In mathematics, a regular polytope is a polytope whose symmetry group
acts transitively on its flags, thus giving it the highest degree of
symmetry. All its elements or jfaces (for all
0 ≤ j ≤ n, where n is the dimension of the
polytope) — cells, faces and so on — are also transitive on the
symmetries of the polytope, and are regular polytopes of dimension
≤ n.
Regular polytopes are the generalized analog in any number of
dimensions of regular polygons (for example, the square or the regular
pentagon) and regular polyhedra (for example, the cube). The strong
symmetry of the regular polytopes gives them an aesthetic quality that
interests both nonmathematicians and mathematicians.
Classically, a regular polytope in n dimensions may be defined as
having regular facets [(n − 1)faces] and regular vertex
figures. These two conditions are sufficient to ensure that all faces
are alike and all vertices are alike. Note, however, that this
definition does not work for abstract polytopes.
A regular polytope can be represented by a
Schläfli symbol
Contents 1 Classification and description 1.1
Schläfli
2 History of discovery 2.1 Convex polygons and polyhedra 2.2 Star polygons and polyhedra 2.3 Higherdimensional polytopes 2.4 Apeirotopes — infinite polytopes 2.5 Regular complex polytopes 2.6 Abstract polytopes 2.6.1 Regularity of abstract polytopes
2.6.2
Vertex figure
3 Constructions 3.1 Polygons 3.2 Polyhedra 3.3 Higher dimensions 4 Regular polytopes in nature 5 See also 6 References 6.1 Notes 6.2 Bibliography 7 External links Classification and description[edit] 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
In two dimensions there are infinitely many regular polygons. In three
and four dimensions there are several more regular polyhedra and
4polytopes 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
A convex regular polygon having n sides is denoted by n . So an
equilateral triangle 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 coprime, 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 5/2, 3 5, 5/2 and 5/2, 5 . p
is the vertex figure of the polyhedron.
A regular
4polytope
Duality of the regular polytopes[edit]
The dual of a regular polytope is also a regular polytope. The
Schläfli symbol
All regular polygons, a .
All regular nsimplexes, 3,3,...,3
The regular
24cell
Regular simplices[edit] Graphs of the 1simplex to 4simplex. Line segment Triangle Tetrahedron Pentachoron
Main article: Simplex 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 a second, orthogonal, dimension at a distance r from both, and join to A and B to form an equilateral triangle. Mark point D in a third, orthogonal, dimension a distance r from all three, and join to form a regular tetrahedron. And so on for higher dimensions. These are the regular simplices or simplexes. Their names are, in order of dimensionality: 0. Point
1. Line segment
2.
Equilateral triangle
Measure polytopes (hypercubes)[edit] Graphs of the 2cube to 4cube. Square Cube Tesseract Main article: Hypercube 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 for higher dimensions. These are the measure polytopes or hypercubes. Their names are, in order of dimensionality: 0. Point
1. Line segment
2.
Square
Cross polytopes (orthoplexes)[edit] Graphs of the 2orthoplex to 4orthoplex. Square Octahedron 16cell Main article: Orthoplex 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 centered 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 for higher dimensions. These are the cross polytopes or orthoplexes. Their names are, in order of dimensionality: 0. Point
1. Line segment
2.
Square
History of discovery[edit]
Convex polygons and polyhedra[edit]
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
Platonic solids Tetrahedron Cube Octahedron Dodecahedron Icosahedron Star polygons and polyhedra[edit]
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 Bradwardine
(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
KeplerPoinsot polyhedra Small stellated dodecahedron Great stellated dodecahedron Great dodecahedron Great icosahedron Higherdimensional polytopes[edit] A 3D projection of a rotating tesseract. This tesseract is initially oriented so that all edges are parallel to one of the four coordinate space axes. The rotation takes place in the xw plane. 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
(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
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 npolytope is an ndimensional 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 (nonrecursive) definition, which states that a polytope is regular if it has a sufficient degree of symmetry. An npolytope is regular if any set consisting of a vertex, an edge containing it, a 2dimensional 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 whose symmetry group 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
The Hemicube is derived from a cube by equating opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces. Grünbaum also discovered the 11cell, a fourdimensional selfdual
object whose facets are not icosahedra, but are "hemiicosahedra" —
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
hemiicosahedron 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
11cell
Fn / V = F V ≤ F ≤ Fn where Fn is the maximal face, i.e. the notional nface which contains
all other faces. Note that each iface, i ≥ 0 of the
original polytope becomes an (i − 1)face of the vertex
figure.
Unlike the case for Euclidean polytopes, an abstract polytope with
regular facets and vertex figures may or may not be regular itself –
for example, the square pyramid, all of whose facets and vertex
figures are regular abstract polygons.
The classical vertex figure will, however, be a realisation of the
abstract one.
Constructions[edit]
Polygons[edit]
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 nsided
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[edit]
Euclid's Elements
Net for icosahedron 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 foldout net. To obtain a foldout 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 rulerandcompass methods for drawing these foldout 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 foldout nets. For example, there are 11 for the cube, and over 900000 for the dodecahedron. Some interesting foldout nets of the cube, octahedron, dodecahedron and icosahedron are available here. Numerous children's toys, generally aimed at the teen or preteen 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 edgetoedge in a large number of different ways. A child playing with such a toy could rediscover 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[edit] Net for tesseract A perspective projection (Schlegel diagram) for tesseract An animated cutaway crosssection of the 24cell. In higher dimensions, it becomes harder to say what one means by
"constructing" the objects. Clearly, in a 3dimensional 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, suitable for four dimensions, uses
fourdimensional stereography.[1] Depth in a third dimension is
represented with horizontal relative displacement, depth in a fourth
dimension with vertical relative displacement between the left and
right images of the stereograph.
The second approach is to embed the higherdimensional objects in
threedimensional space, using methods analogous to the ways in which
threedimensional objects are drawn on the plane. For example, the
fold out nets mentioned in the previous section have
higherdimensional equivalents. Some of these may be viewed at [1].
One might even imagine building a model of this foldout net, as one
draws a polyhedron's foldout net on a piece of paper. Sadly, we could
never do the necessary folding of the 3dimensional structure to
obtain the 4dimensional polytope because of the constraints of the
physical universe. Another way to "draw" the higherdimensional 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 4dimensional polychora
directly into two dimensions is quite confusing. Easier to understand
are 3d 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 threedimensional hyperplane will be a polytope (not
necessarily regular). If the hyperplane is moved through the shape,
the threedimensional 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
fourdimensional structure of the fourdimensional regular polytopes,
via such cutaway cross sections. This is analogous to the way a CAT
scan reassembles twodimensional images to form a 3dimensional
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 threedimensional viewer can comprehend the structure of
a fourdimensional polytope 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 3dimensional
(Euclidean) space. However, a
4polytope
A regular dodecahedral honeycomb, 5,3,4 , of hyperbolic space projected into 3space. Locally, this space seems like the one we are familiar with, and therefore, a virtualreality system could, in principle, be programmed to allow exploration of these "tessellations", that is, of the 4dimensional 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[edit] 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 obviously cannot exist in a threedimensional world. However this might not rule them out altogether. In physical 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 telltale signs of a few regular candidates, so far without definite results.[citation needed] See also[edit] List of regular polytopes Johnson solid Bartel Leendert van der Waerden References[edit] Notes[edit] ^ a b Brisson, David W. (1978), "Visual Comprehension in nDimensions", in Brisson, David W., Hypergraphics: Visualizing Complex Relationships in Art, Science and Technology, AAAS Selected Symposium, 24, Washington, D.C.: AAAS, pp. 109–145 ^ Coxeter (1974) Bibliography[edit] (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). (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 NATOASI Conference on Polytopes ... etc. ... (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic pp. 43–70. (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 ∫n dx dy ... dz, Whose Limits Are p 1 = a 1 x + b 1 y + ⋯ + h 1 z ≥ 0 , displaystyle p_ 1 =a_ 1 x+b_ 1 y+cdots +h_ 1 zgeq 0, p 2 > 0 , … , p n > 0 displaystyle p_ 2 >0,ldots ,p_ n >0 and x 2 + y 2 + ⋯ + z 2 < 1 displaystyle x^ 2 +y^ 2 +cdots +z^ 2 <1 Quarterly Journal of Pure and Applied
Mathematics
External links[edit] Olshevsky, George. "Regular polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007. The Atlas of Small Regular Polytopes  List of abstract regular polytopes. v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square pgon Hexagon Pentagon Uniform polyhedron
Tetrahedron
Octahedron
Dodecahedron
Uniform 4polytope
5cell
16cell
Uniform 5polytope
5simplex
5orthoplex
Uniform 6polytope
6simplex
6orthoplex
Uniform 7polytope
7simplex
7orthoplex
Uniform 8polytope
8simplex
8orthoplex
Uniform 9polytope
9simplex
9orthoplex
Uniform 10polytope
10simplex
10orthoplex
Uniform npolytope nsimplex northoplex • ncube ndemicube 1k2 • 2k1 • k21 npentagonal polytope Topics:
Polytope
