Kepler-Poinsot polyhedron

Kepler-Poinsot polyhedron

The Kepler-Poinsot polyhedra is a popular name for the regular star polyhedra. Each has faces which are congruent regular convex polygons or star polygons and has the same number of faces meeting at each vertex (compare to Platonic solids).

There are four Kepler-Poinsot polyhedra:
*Small stellated dodecahedron
*Great stellated dodecahedron
*Great dodecahedron
*Great icosahedron.

These polyhedra are often referred to as the Kepler-Poinsot solids, though they are perhaps more easily understood as "surfaces".

Geometry

So far, Conway's names have seen some use but have not really caught on.

Regular star polyhedra in art and culture

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. Wenzel Jamnitzer published his book of woodcuts "Perspectiva Corporum Regularium" in 1568. He depicts the great dodecahedron and the great stellated dodecahedron - this second is slightly distorted, probably through errors in method rather than ignorance of the form. However there is no evidence that these artists understood the regularity of such fgures.

In the 20th Century, Artist M. C. Escher's interest in geometric forms often led to works based on or including regular solids; "Gravitation" is based on a small stellated dodecahedron.

A dissection of the great dodecahedron was used for the 1980s puzzle Alexander's Star.

Norwegian artist Vebjørn Sands sculpture " [http://www.vebjorn-sand.com/star.html The Kepler Star] " is displayed near Oslo Airport, Gardermoen. The star spans 14 meters, and consists of an icosahedron and a dodecahedron inside a great stellated dodecahedron.

See also

* Regular polytope
* Regular polyhedron
* List of regular polytopes
* Uniform polyhedron
* Polyhedral compound
* Stellation
* Schläfli-Hess polychoron The 10 4-dimensional star polytopes

References

* 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. "Philos. 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", editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (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
*, pp. 39-41.
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, "The Symmetry of Things" 2008, ISBN 978-1-516881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 3)

External links

*Mathworld | urlname=Kepler-PoinsotSolid | title=Kepler-Poinsot solid
* [http://www.software3d.com/Kepler.php Paper models of Kepler-Poinsot polyhedra]
* [http://www.korthalsaltes.com/Kepler-Poinsot_Polyhedra.html Free paper models (nets) of Kepler-Poinsot polyhedra]
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.georgehart.com/virtual-polyhedra/kepler-poinsot-info.html VRML models of the Kepler-Poinsot polyhedra]
* [http://www.steelpillow.com/polyhedra/StelFacet/history.html Stellation and facetting - a brief history]
* [http://www.software3d.com/Stella.php Stella: Polyhedron Navigator] : Software used to create many of the images on this page.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Kepler–Poinsot polyhedron — 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.… …   Wikipedia

  • Solide de Kepler-Poinsot — Les solides de Kepler Poinsot sont les polyèdres étoilés réguliers. Chacun possède des faces qui sont des polygones convexes réguliers congruents ou des polygones étoilés et possède le même nombre de faces se rencontrant à chaque sommet (comparer …   Wikipédia en Français

  • Polyhedron — Polyhedra redirects here. For the relational database system, see Polyhedra DBMS. For the game magazine, see Polyhedron (magazine). For the scientific journal, see Polyhedron (journal). Some Polyhedra Dodecahedron (Regular polyhedron) …   Wikipedia

  • Polyhedron model — A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material.Since there are 75 uniform polyhedra, including the five regular… …   Wikipedia

  • Regular polyhedron — A regular polyhedron is a polyhedron whose faces are congruent (all alike) regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge transitive, vertex transitive and… …   Wikipedia

  • Uniform polyhedron — A uniform polyhedron is a polyhedron which has regular polygons as faces and is transitive on its vertices (i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high… …   Wikipedia

  • Star polyhedron — In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star like visual quality.There are two general kinds of star polyhedron: *Polyhedra which self intersect in a repetitive way. *Concave… …   Wikipedia

  • Dual polyhedron — The dual of a cube is an octahedron, shown here with vertices at the cube face centers …   Wikipedia

  • Uniform star polyhedron — A display of uniform polyhedra at the Science Museum in London …   Wikipedia

  • Nonconvex uniform polyhedron — In geometry, a nonconvex uniform polyhedron, or uniform star polyhedron, is a self intersecting uniform polyhedron. Each can contain either star polygon faces, star polygon vertex figures or both.The complete set of 57 nonprismatic uniform star… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”