtained will not be regular, but they can still be considered stellations. (See also List of Wenninger polyhedron models) History[edit] Floor mosaic in St Mark's Basilica,
Most, if not all, of the Kepler-Poinsot polyhedra were known of in
some form or other before Kepler. A small stellated dodecahedron
appears in a marble tarsia (inlay panel) on the floor of St. Mark's
Basilica, Venice, Italy. It dates from the 15th century and is
sometimes attributed to Paolo Uccello. In his Perspectiva corporum
regularium (Perspectives of the regular solids), a book of woodcuts
published in the 16th century,
Conway's operational terminology gives a hexagonal diagram of
relations between the 4 regular star polyhedra and 2 convex forms.[3]
A hundred years later, John Conway developed a systematic terminology for stellations in up to four dimensions. Within this scheme, he suggested slightly modified names for one of the regular star polyhedra, dropping the small adjective. Conway's names have seen some use but have not been widely adopted. Cayley's name Conway's name and (abbreviation) small stellated dodecahedron stellated dodecahedron (sD) great dodecahedron (gD) great stellated dodecahedron (gsD) great icosahedron (gI) Regular star polyhedra in art and culture[edit] Alexander's Star Regular star polyhedra first appear in Renaissance art. A small
stellated dodecahedron is depicted in a marble tarsia on the floor of
St. Mark's Basilica, Venice, Italy, dating from ca. 1430 and sometimes
attributed to Paulo Ucello.
Regular polytope
Regular polyhedron
List of regular polytopes
Uniform polyhedron
Uniform star polyhedron
Polyhedral compound
References[edit] ^ Coxeter, Star polytopes and the Schläfli function f(α,β,γ) p. 121 1. The Kepler–Poinsot polyhedra ^ Perspectiva corporum regularium ^ The Symmetries of Things, p.405 Figure 26.1 Relationships among the three-dimensional star-polytopes J. Bertrand, Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de l'Académie des Sciences, 46 (1858), pp. 79–82, 117. Augustin-Louis Cauchy, Recherches sur les polyèdres. J. de l'École Polytechnique 9, 68-86, 1813. Arthur Cayley, On Poinsot's Four New Regular Solids. Phil. Mag. 17, pp. 123–127 and 209, 1859. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetry of Things 2008, ISBN 978-1-56881-220-5 (Chapter 24, Regular Star-polytopes, pp. 404–408) Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1] (Paper 1) H.S.M. Coxeter, The Nine Regular Solids [Proc. Can. Math. Congress 1 (1947), 252–264, MR 8, 482] (Paper 10) H.S.M. Coxeter, Star Polytopes and the Schlafli Function f(α,β,γ) [Elemente der Mathematik 44 (2) (1989) 25–36] P. Cromwell, Polyhedra, Cabridgre University Press, Hbk. 1997, Ppk. 1999. Theoni Pappas, (The Kepler–Poinsot Solids) The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 113, 1989. Louis Poinsot, Memoire sur les polygones et polyèdres. J. de l'École Polytechnique 9, pp. 16–48, 1810. Lakatos, Imre; Proofs and Refutations, Cambridge University Press (1976) - discussion of proof of Euler characteristic Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 0-521-54325-8. , pp. 39–41. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 3) Anthony Pugh (1976). Polyhedra: A Visual Approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 8: Kepler Poisot polyhedra External links[edit] Weisstein, Eric W. "Kepler–Poinsot solid". MathWorld.
Paper models of Kepler–Poinsot polyhedra
Free paper models (nets) of Kepler–Poinsot polyhedra
The Uniform Polyhedra
VRML models of the Kepler–Poinsot polyhedra
v t e Star-polyhedra navigator Kepler-Poinsot polyhedra (nonconvex regular polyhedra) small stellated dodecahedron great dodecahedron great stellated dodecahedron great icosahedron Uniform truncations of Kepler-Poinsot polyhedra dodecadodecahedron truncated great dodecahedron rhombidodecadodecahedron truncated dodecadodecahedron snub dodecadodecahedron great icosidodecahedron truncated great icosahedron nonconvex great rhombicosidodecahedron great truncated icosidodecahedron Nonconvex uniform hemipolyhedra tetrahemihexahedron cubohemioctahedron octahemioctahedron small dodecahemidodecahedron small icosihemidodecahedron great dodecahemidodecahedron great icosihemidodecahedron great dodecahemicosahedron small dodecahemicosahedron Duals of nonconvex uniform polyhedra medial rhombic triacontahedron small stellapentakis dodecahedron medial deltoidal hexecontahedron small rhombidodecacron medial pentagonal hexecontahedron medial disdyakis triacontahedron great rhombic triacontahedron great stellapentakis dodecahedron great deltoidal hexecontahedron great disdyakis triacontahedron great pentagonal hexecontahedron Duals of nonconvex uniform polyhedra with infinite stellations tetrahemihexacron hexahemioctacron octahemioctacron small dodecahemidodecacron small icosihemidodecacron great dodecahemidodecacron great icosihemidodecacron great dodecahemicosacron small dodeca |