Solid of Kepler-Poinsot

The solid of Kepler-Poinsot are the spangled polyhedral regular. Each one has faces which are convex polygons regular congruent or polygons spangled and has the same number of faces meeting at each top (to compare with the solid of Plato).

There exist four Solides of Kepler-Poinsot :

It is unhappy that these figures were known as “solid” because they are included/understood better as a surfaces .

Geometry

The small one and the large spangled dodecahedron has faces in form of Pentagramme S nonconvex regular. The large dodecahedron and the large icosahedron have faces in form of pentagon S convex, but have figures of tops in form of Pentagramme S. the first pair and the second is duaux the ones the others.

These figures can induce in error, because they include the Pentagramme S like faces and figures of tops. The faces and the tops can be supposed in an erroneous way where the faces are cut, but they are not counted.

If the intersections are counted like new edges and new tops, they will not be regular, but they can still be considered in the Stellation S. (see also the Liste of the models of polyhedrons of Wenninger)

History

A small spangled dodecahedron appears in mosaic of the ground of the Basilique Saint-Marc of Venice in Italy. It dates from the XVe century and is sometimes allotted to Paolo Uccello.

In its Perspectiva corporum regularium (Perspective of the regular solids), a book of engravings on wood published in the XVIe century, Wenzel Jamnitzer depicts the large dodecahedron. It is clear, starting from the general arrangement of the book, which it regards the five solids of Plato as regular, without including/understanding the regular nature of its large dodecahedron. It depicts also a figure often confused with the large spangled dodecahedron, although triangular surfaces of the arms are not completely coplanar, it has 60 triangular faces.

The solid of Kepler were discovered by Johannes Kepler in 1619. It obtained them by Stellation regular convex dodecahedron, initially by treating it like a surface rather than a solid. It foot-note that by extending the edges or the faces of the convex dodecahedron until they meets again, he could obtain spangled pentagons. Moreover, he recognized that these spangled pentagons were also regular. He found two dodecahedrons spangled in this manner, the small one and the large one. Each one has the central convex area of each face " cachée" with the interior, only the visible triangular arm. The final stage of Kepler was to recognize that these polyhedrons coincided with the definition of the regular solids, even if they were not convex, like were the solid of Plato traditional.

In 1809, Louis Poinsot redécouvrit these two figures. He also considered the spangled tops as well as the faces spangled, and thus discovered two regular stars moreover, the large icosahedron and the large dodecahedron. Certain people call those the solid of Poinsot . Poinsot did not know if he had discovered all the regular spangled polyhedrons.

Three years later, Augustin Cauchy showed that the list was complete, and almost a half-century later Bertrand provides a more elegant demonstration in faceting the solid of Plato.

The solids of Kepler-Poinsot accepted their names the following year, in 1859, by Arthur Cayley.

The characteristic of Euler

A solid of Kepler-Poinsot covers its more circumscribed sphere once. Because of that, they are not necessarily topologically equivalent with the sphere like are the solids of Plato, and in particular, the Caractéristique of Euler

S - HAS + F = 2

is not always valid.

The value of the Caractéristique of Euler χ depends on the form on the polyhedron. Let us consider for example the small spangled dodecahedron. It consists of a Dodécaèdre with a pentagonal pyramid on each one of its twelve faces. Each of the twelve faces is a Pentagramme with the pentagonal part hidden in the solid. The part external of each face makes up of five triangles which are touched only in five points. Alternatively, we could count these triangles as separate faces - it there of 60 (but they are only isosceles triangles, and not of the regular polygons). In a similar way, each edge would be now divided into three edges (but then, they are of two kinds). " five points" that one has just mentioned, the 20 additional tops form together, thus, we have a total of 32 tops (of kinds, again). The hidden internal pentagons are not necessary to form the surface of the polyhedron and can disappear. Now, the relation of Euler is valid: 60 - 90 + 32 = 2. Nevertheless, this polyhedron that is not described by the symbol of Schläfli {5/2,5}, and thus, cannot be a solid of Kepler-Poinsot even if it still resembles it of appearance.

Trivia

a dissection of a large dodecahedron was used for the puzzle of the Eighties, the star of Alexandre.
interest of the artist M.C. Escher in the geometrical forms often resulted basing its work on regular solids or in including them; Gravitation is based on a small spangled dodecahedron.

See too

regular Polytope
Polyhedral regular
List of polytopes regular
Polyhedral made up uniform
Polyhedral
Stellation

References

J. Bertrand, Note on the theory of the regular polyhedrons, Reports of the meetings of the Academy of Science , 46 (1858), pp. 79-82, 117.
Augustin Louis Cauchy, Research on the polyhedrons. J. Polytechnic school 9,68-86, 1813.
Arthur Cayley, One Poinsot' S Furnace New Regular Solids. Philosophies. Mag. 17 , pp. 123-127 and 209,1859.
P. Cromwell, Polyhedra , Cabridgre University Near, Hbk. 1997, km No. 1999.
Theoni Dads, (The Kepler-Poinsot Solids) The Joy off Mathematics . San Carlos, CA: Wide World Publ. will /Tetra, p. 113,1989.
Louis Poinsot, Memoire on the polygons and polyhedrons. J. Polytechnic school 9 , pp. 16-48, 1810.
Magnus Wenninger Dual Models Cambridge, England: Cambridge University Close, pp. 39-41, 1983.

External bonds

Solid of Kepler-Poinsot
Model paper of the Model polyhedrons of Kepler-Poinsot
paper plain of polyhedrons of Kepler-Poinsot
uniform polyhedrons
Model vrml of the polyhedrons of Kepler-Poinsot
Stellation and to facet - a short history

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