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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
and
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
, and differ from these in having regular
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...
mic
face The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect th ...
s or
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
s. They can all be seen as three-dimensional analogues of the pentagram in one way or another.


Characteristics


Sizes

The great icosahedron edge length is \phi^4 = \tfrac12\bigl(7+3\sqrt5\,\bigr) times the original icosahedron edge length. The small stellated dodecahedron, great dodecahedron, and great stellated dodecahedron edge lengths are respectively \phi^3 = 2+\sqrt5, \phi^2 = \tfrac12\bigl(3+\sqrt5\,\bigr), and \phi^5 = \tfrac12\bigl(11+5\sqrt5\,\bigr) times the original dodecahedron edge length.


Non-convexity

These figures have
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...
s (star pentagons) as faces or vertex figures. The
small Small means of insignificant size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to three geometrical measures: length, area, or ...
and
great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at eac ...
have nonconvex regular
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...
faces. The
great dodecahedron In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vert ...
and great icosahedron have
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
polygonal faces, but pentagrammic
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
s. 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 spheres at the true vertices, and blue rods along the true edges. For example, the
small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentag ...
has 12
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...
faces with the central
pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
al part hidden inside the solid. The visible parts of each face comprise five
isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...
s 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
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for ...
holds: 60 − 90 + 32 = 2. However, this polyhedron is no longer the one described by the
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
, and so can not be a Kepler–Poinsot solid even though it still looks like one from outside.


Euler characteristic χ

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 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 :\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 Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
(''D'') of the
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
s (d_v) and faces (d_f) was given by
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...
, 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 = 2D.


Duality and Petrie polygons

The Kepler–Poinsot polyhedra exist in dual pairs. Duals have the same
Petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...
, or more precisely, Petrie polygons with the same two dimensional projection. The following images show the two dual compounds with the same edge radius. They also show that the Petrie polygons are skew. Two relationships described in the article below are also easily seen in the images: That the violet edges are the same, and that the green faces lie in the same planes.


Summary


Relationships among the regular polyhedra


Conway's operational terminology

John Conway defines the Kepler–Poinsot polyhedra as ''greatenings'' and ''stellations'' of the convex solids.
In his naming convention, the
small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentag ...
is just the ''stellated dodecahedron''. ''Stellation'' changes pentagonal faces into pentagrams. (In this sense stellation is a unique operation, and not to be confused with the more general
stellation In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific ...
described below.) ''Greatening'' maintains the type of faces, shifting and resizing them into parallel planes.


Stellations and facetings

The great icosahedron is one of the
stellation In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific ...
s of the
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
. (See ''
The Fifty-Nine Icosahedra ''The Fifty-Nine Icosahedra'' is a book written and illustrated by Harold Scott MacDonald Coxeter, H. S. M. Coxeter, Patrick du Val, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic re ...
'')
The three others are all the stellations of the
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
. The
great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at eac ...
is a
faceting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new Vertex (geometry), vertices. New edges of a faceted po ...
of the dodecahedron.
The three others are facetings of the icosahedron. If the intersections are treated as new edges and vertices, the figures obtained will not be regular, but they can still be considered
stellation In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific ...
s. (See also List of Wenninger polyhedron models)


Shared vertices and edges

The great stellated dodecahedron shares its vertices with the dodecahedron. The other three Kepler–Poinsot polyhedra share theirs with the icosahedron.


The stellated dodecahedra


Hull and core

The
small Small means of insignificant size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to three geometrical measures: length, area, or ...
and
great Great may refer to: Descriptions or measurements * Great, a relative measurement in physical space, see Size * Greatness, being divine, majestic, superior, majestic, or transcendent People * List of people known as "the Great" * Artel Great (bo ...
stellated dodecahedron can be seen as a regular and a
great dodecahedron In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vert ...
with their edges and faces extended until they intersect.
The pentagon faces of these cores are the invisible parts of the star polyhedra's pentagram faces.
For the small stellated dodecahedron the hull is \varphi times bigger than the core, and for the great it is \varphi + 1 = \varphi^2 times bigger.
(The midradius is a common measure to compare the size of different polyhedra.)


Augmentations

Traditionally the two star polyhedra have been defined as ''augmentations'' (or ''cumulations''), Kepler calls the small stellation an ''augmented dodecahedron'' (then nicknaming it ''hedgehog''). These
naïve Naivety (also spelled naïvety), naiveness, or naïveté is the state of being naive. It refers to an apparent or actual lack of experience and sophistication, often describing a neglect of pragmatism in favor of moral idealism. A ''naïve'' may ...
definitions are still used. E.g.
MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...
states that the two star polyhedra can be constructed by adding pyramids to the faces of the Platonic solids.


Symmetry

All Kepler–Poinsot polyhedra have full
icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
, just like their convex hulls. The great icosahedron and its dual resemble the icosahedron and its dual in that they have faces and vertices on the 3-fold (yellow) and 5-fold (red) symmetry axes.
In the
great dodecahedron In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vert ...
and its dual all faces and vertices are on 5-fold symmetry axes (so there are no yellow elements in these images). The following table shows the solids in pairs of duals. In the top row they are shown with pyritohedral symmetry, in the bottom row with icosahedral symmetry (to which the mentioned colors refer). The table below shows
orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Plane (mathematics), two dimensions. Orthographic projection is a form of parallel projection in ...
s from the 5-fold (red), 3-fold (yellow) and 2-fold (blue) symmetry axes.


History

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 Venice ( ; ; , formerly ) is a city in northeastern Italy and the capital of the Veneto Regions of Italy, region. It is built on a group of 118 islands that are separated by expanses of open water and by canals; portions of the city are li ...
, Italy. It dates from the 15th century and is sometimes attributed to
Paolo Uccello Paolo Uccello ( , ; 1397 – 10 December 1475), born Paolo di Dono, was an Italian Renaissance painter and mathematician from Florence who was notable for his pioneering work on visual Perspective (graphical), perspective in art. In his book ''Liv ...
. In his '' Perspectiva corporum regularium'' (''Perspectives of the regular solids''), a book of woodcuts published in 1568, Wenzel Jamnitzer depicts the
great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at eac ...
and a
great dodecahedron In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vert ...
(both shown below). There is also a truncated version of the
small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentag ...
. It is clear from the general arrangement of the book that he regarded only the five Platonic solids as regular. The small and great stellated dodecahedra, sometimes called the Kepler polyhedra, were first recognized as regular by
Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...
around 1619.H.S.M. Coxeter, P. Du Val, H.T. Flather and J.F. Petrie; ''The Fifty-Nine Icosahedra'', 3rd Edition, Tarquin, 1999. p.11 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 Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
, as the traditional
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s were. In 1809,
Louis Poinsot Louis Poinsot (; 3 January 1777 – 5 December 1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a ...
rediscovered Kepler's figures, by assembling star pentagons around each vertex. He also assembled convex polygons around star vertices to discover two 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 Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
proved the list complete by stellating the
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s, and almost half a century after that, in 1858, Bertrand provided a more elegant proof by
faceting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new Vertex (geometry), vertices. New edges of a faceted po ...
them. The following year,
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...
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 the
small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentag ...
is just the ''stellated dodecahedron''.


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 Uccello. In the 20th century, artist
M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...
's interest in geometric forms often led to works based on or including regular solids; ''
Gravitation In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
'' is based on a small stellated dodecahedron. A
dissection Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause of ...
of the great dodecahedron was used for the 1980s puzzle Alexander's Star. Norwegian artist Vebjørn Sand's sculpture ''The Kepler Star'' is displayed near Oslo Airport, Gardermoen. The star spans 14 meters, and consists of an
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
and a
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
inside a great stellated dodecahedron.


See also

*
Regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...
*
Regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitive group action, transitively on its Flag (geometry), flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In ...
*
List of regular polytopes This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. Overview This table shows a summary of regular polytope counts by rank. There are no Euclidean regular star tessellations in any number of dimensions. ...
*
Uniform polyhedron In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform po ...
* Uniform star polyhedron *
Polyhedral compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common Centroid, centre. They are the three-dimensional analogs of star polygon#Regular compounds, polygonal compounds such as the hexagram. The oute ...
* Regular star 4-polytope – the ten regular star
4-polytope In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: Vertex (geometry), vertices, Edge (geo ...
s, 4-dimensional analogues of the Kepler–Poinsot polyhedra


References


Notes


Bibliography

* 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 Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
, ''Recherches sur les polyèdres.'' J. de l'École Polytechnique 9, 68–86, 1813. *
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...
, On Poinsot's Four New Regular Solids. ''Phil. Mag.'' 17, pp. 123–127 and 209, 1859. * John H. Conway, Heidi Burgiel,
Chaim Goodman-Strauss Chaim Goodman-Strauss (born June 22, 1967 in Austin, Texas) is an American mathematician who works in convex geometry, especially aperiodic tiling. He retired from the faculty of the University of Arkansas and currently serves as outreach mathem ...
, ''The Symmetry of Things'' 2008, (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,

** (Paper 1) H.S.M. Coxeter, ''The Nine Regular Solids'' roc. 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)'' lemente der Mathematik 44 (2) (1989) 25–36* Theoni Pappas, (The Kepler–Poinsot Solids) ''The Joy of Mathematics''. San Carlos, CA: Wide World Publ./Tetra, p. 113, 1989. *
Louis Poinsot Louis Poinsot (; 3 January 1777 – 5 December 1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a ...
, 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-Strauss Chaim Goodman-Strauss (born June 22, 1967 in Austin, Texas) is an American mathematician who works in convex geometry, especially aperiodic tiling. He retired from the faculty of the University of Arkansas and currently serves as outreach mathem ...
, ''The Symmetries of Things'' 2008, (Chapter 26. pp. 404: Regular star-polytopes Dimension 3) * Chapter 8: Kepler Poisot polyhedra


External links

*
Paper models of Kepler–Poinsot polyhedraFree paper models (nets) of Kepler–Poinsot polyhedraThe Uniform Polyhedra
in Visual Polyhedra
Stella: Polyhedron Navigator
Software used to create many of the images on this page. {{DEFAULTSORT:Kepler-Poinsot Polyhedron Johannes Kepler Nonconvex polyhedra