A dodecahedron, one of the five Platonic solids.

In mathematics, a regular polytope is a geometric figure with a high degree of symmetry. Examples in two dimensions include the square, the regular pentagon and hexagon, and so on. In three dimensions the regular polytopes include the cube, the dodecahedron, and all Platonic solids. There exist examples in higher dimensions also. Circles and spheres, although highly symmetric, are not considered polytopes because they do not have flat faces. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians. Download high resolution version (847x829, 63 KB)Dodecahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (847x829, 63 KB)Dodecahedron, made by me using POV-Ray, see image:poly. ... Jump to: navigation, search A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... Jump to: navigation, search In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Jump to: navigation, search Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... Jump to: navigation, search Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... In plane geometry, a square is a polygon with four equal sides and equal angles. ... Jump to: navigation, search In geometry, a pentagon is any five-sided polygon. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... Jump to: navigation, search A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... Jump to: navigation, search In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ... Jump to: navigation, search In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ... A sphere is a perfectly symmetrical geometrical object. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... Jump to: navigation, search Aesthetics (also esthetics and æsthetics) is the philosophy of beauty and art. ...

Many regular polytopes, at least in two and three dimensions, exist in nature and have been known since prehistory. The earliest surviving mathematical treatment of these objects comes to us from ancient Greek mathematicians such as Euclid. Indeed, Euclid wrote a systematic study of mathematics, publishing it under the title Elements, which built up a logical theory of geometry and number theory. His work concluded with mathematical descriptions of the five Platonic solids. Jump to: navigation, search Ancient Greece is the term used to describe the Greek-speaking world in ancient times. ... Jump to: navigation, search Euclid Euclid of Alexandria (Greek: ) (ca. ... Euclids Elements (Greek Στοιχεία) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. ... Jump to: navigation, search Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ...

The definition of the regular polytopes remained static for many centuries after Euclid. Overall however, the history of the study of regular polytopes has been one where the definition, in fits and starts, was gradually "relaxed", allowing more and more different objects to be considered among their number. The five Platonic solids were joined, towards the middle of the second millennium, by the Kepler-Poinsot polyhedra. By the end of the 19th century, mathematicians had begun to consider regular polytopes in four and higher dimensions, such as the tesseract and the 24-cell. The latter are quite hard to visualise, but still retain the aesthetically pleasing symmetry of their lower dimensional cousins. Harder still to imagine are the more modern abstract regular polytopes such as the 57-cell or the 11-cell. Mathematicians who study such objects insist, however, that the aesthetic qualities of these objects remain. A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Jump to: navigation, search In geometry, the tesseract, or hypercube, is a regular convex polychoron with eight cubical cells. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... The hemicube is constructed from the cube by treating opposite edges (likewise faces and corners) as really the same edge. ... In mathematics, the 57-cell is a four-dimensional self-dual abstract regular polytope. ... In mathematics, the 11-cell is a four-dimensional self-dual abstract regular polytope. ...

History of discovery

The history of discovery of the regular polytopes can be characterised by a gradual broadening of widespread understanding of the term. Gradually, the term "regular polytope" has been given successively wider meaning, allowing more different geometric objects to be so labeled. With each widening, new geometric figures are uncovered — these new figures usually being completely unknown to previous generations.

Prehistory

Greeks are usually credited with being the first to discover the regular polyhedra. The earliest known written records of these shapes come from Greek authors, who also gave the first known mathematical description of them.

Across the Mediterranean Sea was another civilisation, the Etruscan. There is a possibility that these people predated the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua (in Northern Italy) in the late 1800s of a dodecahedron made of soapstone, and dating back more than 2,500 years (Lindemann, 1987). It may be argued, however, that the construction of this form was inspired by the pyritohedron (mentioned elsewhere in this article), as pyrite minerals are relatively abundant in that part of the world. Jump to: navigation, search Satellite image The Mediterranean Sea is a part of the Atlantic Ocean almost completely enclosed by land, on the north by Europe, on the south by Africa, and on the east by Asia. ... Jump to: navigation, search Map showing the extent of the Etruscan civilization and the twelve Etruscan League cities. ... Location within Italy Tronco Maestro Riviera: a pedestrian walk along a section of the inland waterway or naviglio interno of Padua The city of Padua (Lat. ... Jump to: navigation, search Events and Trends Beginning of the Napoleonic Wars (1803 - 1815). ... The lid of a soapstone box to show the characteristic look of the stone. ...

Even pre-dating the Etruscans however, come discoveries from Scotland of stones carved in shapes showing the symmetry of all five of the platonic solids. These stones, dating back to perhaps 4,000 years, show not only the form of each of the five platonic solids, but also the relations of duality amongst them (that is, that the centres of the faces of the cube gives the vertices of an octahedron, and so on). Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. It is impossible to know why these objects were made, or how the sculptor gained the inspiration for them. Timeline of Scottish history Caledonia List of not fully sovereign nations Subdivisions of Scotland National parks (Scotland) Traditional music of Scotland Flower of Scotland Wars of Scottish Independence National Trust for Scotland Historic houses in Scotland Castles in Scotland Museums in Scotland Abbeys and priories in Scotland Gardens in Scotland... The Ashmolean Museum (in full the Ashmolean Museum of Art and Archaeology) in Oxford, England is the worlds first university museum. ... The University of Oxford, located in the city of Oxford in England, is the oldest university in the English-speaking world. ...

There is no proof that the Etruscans or ancient Scots had any mathematical understanding of the regular solids — nor is there any proof that they did not. The root of the human discovery of the three-dimensional polytopes, particularly of the simpler ones, is probably impossible to trace. In any case, it is the Greek mathematical treatment of the platonic solids that has come down to us, and inspired our modern mathematical treatment of them.

Greeks

Some authors (Sanford, 1930) credit Pythagoras (550 BC) with being familiar with the Platonic solids, whereas others indicate that he may only have been familiar with the tetrahedron, cube, and dodecahedron, crediting the discovery of the other two to Theaetetus (an Athenian), who in any case gave a mathematical description of all five (Van der Waerden, 1954), (Euclid, book XIII). H.S.M. Coxeter (Coxeter, 1948, Section 1.9) credits Plato (400 BC) with having made models of them, and mentions that one of the earlier Pythagoreans used all five in a correspondence between the polyhedra and the nature of the universe as it was then perceived. It is from Plato's name that the term Platonic Solids is derived. Jump to: navigation, search This topic is considered to be an essential subject on Wikipedia. ... Centuries: 7th century BC - 6th century BC - 5th century BC Decades: 600s BC - 590s BC - 580s BC - 570s BC - 560s BC - 550s BC - 540s BC - 530s BC - 520s BC - 510s BC - 500s BC Events and Trends Carthage conquers Sicily, Sardinia and Corsica 559 BC - King Cambyses I of Anshan dies... Theaetetus ( 417 B.C. – 369 B.C.) was a Greek mathematician of Geometry. ... Jump to: navigation, search The Acropolis in central Athens, one of the most important landmarks in world history. ... H(arold). ... Jump to: navigation, search Statue of a philosopher, presumably Plato, in Delphi. ... Centuries: 5th century BC - 4th century BC - 3rd century BC Decades: 450s BC 440s BC 430s BC 420s BC 410s BC - 400s BC - 390s BC 380s BC 370s BC 360s BC 350s BC Years: 405 BC 404 BC 403 BC 402 BC 401 BC - 400 BC - 399 BC 398 BC... Jump to: navigation, search The Pythagoreans were a Hellenic organization of astronomers, musicians, mathematicians, and philosophers who believed that all things are, essentially, numeric. ...

Star polyhedra

For almost 2000 years, the concept of a regular polytope remained as developed by the ancient Greek mathematicians. One might characterise the Greek definition as follows:

• A regular polygon is a (convex) planar figure with all edges equal and all corners equal
• A regular polyhedron is a solid (convex) figure with all faces being congruent regular polygons, the same number arranged around each vertex.

This definition rules out, for example, the square pyramid (since although all the faces are regular, the square base is not congruent to the triangular sides), or the shape formed by joining two tetrahedra together (since although all faces would be equilateral triangles, that is, congruent and regular, some vertices have 3 triangles and others have 4). In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... In geometry, the square pyramid, a pyramid with a square base and equilateral sides, is one of the Johnson solids (J1). ...

Finally, starting in the 15th century, the next generation of regular polytopes began to surface. The regular star polyhedra are called the Kepler-Poinsot solids, after Johannes Kepler and Louis Poinsot. They have non-convex regular polygons, typically pentagrams, either as faces or as vertex figures (circuits around each corner). Kepler's two polyhedra were constructed by others before his time, but Kepler was the first to recognise that they could be considered "regular" if one removed the restriction that regular polytopes be convex. Later, Poinsot discovered the remaining two. Cayley gave them English names which have become accepted. They are: (Kepler's) the small stellated dodecahedron and great stellated dodecahedron, and (Poinsot's) the great icosahedron and great dodecahedron. A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ... Jump to: navigation, search Johannes Kepler Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German mathematician, astronomer and astrologer of famed brilliance. ... Louis Poinsot (1777 - 1859) was a French mathematician and physicist. ...

The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the platonic solids was given a big push by H. S. M. Coxeter and others in 1938, with the now famous paper The 59 icosahedra. This work has recently been re-published (Coxeter, 1999). Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ... Harold Scott MacDonald Donald Coxeter, CC , Ph. ... Jump to: navigation, search 1938 was a common year starting on Saturday (link will take you to calendar). ...

The reciprocal process to stellation is called facetting (or faceting). Every stellation of one polytope is dual, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. N. J. Bridge (1974) listed the simpler facettings of the dodecahedron, and reciprocated them to discover a stellation of the icosahedron that was missing from the famous "59". More have been discovered since, and the story is not yet ended.

Higher-dimensional polytopes

It was not until the 19th century that a Swiss mathematician, Ludwig Schläfli, examined and characterised the regular polytopes in higher dimensions. His efforts were first published in full in (Schläfli, 1901), six years posthumously, although parts of it were published in 1855 and 1858 (Schläfli, 1855), (Schläfli, 1858). Interestingly, between 1880 and 1900, Schläfli's results were rediscovered independently by at least nine other mathematicians — see (Coxeter, 1948, pp143–144) for more details. This article needs to be cleaned up to conform to a higher standard of quality. ... 1855 was a common year starting on Monday (see link for calendar). ... Jump to: navigation, search 1858 is a common year starting on Friday. ...

The latter reference is probably the most comprehensive printed treatment of Schläfli's and similar results to date. Schläfli showed that there are six regular convex polytopes in 4 dimensions, and exactly three in each higher dimension. Descriptions of these may be found in the List of regular polytopes. Also of interest are the regular stellated 4-polytopes, not discovered by Schläfli. They are also described on the List of regular polytopes. In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ... Jump to: navigation, search This page lists the regular polytopes in Euclidean space. ... Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ...

At the start of the 20th century, the definition of a regular polytope was as follows.

• A regular polygon is a polygon whose edges are all equal and whose angles are all equal.
• A regular polyhedron is a polyhedron whose faces are all congruent regular polygons, and whose vertex figures are all congruent and regular.
• And so on, a regular n-polytope is an n-dimensional polytope whose (n − 1)-dimensional faces are all regular and congruent, and whose vertex figures are all regular and congruent.

This is a "recursive" definition. It defines regularity of higher dimensional figures in terms of regular figures of a lower dimension. There is an equivalent (non-recursive) definition, which states that a polytope is regular if it has a sufficient degree of symmetry. In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...

• An n-polytope is regular if any list consisting of:

a vertex, an edge containing it, a 2-dimensional face containing them, and so on up to n − 1 dimensions can be mapped to any other by a symmetry of the polytope.

So for example, the cube is regular because if we choose a vertex of the cube, and one of the three edges it is on, and one of the two faces containing the edge, then this triplet (vertex, edge, face) can be mapped to any other such triplet by a suitable symmetry of the cube.

Abstract regular polytopes

In the 20th century, some important developments were made. The symmetry groups of the classical regular polytopes were generalised into what are now called Coxeter groups. Coxeter groups also include the symmetry groups of regular tessellations of space or of the plane. For example, the symmetry group of an infinite chessboard would be a Coxeter group. Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... Jump to: navigation, search In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a Coxeter group is a group with a presentation of the form where mi,j ≥ 2; the condition mi,j = ∞ means no relation of the form (xixj)m should be imposed. ... A tessellated plane A tessellation of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps. ...

In the 1960s Branko Grünbaum issued a call to the geometric community to consider more abstract types of regular polytopes that he called polystromata. He developed the theory of polystromata, showing examples of new objects he called regular apeirotopes, that is, regular polytopes with infinitely many faces. A simple example of an apeirogon would be a zig-zag. It seems to satisfy the definition of a regular polygon — all the edges are the same length, and all the angles are the same. More importantly, perhaps, there are symmetries of the zig-zag that can map any pair of a vertex and attached edge to any other. Jump to: navigation, search The 1960s in its most obvious sense refers to the decade between 1960 and 1969, but the expression has taken on a wider meaning over the past twenty years. ... Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ... Jump to: navigation, search Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. ...

The "hemicube" is constructed from the cube by treating opposite edges (likewise faces and corners) as really the same edge. It has 3 faces, 6 edges, and 4 corners.

Grünbaum also discovered the 11-cell, a beautiful four-dimensional self-dual object whose facets are not icosahedra, but are "hemi-icosahedra" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the same face (Grünbaum, 1977). The hemi-icosahedron has only 10 triangular faces, and 6 vertices, unlike the icosahedron, which has 20 and 12. Image File history File links Image showing how a hemicube is constructed. ... In mathematics, the 11-cell is a four-dimensional self-dual abstract regular polytope. ... Jump to: navigation, search Aesthetics (also esthetics and æsthetics) is the philosophy of beauty and art. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...

This concept may be easier for the reader to grasp if one considers the relationship of the cube and the hemicube. An ordinary cube has 8 corners, they could be labeled A to H, with A opposite H, B opposite G, and so on. In a hemicube, A and H would be treated as the same corner. So would B and G, and so on. The edge AB would become the same edge as GH, and the face ABEF would become the same face as CDGH. The new shape has only three faces, 6 edges and 4 corners.

A few years after Grünbaum's discovery of the 11-cell, H. S. M. Coxeter independently discovered the same shape. He had earlier discovered a similar polytope, the 57-cell (Coxeter 1982, 1984). In mathematics, the 11-cell is a four-dimensional self-dual abstract regular polytope. ... Harold Scott MacDonald Donald Coxeter, CC , Ph. ... In mathematics, the 57-cell is a four-dimensional self-dual abstract regular polytope. ...

Unfortunately perhaps, the study of polystromata fell by the wayside, as mathematicians turned their interest to other similar abstract geometric concepts, including the concepts of buildings and geometries, abstract polytopes, Euler posets and others. The 11-cell and 57-cell remain important examples of (in particular) abstract regular polytopes.

An abstract regular polytope is defined as a set, supposed to represent the set of vertices, edges, faces and so on of a polytope, with an idea of which of these "lie on" which others. Certain restrictions are imposed on the set that are similar to properties satisfied by the classical regular polytopes (including the platonic solids). The restrictions, however, are loose enough that regular tessellations, hemicubes, and even objects as strange as the 11-cell or stranger, are all examples of regular polytopes. The theory has been developed largely by Egon Schulte and Peter McMullen (McMullen, 2002), but other researchers have also made contributions.

Constructions

Polygons

The traditional way to "construct" a regular polygon, or indeed any other figure on the plane, is by use of a ruler (more properly, a straightedge) and a compass. Constructing some regular polygons is very easy (the easiest is perhaps the equilateral triangle), some harder, or "impossible". The simplest few regular polygons that are "impossible" to construct using just a ruler and a compass are the n-sided polygons with n equal to 7, 9, 11, 13, 14, 18, 19, 21, and so forth. (See Ruler-and-compass constructions for more information.) A ruler is an instrument used in geometry and technical drawing to measure short distances and/or to rule straight lines. ... A straightedge is a tool similar to a ruler, but without markings. ... a compass In drafting, a compass (or pair of compasses) is an instrument]] used by mathematicians and craftsmen in for drawing or inscribing a circle or arc. ... A number of ancient problems in plane geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ...

One thing that is commonly overlooked about ruler-and-compass constructions is that for many of the polygons, the constructions are impossible to perform using a real ruler and compass. For example, it has been shown that it is "possible" to construct a regular 65537-gon using only these tools. However, if one were to do so, making each edge 1cm long, the polygon would need to be over 200m across, and the radii of the incircle and the circumcircle would differ by less than a quarter of a micrometre — approximately the wavelength of ultraviolet light! One would need an ultraviolet camera to distinguish between this polygon and a circle — not to mention a very sharp pencil to draw it! The construction in any case would be extremely complex, and is only of theoretical interest. Jump to: navigation, search Ultraviolet (UV) radiation is electromagnetic radiation of a wavelength shorter than that of the visible region, but longer than that of soft X-rays. ...

Another thing that is commonly overlooked is that even "unconstructable" polygons, can be constructed, if one is satisfied with an approximation to the desired polygon, rather than with an exact representation. Indeed, with real compasses and rulers held by real hands and drawn on real paper, approximations are the best that can be achieved even for the so-called "constructable" polygons. An example that illustrates this very clearly is the following simple construction of a regular heptagon: A heptagon is a plane figure with seven sides and seven angles. ...

• Use the compass to draw a circle.
• Choose a point B on the circle.
• Without adjusting the compass, place the compass point on B, and mark two more points A and C on the circle (on opposite sides of B)
• Bisect the chord AC, to find the midpoint D
• Set the compass to distance AD
• Use the new compass setting to mark 7 points around the original circle

This construction will, to the accuracy of a typical pencil, construct a regular heptagon. If the radius of the circle is 5cm, the distance AD will be 4.3301cm. The edges of a regular heptagon should be 4.3388cm, a difference of less than 0.1mm. Very few school students, or even draftsmen, have pencils or compasses with points as sharp as that. Jump to: navigation, search In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ... For the numerical analysis algorithm, see bisection method. ... A chord is a geometric figure. ... In mathematics, a line segment is a part of a line that is bounded by two end points. ... An example of a technical drawing with orthographic and isometric view. ...

Polyhedra

Euclid's elements (see for example Euclid's Elements) gave what amount to ruler-and-compass constructions for the five platonic solids. However, the merely practical question of how one might draw a straight line in space, even with a ruler, might lead one to question what exactly it means to "construct" a regular polyhedron. (One could ask the same question about the polygons, of course.)

The English word "construct" has the connotation of systematically building the thing constructed. The most common way presented to construct a regular polyhedron is via a fold-out net. To obtain a fold-out net of a polyhedron, one takes the surface of the polyhedron and cuts it along just enough edges so that the surface may be laid out flat. This gives a plan for the net of the unfolded polyhedron. Since the platonic solids have only triangles, squares and pentagons for faces, and these are all constructible with a ruler and compass, there exist ruler-and-compass methods for drawing these fold-out nets. The same applies to Kepler's polyhedra.

If this net is drawn on cardboard, or similar foldable material (for example, sheet metal), the net may be cut out, folded along the uncut edges, joined along the appropriate cut edges, and so forming the polyhedron for which the net was designed. For a given polyhedron there may be many fold-out nets. For example, there are 11 for the cube, and over 40000 for the dodecahedron. Some interesting fold-out nets of the cube, octahedron, dodecahedron and icosahedron are available here.

Numerous children's toys, generally aimed at the teen or pre-teen age bracket, allow experimentation with regular polygons and polyhedra. For example, klikko provides sets of plastic triangles, squares, pentagons and hexagons that can be joined edge-to-edge in a large number of different ways. A child playing with such a toy could re-discover the platonic solids (or the archimedean solids), especially if given a little guidance from a knowledgeable adult. In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...

In theory, almost any material may be used to construct regular polyhedra. Instructions for building origami models may be found here or here, for example. They may be carved out of wood, modeled out of wire, formed from stained glass. The imagination is the limit. Jump to: navigation, search A crane and papers of the same size used to fold it Origami (折り紙 or 折紙 origami paper folding) is the artof japanese hACKSJapanese paper folding. ...

Higher dimensions

In higher dimensions, it becomes harder to say what one means by "constructing" the objects. Clearly, in a 3-dimensional universe, it is impossible to build a physical model of a 4-dimensional object. There are several approaches normally taken to overcome this matter.

The first approach is to embed the higher-dimensional objects in three-dimensional space, using methods analogous to the ways in which three-dimensional objects are drawn on the plane. For example, the fold out nets mentioned in the previous section have higher-dimensional equivalents. Some of these may be viewed at [1]. One might even imagine building a model of this fold-out net, as one draws a polyhedron's fold-out net on a piece of paper. Sadly, we could never do the necessary folding of the 3-dimensional structure to obtain the 4-dimensional polytope, because of the constraints of the physical universe. Another way to "draw" the higher-dimensional shapes in 3 dimensions is via some kind of projection, for example, the analogue of either orthographic or perspective projection. Coxeter's famous book on polytopes (Coxeter, 1948) has some examples of such orthographic projections. Other examples may be found on the web (see for example [2]). Note that immersing the 4-dimensional objects directly into two dimensions is quite confusing. Easier to understand are 3-d models of the projections. Such models are occasionally found in science museums or mathematics departments of universities (such as that of the Université Libre de Bruxelles). Jump to: navigation, search This article is about technical drawings. ... Jump to: navigation, search This article needs to be cleaned up to conform to a higher standard of quality. ... The Université Libre de Bruxelles (or ULB) is a French-speaking university in Brussels, Belgium. ...

The intersection of a four (or higher) dimensional regular polytope with a three-dimensional hyperplane will be a polytope (not necessarily regular). If the hyperplane is moved through the shape, the three-dimensional slices can be combined, animated into a kind of four dimensional object, where the fourth dimension is taken to be time. In this way, we can see (if not fully grasp) the full four-dimensional structure of the four-dimensional regular polytopes, via such cutaway cross sections. This is analogous to the way a CAT scan reassembles two-dimensional images to form a 3-dimensional representation of the organs being scanned. The ideal would be an animated hologram of some sort, however, even a simple animation such as the one shown can already give some limited insight into the structure of the polytope. Animated cross section of a 24-cell. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... Jump to: navigation, search Animation (plural: Animations) is the illusion of motion created by the consecutive display of images of static elements. ... CAT apparatus in a hospital Computed axial tomography (CAT), computer-assisted tomography, computed tomography, CT, or body section roentgenography is the process of using digital processing to generate a three-dimensional image of the internals of an object from a large series of two-dimensional X-ray images taken around... This article is about the photographic technique. ...

Another way a three-dimensional viewer can comprehend the structure of a four-dimensional object is through being "immersed" in the object, perhaps via some form of virtual reality technology. To understand how this might work, imagine what one would see if space were filled with cubes. The viewer would be inside one of the cubes, and would be able to see cubes in front of, behind, above, below, to the left and right of himself. If one could travel in these directions, one could explore the array of cubes, and gain an understanding of its geometrical structure. An infinite array of cubes is not a polytope in the traditional sense. In fact, it is a tessellation of 3-dimensional (Euclidean) space. However, a 4-dimensional polytope can be considered a tessellation of a 3-dimensional non-Euclidean space, namely, a tessellation of the surface of a four-dimensional sphere. Locally, this space seems like the one we are familiar with, and therefore, a virtual-reality system could, in principle, be programmed to allow exploration of these "tessellations", that is, of the 4-dimensional regular polytopes. The mathematics department at UIUC has a number of pictures of what one would see if embedded in a tessellation of hyperbolic space with dodecahedra. Such a tessellation forms an example of an infinite abstract regular polytope. An example may be seen at this page. Jump to: navigation, search Virtual reality (VR) is an environment that is simulated by a computer. ... Jump to: navigation, search In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... A sphere is a perfectly symmetrical geometrical object. ... Jump to: navigation, search University of Illinois at Urbana-Champaign   The University of Illinois at Urbana-Champaign, also known as UIUC and the U of I (the officially preferred abbreviation), is the largest campus in the University of Illinois system. ... A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ...

Normally, for abstract regular polytopes, a mathematician considers that the object is "constructed" if the structure of its symmetry group is known. This is because of an important theorem in the study of abstract regular polytopes, providing a technique that allows the abstract regular polytope to be constructed from its symmetry group in a standard and straightforward manner. The symmetry group of an object (e. ...

Polytopes in nature

Polygons

The Giant's Causeway, in Ireland

Numerous regular polygons may be seen in nature. In the world of minerals, crystals often have faces which are triangular, square or hexagonal. Quasicrystals can even have regular pentagons as faces. Another fascinating example of regular polygons occurring as a result of geological processes may be seen at the Giant's Causeway in Ireland, or at the Devil's Postpile in California, where the cooling of lava has formed areas of tightly packed hexagonal columns of basalt. Close up of Giants Causeway. ... Basalt columns Giants Causeway A plane of columns The Giants Boot The Giants Causeway is an area of 40,000 tightly packed basalt columns resulting from a volcanic eruption 60 million years ago. ... Quasicrystals are a peculiar form of solid in which the atoms of the solid are arranged in a seemingly regular, yet non-repeating structure. ... Basalt columns Giants Causeway A plane of columns The Giants Boot The Giants Causeway is an area of 40,000 tightly packed basalt columns resulting from a volcanic eruption 60 million years ago. ... The longer fragments of basalt at the base of the cliff can be larger than a person. ... Jump to: navigation, search State nickname: The Golden State Other U.S. States Capital Sacramento Largest city Los Angeles Governor Arnold Schwarzenegger (R) Senators Dianne Feinstein (D) Barbara Boxer (D) Official languages English Area 410,000 km² (3rd)  - Land 404,298 km²  - Water 20,047 km² (4. ... Jump to: navigation, search Look up Lava, ‘A‘a, or Pāhoehoe in Wiktionary, the free dictionary Lava is molten rock that a volcano expels during an eruption. ... Basalt Basalt is an extrusive igneous rock, sometimes porphyritic, and is often both fine-grained and dense. ...

Starfruit, a popular fruit in Southeast Asia

The most famous hexagons in nature are found in the animal kingdom. The wax honeycomb made by bees is an array of hexagons used to store honey and pollen, and as a secure place for the larvae to grow. There also exist animals who themselves take the approximate form of regular polygons (or at least have the same symmetry) for example starfish and sometimes other echinoderms (such as sea urchins) display the symmetry of a pentagon or sometimes other polygons (such as the heptagon). In fact, echinoderms do not display exact radial symmetry. However, Jellyfish and Comb jellies do, usually four-fold (like the square) or eight-fold. Download high resolution version (640x797, 110 KB)Carambolas, Arkin variety. ... Download high resolution version (640x797, 110 KB)Carambolas, Arkin variety. ... Binomial name Averrhoa carambola Carambola or star fruit (Averrhoa carambola, Averrhoaceae or Oxalidaceae) is native to Sri Lanka and popular throughout Southeast Asia. ... Location of Southeast Asia Southeast Asia is a subregion of Asia. ... Honeycomb on a Langstroth frame A honeycomb is a mass of hexagonal wax cells built by honeybees in their nests to contain their larvae and stores of honey and pollen. ... Jump to: navigation, search Families Andrenidae Apidae Colletidae Halictidae Heterogynaidae Megachilidae Melittidae Oxaeidae Sphecidae Stenotritidae Bees (Apoidea superfamily) are flying insects, closely related to wasps and ants. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ... NON TECHNICAL AND OF LOW INTELLIGENCE COMPUTER USER CALLING TECH SUPPORT. SEE S.E.C.S. ALSO This page is a candidate for speedy deletion. ... Classes Asteroidea Concentricycloidea Crinoidea Echinoidea Holothuroidea Ophiuroidea Echinoderms (Echinodermata) is a phylum of marine animals found in the ocean at all depths. ... Subclasses Euechinoidea Superorder Atelostomata Order Cassiduloida Order Spatangoida (heart urchins) Superorder Diadematacea Order Diadematoida Order Echinothurioida Order Pedinoida Superorder Echinacea Order Arbacioida Order Echinoida Order Phymosomatoida Order Salenioida Order Temnopleuroida Superorder Gnathostomata Order Clypeasteroida (sand dollars) Order Holectypoida Perischoechinoidea Order Cidaroida (pencil urchins) Slate pencil urchin (cidaroid) Group of black... Jump to: navigation, search In geometry, a pentagon is any five-sided polygon. ... A heptagon is a plane figure with seven sides and seven angles. ... In biology, radial symmetry is a property of some multicellular organisms. ... Jump to: navigation, search Orders Stauromedusae Coronatae Semaeostomae - Disc jellyfish Rhizostomae Jellyfish (also called jellies or sea jellies as they are not true fish) are animals that belong to Phylum Cnidaria, included in the class Scyphozoa (from Greek skyphos cup and zoon animal). The name jellyfish is also sometimes used... Classes Tentaculata Nuda Ctenophores are jellyfish-like animals commonly called comb jellies, sea gooseberries, sea walnuts, or Venus girdles. ...

Radial symmetry (and other symmetry) is also widely observed in the plant kingdom, particularly amongst flowers, and (to a lesser extent) seeds and fruit, the most common form of such symmetry being pentagonal. A particularly striking example is the Starfruit, a slightly tangy fruit popular in Southeast Asia, whose cross-section is shaped like a pentagonal star. Binomial name Averrhoa carambola Carambola or star fruit (Averrhoa carambola, Averrhoaceae or Oxalidaceae) is native to Sri Lanka and popular throughout Southeast Asia. ...

Moving off earth into space, early mathematicians doing calculations using Newton's law of gravitation discovered that if two bodies (such as the sun and the earth) are orbiting one another, there exist certain points in space where a smaller body (such as an asteroid or a space station)will remain in a stable orbit, following (for example) the earth but never catching up or falling behind. These points are called Lagrangian points. The sun-earth system has five Lagrangian points. The two most stable are exactly 60 degrees ahead and behind the earth in its orbit. That is, joining the centre of the sun and the earth and one of these stable Lagrangian points forms an equilateral triangle. Astronomers have already found asteroids at these points. It is still debated whether it is practical to keep a space station at the Lagrangian point — although it would never need course corrections, it would have to frequently dodge the asteroids that are already present there! (Already there are satellites and space observatories at the less stable Lagrangian points, which do not form the point of an equilateral triangle with the earth and the sun.) Jump to: navigation, search Sir Isaac Newton at 46 in Godfrey Knellers 1689 portrait Sir Isaac Newton, PRS (25 December 1642 (OS) – 20 March 1727 (OS) / 4 January 1643 (NS) – 31 March 1727 (NS)) was an English physicist, mathematician, astronomer, philosopher, and alchemist. ... Jump to: navigation, search The Lagrangian points (IPA: ; also Lagrange point, L-point, or libration point), are the five positions in space where a small object can be stationary with respect to two larger objects (such as a satellite with respect to the Earth and Moon). ... Image of the Trojan asteroids in front of and behind Jupiter along its orbital path. ...

Polyhedra

The credit for the first constructions of the platonic solids does not go to the human race — each of them occurs naturally in one form or another, although not all of these occurrences are visible to the naked eye. The tetrahedron, cube, and octahedron all occur as crystals. These by no means exhaust the numbers of possible forms of crystals (Smith, 1982, p212), of which there are 48. Neither the regular icosahedron nor the regular dodecahedron are amongst them, although one of the forms, called the pyritohedron (named for the group of minerals of which it is typical) is has twelve pentagonal faces, arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular. Quartz crystal A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ... Jump to: navigation, search The mineral pyrite, or iron pyrite, is iron disulfide, FeS2. ...

Circogonia icosahedra, a species of Radiolaria.

In the early 20th century, Ernst Haeckel described (Haeckel, 1904) a number of species of Radiolaria, some of whose skeletons are shaped like various regular polyhedra. Examples include Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus and Circorrhegma dodecahedra. The shapes of these creatures should be obvious from their names. Circogonia Icosahedra from Haeckels 1904 Kunstformen der Natur. 157 by 175 pixels, 6172 bytes. ... Possible classes Polycystinea Acantharea Taxopodea Radiolaria are amoeboid protozoa that produce intricate mineral skeletons, typically with a central capsule dividing the cell into inner and outer portions, called endoplasm and ectoplasm. ... Ernst Haeckel Ernst Heinrich Philipp August Haeckel (February 16, 1834 - August 8, 1919), also written von Haeckel, was a German biologist and philosopher who popularized Charles Darwins work in Germany. ... Possible classes Polycystinea Acantharea Taxopodea Radiolaria are amoeboid protozoa that produce intricate mineral skeletons, typically with a central capsule dividing the cell into inner and outer portions, called endoplasm and ectoplasm. ...

A more recent discovery is of a series of new types of carbon molecule, known as the fullerenes (see (Curl, 1991) for an easy to read exposition of this discovery). Although C₆₀, the most easily produced fullerene, looks more or less spherical, some of the larger varieties (such as C₂₄₀, C₄₈₀ and C₉₆₀) are hypothesised to take on the form of slightly rounded icosahedra, a few nanometres across. Jump to: navigation, search General Name, Symbol, Number carbon, C, 6 Chemical series nonmetals Group, Period, Block 14, 2, p Appearance black (graphite) colorless (diamond) Atomic mass 12. ... Jump to: navigation, search Fullerene C540 Fullerenes are one of only four types of naturally occurring forms of carbon (the other three being diamond, graphite and ceraphite). ...

As an aside: In ancient times the Pythagoreans believed that there was a harmony between the regular polyhedra and the orbits of the planets. In the 17th century, Johannes Kepler studied data on planetary motion compiled by Tycho Brahe and for a decade tried to establish the pythagorean ideal by finding a match between the sizes of the polyhedra and the sizes of the planets' orbits. His search failed in its original objective, but out of this research came Kepler's discoveries of the Kepler solids as regular polytopes, the realisation that the orbits of planets are not circles, and of Kepler's laws of planetary motion for which he is now famous. In Kepler's time only five planets (excluding the earth) were known, nicely matching the number of platonic solids. Kepler's work, and the discovery since that time of Uranus, Neptune and Pluto, have thrown the Pythagorean idea well and truly into the dustbins of scientific history. The Pythagoreans were an Hellenic organization of astronomers, musicians, mathematicians, and philosophers; who believed that all things are, essentially, numeric. ... Jump to: navigation, search A planet in common parlance is a large object in orbit around a star that is not a star itself. ... Jump to: navigation, search Johannes Kepler Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German mathematician, astronomer and astrologer of famed brilliance. ... Jump to: navigation, search Tycho Brahe (born Tyge Ottesen Brahe) (December 14, 1546 – October 24, 1601) was a Danish nobleman known primarily for his work as an astronomer and an astrologer (the two were highly related in his day), as well as an alchemist. ... Jump to: navigation, search Johannes Keplers primary contributions to astronomy/astrophysics were his three laws of planetary motion. ... Atmospheric characteristics Atmospheric pressure 120 kPa Hydrogen 83% Helium 15% Methane 1. ... Atmospheric characteristics Surface pressure ≫100 MPa Hydrogen - H2 80% ±3. ... Atmospheric characteristics Atmospheric pressure 0. ...

References

• (Bridge, 1974) Bridge, N. J.; Facetting the Dodecahedron Acta Crystallographica A30 pp548–552.
• (Coxeter, 1948) Coxeter, H. S. M.; Regular Polytopes, (Methuen and Co., 1948).
• (Coxeter, 1982) Coxeter, H. S. M.; Ten Toroids and Fifty-Seven hemi-Dodecahedra Geometrica Dedicata 13 pp87–99.
• (Coxeter, 1984) Coxeter, H. S. M.; A Symmetrical Arrangement of Eleven hemi-Icosahedra Annals of Discrete Mathematics 20 pp103–114.
• (Coxeter, 1999) Coxeter, H. S. M.; Du Val, P.; Flather, H. T.; Petrie, J. F.; The Fifty-Nine Icosahedra (Tarquin Publications, Stradbroke, England, 1999)
• (Curl, 1991) Curl, R. F.; Smalley, R. E.; Fullerenes, Scientific American 265 4 (1991) pp32–41.
• (Euclid) Euclid, Elements, English Translation by Heath, T. L.; (Cambridge University Press, 1956).
• (Grünbaum, 1977) Grünbaum, B.; Regularity of Graphs, Complexes and Designs, in Problèmes Combinatoires et Théorie des Graphes, Colloquium Internationale CNRS, Orsay, 260 pp191–197.
• (Haeckel, 1904) Haeckel, E.; Kunstformen der Natur (1904). Available as Haeckel, E.; Art forms in nature (Prestel USA, 1998), ISBN 3791319906, or online at http://caliban.mpiz-koeln.mpg.de/~stueber/haeckel/kunstformen/natur.html
• (Lindemann, 1987) Lindemann F.; Sitzunger Bayerische Akademie Der Wissenschaften 26 (1987) pp625–768.
• (McMullen, 2002) McMullen, P.; Schulte, S.; Abstract Regular Polytopes; (Cambridge University Press, 2002)
• (Sanford, 1930) Sanford, V.; A Short History Of Mathematics, (The Riverside Press, 1930).
• (Schläfli, 1855), Schläfli, L.; Reduction D'Une Integrale Multiple Qui Comprend L'Arc Du Cercle Et L'Aire Du Triangle Sphérique Comme Cas Particulières, Journal De Mathematiques 20 (1855) pp359–394.
• (Schläfli, 1858), Schläfli, L.; On The Multiple Integral ∫ⁿdxdy...dz, Whose Limits Are p₁=a₁x+b₁y+ ... +h₁z>0, p₂ > 0,...,p_n > 0 and x² + y² + ... + z² < 1 Quarterly Journal Of Pure And Applied Mathematics 2 (1858) pp269–301, 3 (1860) pp54–68, 97–108.
• (Schläfli, 1901), Schläfli, L.; Theorie Der Vielfachen Kontinuität, Denkschriften Der Schweizerischen Naturforschenden Gesellschaft 38 (1901) pp1–237.
• (Smith, 1982) Smith, J. V.; Geometrical And Structural Crystallography, (John Wiley and Sons, 1982).
• (Van der Waerden, 1954) Van der Waerden, B. L.; Science Awakening, (P Noordhoff Ltd, 1954), English Translation by Arnold Dresden.

See also

• H. S. M. Coxeter
• List of regular polytopes
• Platonic solid
• Johnson solid

Harold Scott MacDonald Donald Coxeter, CC , Ph. ... Jump to: navigation, search This page lists the regular polytopes in Euclidean space. ... Jump to: navigation, search In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ... The elongated square gyrobicupola (J37), a Johnson solid In geometry, a Johnson solid is a convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. ...

External links

• Stella: Polyhedron Navigator Tool for exploring polyhedra and printing nets
• Ernst Haeckel's Kunstformen der Natur online (German)
• Interesting fold-out nets of the cube, octahedron, dodecahedron and icosahedron