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 . 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 NON-CONVEXITY 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
EULER CHARACTERISTIC χ A
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
DUALITY The Kepler–Poinsot polyhedra exist in dual pairs: *
SUMMARY NAME
PICTURE
Spherical
tiling
{5/2,5} 12 {5/2} 30 12 {5} (5/2)5 -6 3 Ih Great dodecahedron
{5/2,3} 12 {5/2} 30 20 {3} (5/2)3 2 7 Ih Great icosahedron
{5,5/2} 12 {5} 30 12 {5/2} (55)/2 -6 3 Ih Small stellated dodecahedron
{3,5/2} 20 {3} 30 12 {5/2} (35)/2 2 7 Ih Great stellated dodecahedron RELATIONSHIPS AMONG THE REGULAR POLYHEDRA 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 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 ,
The small and great stellated dodecahedra, sometimes called the KEPLER POLYHEDRA, were first recognized as regular by Johannes Kepler in 1619. He obtained them by stellating the regular convex dodecahedron, for the first time treating it as a surface rather than a solid. He noticed that by extending the edges or faces of the convex dodecahedron until they met again, he could obtain star pentagons. Further, he recognized that these star pentagons are also regular. In this way he constructed the two stellated dodecahedra. Each has the central convex region of each face "hidden" within the interior, with only the triangular arms visible. Kepler's final step was to recognize that these polyhedra fit the definition of regularity, even though they were not convex , as the traditional Platonic solids were. In 1809,
Three years later,
The following year,
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 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.
In the 20th Century, Artist
A dissection of the great dodecahedron was used for the 1980s puzzle Alexander\'s Star . Norwegian artist Vebjørn Sands sculpture The Kepler Star is
displayed near
SEE ALSO Media related to Kepler-Poinsot solids at Wikimedia Commons *
REFERENCES * ^ 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'Acad |