Kepler-Poinsot Solids. The stellations of a dodecahedron are often referred to as Kepler-Solids. The Kepler-Poinsot solids or polyhedra is a popular name for the. The four Kepler-Poinsot polyhedra are regular star polyhedra. For nets click on the links to the right of the pictures. Paper model Great Stellated Dodecahedron. A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have.
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In geometrya Kepler—Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular keplrr dodecahedron and icosahedronand differ from these in having regular pentagrammic faces or vertex figures.
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 keplr such lines intersect at a point that is not a corner of any face, these points are false keepler.
The images below pojnsot 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.
Kepler-Poinsot Solid — from Wolfram MathWorld
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 kindsand 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: A Kepler—Poinsot polyhedron covers its circumscribed sphere more than once, with kepled centers of faces acting as winding points in the figures which have pentagrammic faces, and the vertices in the others.
Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the Euler relation.
This pionsot was never widely held. A modified form of Euler’s formula, using density D of the vertex figures and faces was given by Arthur Cayleyand holds both for convex polyhedra where the correction factors are all 1and the Kepler—Poinsot polyhedra:. The Kepler—Poinsot polyhedra exist in dual pairs:. 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 regularbut they can still be considered stellations. See also List of Wenninger polyhedron models. 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 BasilicaVeniceItaly. It dates from the 15th century and is sometimes attributed to Paolo Uccello. In his Perspectiva corporum regularium Perspectives of the regular solidsa book of woodcuts published in the 16th century, Wenzel Jamnitzer depicts the great dodecahedron and the great stellated dodecahedron.
The small and great stellated dodecahedra, sometimes called the Kepler polyhedrawere first recognized as regular by Johannes Kepler in He obtained them by stellating the regular convex dodecahedron, for the first time treating it as a surface rather than a solid.
Kepler-Poinsotov polieder – Wikipedija, prosta enciklopedija
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 convexas the traditional Platonic solids were.
InLouis Poinsot rediscovered Kepler’s figures, by assembling star pentagons around each vertex. He also assembled convex polygons poineot star vertices to discover keplr more regular stars, the great icosahedron and great dodecahedron.
Some people call these two the Poinsot polyhedra. Poinsot did not know if he had discovered all the regular star polyhedra. Three years later, Augustin Cauchy proved the list complete by stellating the Platonic solidsand almost half a century after that, inBertrand provided a more elegant proof by faceting them.
The following year, Arthur Cayley gave the Kepler—Poinsot polyhedra the names by which they are generally known today. 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 two of the regular star polyhedra:. Regular star polyhedra first appear in Renaissance art.
;oinsot small stellated dodecahedron is depicted in a marble tarsia on the floor of St.
Mark’s Basilica, Venice, Italy, dating from ca. Wenzel Jamnitzer published his book of woodcuts Perspectiva Corporum Regularium in 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. In the 20th Century, Artist M. 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 s puzzle Alexander’s Star. The star spans 14 meters, and consists of an icosahedron and a dodecahedron inside a great stellated dodecahedron. Media related to Kepler-Poinsot solids at Wikimedia Commons. Kepler—Poinsot polyhedron The four Kepler—Poinsot polyhedra are illustrated above. One face of each figure is shown yellow and outlined in red. Characteristics Non-convexity These figures have pentagrams star pentagons as faces or vertex figures.
Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the Euler relation does not always hold.
A modified form of Euler’s formula, using density D of the vertex figures and faces was given by Arthur Cayleyand holds both for convex polyhedra where the correction factors are all 1and the Kepler—Poinsot polyhedra: Duality The Kepler—Poinsot polyhedra exist in dual pairs: Small stellated dodecahedron and great dodecahedron and Great stellated dodecahedron and great icosahedron.
These share the same vertex and edge arrangements: The icosahedronsmall stellated dodecahedrongreat icosahedronand great dodecahedron. The small stellated dodecahedron and great icosahedron. The dodecahedron and great stellated dodecahedron. The icosahedron and great dodecahedron. The four Kepler—Poinsot polyhedra are illustrated above.
Small stellated dodecahedron sissid. Great stellated dodecahedron gissid.