In mathematics, a regular polytope is a polytope with a high degree of symmetry. It is a higher-dimensional analogue of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). Circles and spheres, although highly symmetric, are not considered polytopes because they do not have plane faces. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians.

Classification and description

Regular polytopes are classified primarily according to their dimensionality.

They can be further classified according to symmetry. For example the cube and the regular octahedron share the same symmetry, as do the regular dodecahedron and icosahedron. Indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries.

Three special classes of regular polytope exist in every dimensionality:

* Regular simplex
* Measure polytope
* Cross polytope

In two dimensions there are infinitely many regular polygons. In three and four dimensions there are several more regular polyhedra and polychora besides these three. In five dimensions and above, these are the only ones. See also the list of regular polytopes.

The idea of a polytope is sometimes generalised to include related kinds of geometrical object. Some of these have regular examples, as discussed in the section on historical discovery below.

Schläfli symbols

Main article: Schläfli symbol

A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th Century, and a slightly modified form has become standard. The notation is best explained by adding one dimension at a time.

* A convex regular polygon having n sides is denoted by {n}. So an equilateral triangles is {3}, a square {4}, and so on indefinitely. A regular star polygon which winds m times around its centre is denoted by the fractional value {n/m}, where n and m are co-prime, so a regular pentagram is {5/2}.

* A regular polyhedron having faces {n} with p faces joining around a vertex is denoted by {n, p}. The nine regular polyhedra are {3, 3} {3, 4} {4, 3} {3, 5} {5, 3} {3, 5/2} {3/2, 5} {5, 5/2} and {5/2, 5}. {p} is the vertex figure of the polyhedron.

* A regular polychoron or polycell having cells {n, p} with q cells joining around an edge is denoted by {n, p, q}. The vertex figure of the polychoron is a {p, q}.

* A five-dimensional regular polytope is an {n, p, q, r}. And so on.

Duality of the regular polytopes

The dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original symbol written backwards: {3, 3} is self-dual, {3, 4} is dual to {4, 3}, {4, 3, 3} to {3, 3, 4} and so on.

The vertex figure of a regular polytope dualises to a 'hyperface' of the dual polytope, and vice versa. For example the vertex figure of {3, 3, 4} is {3, 4}, which dualises to {4, 3} - a cell of {4, 3, 3}.

A regular simplex is self-dual, while the measure and cross polytopes are dual to each other.

Any regular polygon is self-dual.

Regular simplices

Begin with a point A. Mark point B at a distance r from it, and join to form a line segment.

Mark point C in the second dimension, at a distance r from both, and join to A and B to form an equilateral triangle.

Mark point D in the third dimension, a distance r from all three, and join to form a regular tetrahedron.

And so on. These are the regular simplices or simplexes:

Graphs of the 1-simplex to 4-simplex. An n-simplex has n+1 vertices

* Point
* Closed Line segment
* Equilateral triangle
* Regular tetrahedron
* Pentachoron
* 5-simplex

* ....

Measure polytopes (Hypercubes)

The first five hypercubes as edge diagrams. An n-cube has 2n vertices.

Begin with a point A. Extend a line to point B at distance r, and join to form a line segment. Extend a second line of length r, orthogonal to AB, from B to C, and likewise from A to D, to form a square ABCD. Extend lines of length r respectively from each corner, orthogonal to both AB and BC (i.e. upwards). Mark new points E,F,G,H to form the cube ABCDEFGH. And so on. These are the measure polytopes.

* Point
* Line segment
* Square
* Cube
* Tesseract
* Penteract
* Hexeract

Cross polytopes (Orthoplexes)

Graphs of the 2-orthplex to 5-orthoplex. An n-orthoplex has 2n vertices.

Begin with a point O. Extend a line in opposite directions to points A and B a distance r from O and 2r apart. Draw a line COD of length 2r, centred on O and orthogonal to AB. Join the ends to form a square ACBD. Draw a line EOF of the same length and centred on 'O', orthogonal to AB and CD (i.e. upwards and downwards). Join the ends to the square to form a regular octahedron. And so on. These are the cross polytopes.

* Point
* Line segment
* Square
* Octahedron
* 16-cell
* Pentacross
* Hexacross

History of discovery

Convex polygons and polyhedra

The earliest surviving mathematical treatment of regular polygons and polyhedra comes to us from ancient Greek mathematicians. The five Platonic solids were known to them. Pythagoras knew of at least three of them and Theaetetus (ca. 417 B.C. – 369 B.C.) described all five. Later, 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.

Star polygons and polyhedra

Our understanding remained static for many centuries after Euclid. The subsequent history of the regular polytopes can be characterised by a gradual broadening of the basic concept, allowing more and more objects to be considered among their number. Thomas Bredwardine (Bradwardinus) was the first to record a serious study of star polygons. Various star polyhedra appear in Renaissance art, but it was not until Johannes Kepler studied the small stellated dodecahedron and the great stellated dodecahedron in 1619 that he realised these two were regular. Louis Poinsot discovered the great dodecahedron and great icosahedron in 1809, and Augustin Cauchy prove the list complete in 1812. These polyhedra are known as collectively as the Kepler-Poinsot polyhedra.

Main article Regular polyhedron - History.

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. Schläfli called such a figure a "polyschem" (in English, "polyscheme" or "polyschema"). The term "polytope" was introduced by Hoppe in 1882, and first used in English by Mrs. Stott some twenty years later. The term "polyhedroids" was also used in earlier literature (Hilbert, 1952).

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, five of these correspond to the Platonic solids and the other one is the 24-cell. There are exactly three in each higher dimension, which correspond to the tetrahedron, cube and octahedron: these are the regular simplices, measure polytopes and cross polytopes. Descriptions of these may be found in the List of regular polytopes. Also of interest are the nonconvex regular 4-polytopes, partially discovered by Schläfli.

By the end of the 19th century, mathematicians such as Arthur Cayley and Ludwig Schläfli had developed the theory of regular polytopes in four and higher dimensions, such as the tesseract and the 24-cell. The latter are 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. From the mathematical point of view, however, these objects have the same aesthetic qualities as their more familiar two and three-dimensional relatives.

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.

* An n-polytope is regular if any set consisting of a vertex, an edge containing it, a 2-dimensional face containing the edge, and so on up to n−1 dimensions, can be mapped to any other such set 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, or flag, (vertex, edge, face) can be mapped to any other such flag by a suitable symmetry of the cube. Thus we can define a regular polytope very succinctly:

* A regular polytope is one which is transitive on its flags.

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 the Coxeter group [4,4].

Apeirotopes - Infinite polytopes

In the first part of the 20th Century, Coxeter and Petrie discovered three infinite structures {4, 6} {6, 4} and {6, 6}. They called them regular skew polyhedra, because they seemed to satisfy the definition of a regular polyhedron — all the vertices, edges and faces are alike, all the angles are the same, and the figure has no free edges (because they can never be reached). Nowadays we call them infinite polyhedra or apeirohedra. The regular tilings of the plane {4, 4} {3, 6} and {6, 3} can also be regarded as infinite polyhedra.

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, all the angles are the same, and the figure has no loose ends (because they can never be reached). More importantly, perhaps, there are symmetries of the zig-zag that can map any pair of a vertex and attached edge to any other. Since then, other regular apeirogons and higher apeirotopes have continued to be discovered.

Regular complex polytopes

Main article: Complex polytope

A complex number has a real part, which is the bit we are all familiar with, and an imaginary part, which is a multiple of the square root of minus one. A unitary space has its x, y, z, etc. coordinates as complex numbers. This effectively doubles the number of dimensions. A polytope constructed in such a unitary space is called a complex polytope.

Abstract polytopes

Grünbaum also discovered the 11-cell, a 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.

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.

The 11-cell cannot be formed with regular geometry in flat (Euclidean) hyperspace, but only in positively-curved (elliptic) hyperspace.

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).

By 1994 Grünbaum had refined his ideas and now called them abstract polytopes, which he treated as partially ordered sets, or posets of elements such as vertices. The 11-cell and 57-cell remain important regular examples of such abstract 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 as to 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.

A geometric polytope is understood to be a realization of the abstract polytope, such that there is a one-to-one mapping from the abstract elements to the geometric. Thus, any geometric polytope may be described by the appropriate abstract poset, though not all abstract polytopes have proper geometric realizations.

The theory has since been further 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 compass and straightedge. Constructing some regular polygons in this way is very simple (the easiest is perhaps the equilateral triangle), some are more complex, and some are impossible ("not constructible"). The simplest few regular polygons that are impossible to construct are the n-sided polygons with n equal to 7, 9, 11, 13, 14, 18, 19, 21,...

Constructibility in this sense refers only to ideal constructions with ideal tools. Of course reasonably accurate approximations can be constructed by a range of methods; while theoretically possible constructions may be impractical.

Polyhedra

Euclid's Elements gave what amount to ruler-and-compass constructions for the five Platonic solids. (See, for example, Euclid's Elements.) 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 star polyhedra, although here we must be careful to make the net for only the visible outer surface.

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 900000 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 theory, almost any material may be used to construct regular polyhedra. Instructions for building origami models may be found here, for example. They may be carved out of wood, modeled out of wire, formed from stained glass. The imagination is the limit.

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 an object having 4 or more dimensions. 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, or polychoron, 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 even 4-dimensional polychora 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).

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.

Another way a three-dimensional viewer can comprehend the structure of a four-dimensional polychoron 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 polychoron 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.

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.

Regular polytopes in nature

For examples of polygons in nature, see:

Main article: Polygon

Each of the Platonic solids occurs naturally in one form or another:

Main article: Regular polyhedron

Higher polytopes can obviously not exist in a three-dimensional world. In cosmology and in string theory, physicists commonly model the Universe as having many more dimensions.

It is possible that the Universe itself has the form of some higher polytope, regular or otherwise. Astronomers have even searched the sky in the last few years, for tell-tale signs of a few regular candidates, so far without success.

See also

* List of regular polytopes

* Johnson solid

* Bartel Leendert van der Waerden

References

* (Coxeter, 1948) Coxeter, H. S. M.; Regular Polytopes, (Methuen and Co., 1948).

* (Coxeter, 1974) Coxeter, H. S. M.; Regular Complex Polytopes, (Cambridge University Press, 1974).

* (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)

* (Cromwell, 1997) Cromwell, Peter R.; Polyhedra (Cambridge University Press, 1997)

* (Euclid) Euclid, Elements, English Translation by Heath, T. L.; (Cambridge University Press, 1956).

* (Grünbaum, 1977) Grünbaum, B.; Regularity of Graphs, Complexes and Designs, Problèmes Combinatoires et Théorie des Graphes, Colloquium Internationale CNRS, Orsay, 260 pp191–197.

* (Grünbaum, 1994) B. Grünbaum, Polyhedra with hollow faces, Proc of NATO-ASI Conference on Polytopes ... etc. ... (Toronto 1993), ed T. Bisztriczky et al, Kluwer Academic pp. 43-70.

* (Hilbert, 1952) Hilbert, D.; Cohn-Vossen, S. Geometry and the imagination, (Chelsea, 1952) p144.

* (Haeckel, 1904) Haeckel, E.; Kunstformen der Natur (1904). Available as Haeckel, E.; Art forms in nature (Prestel USA, 1998), ISBN 3-7913-1990-6, 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,...,pn > 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.

* (Shephard, 1952) Shephard, G.C.; Regular Complex Polytopes, Proc. London Math. Soc. Series 3, 2 (1952) pp82–97.

* (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.

* D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes

Links

* Olshevsky, George, Regular polytope at Glossary for Hyperspace.

* Stella: Polyhedron Navigator Tool for exploring 3D polyhedra, 4D polytopes, and printing nets

* Ernst Haeckel's Kunstformen der Natur online (German)

* Interesting fold-out nets of the cube, octahedron, dodecahedron and icosahedron

Geometry