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    Regular Polytopes
    Regular Polytopes
    Regular Polytopes
    Ebook545 pages4 hours

    Regular Polytopes

    By H. S. M. Coxeter

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    Polytopes are geometrical figures bounded by portions of lines, planes, or hyperplanes. In plane (two dimensional) geometry, they are known as polygons and comprise such figures as triangles, squares, pentagons, etc. In solid (three dimensional) geometry they are known as polyhedra and include such figures as tetrahedra (a type of pyramid), cubes, icosahedra, and many more; the possibilities, in fact, are infinite! H. S. M. Coxeter's book is the foremost book available on regular polyhedra, incorporating not only the ancient Greek work on the subject, but also the vast amount of information that has been accumulated on them since, especially in the last hundred years. The author, professor of Mathematics, University of Toronto, has contributed much valuable work himself on polytopes and is a well-known authority on them.
    Professor Coxeter begins with the fundamental concepts of plane and solid geometry and then moves on to multi-dimensionality. Among the many subjects covered are Euler's formula, rotation groups, star-polyhedra, truncation, forms, vectors, coordinates, kaleidoscopes, Petrie polygons, sections and projections, and star-polytopes. Each chapter ends with a historical summary showing when and how the information contained therein was discovered. Numerous figures and examples and the author's lucid explanations also help to make the text readily comprehensible. Although the study of polytopes does have some practical applications to mineralogy, architecture, linear programming, and other areas, most people enjoy contemplating these figures simply because their symmetrical shapes have an aesthetic appeal. But whatever the reasons, anyone with an elementary knowledge of geometry and trigonometry will find this one of the best source books available on this fascinating study.
    LanguageEnglish
    PublisherDover Publications
    Release dateMay 23, 2012
    ISBN9780486141589
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    Regular Polytopes - H. S. M. Coxeter

    18

    CHAPTER I

    POLYGONS AND POLYHEDRA

    TWO-DIMENSIONAL polytopes are merely polygons ; these are treated in § 1·1. Three-dimensional polytopes are polyhedra ; these are defined in § 1·2 and developed throughout the first six chapters. § 1·3 contains a version of Euclid’s proof that there cannot be more than five regular solids, and a simple construction to show that each of the five actually exists. The rest of Chapter I is mainly topological : a regular polyhedron is regarded as a map, and later as a configuration. In § 1·5 we take an excursion into recreational mathematics, as a preparation for the notion of a tree of edges in von Staudt’s elegant proof of Euler’s Formula.

    1·1. Regular polygons. Everyone is acquainted with some of the regular polygons : the equilateral triangle which Euclid constructs in his first proposition, the square which confronts us all over the civilized world, the pentagon which can be obtained by making a simple knot in a strip of paper and pressing it carefully flat,³ the hexagon of the snowflake, and so on. The pentagon and the enneagon have been used as bases for the plans of two American buildings : the Pentagon Building near Washington, and the Bahá’í Temple near Chicago. Dodecagonal coins have been made in England and Canada.

    To be precise, we define a p-gon as a circuit of p line-segments A1 A2, A2 A3, . . . , Ap A1, joining consecutive pairs of p points A1, A2, . . . , Ap. The segments and points are called sides and vertices. Until we come to Chapter VI we shall insist that the sides do not cross one another. If the vertices are all coplanar we speak of a plane polygon, otherwise a skew polygon.

    A plane polygon decomposes its plane into two regions, one of which, called the interior, is finite. We shall often find it convenient to regard the p-gon as consisting of its interior as well as its sides and vertices. We can then re-define it as a simply-connected region bounded by p distinct segments. ( Simply-connected means that every simple closed curve drawn in the region can be shrunk to a point without leaving the region, i.e., that there are no holes.)

    The most important case is when none of the bounding lines (or sides produced ) penetrate the region. We then have a convex p-gon, which may be described (in terms of Cartesian coordinates) by a system of p linear inequalities

    These inequalities must be consistent but not redundant, and must provide the range for a finite integral

    (which measures the area).

    A polygon is said to be equilateral if its sides are all equal, equiangular if its angles are all equal. If p > 3, a p-gon can be equilateral without being equiangular, or vice versa ; e.g., a rhomb is equilateral, and a rectangle is equiangular. A plane p-gon is said to be regular if it is both equilateral and equiangular. It is then denoted by {p} ; thus {3} is an equilateral triangle, {4} is a square, {5} is a regular pentagon, and so on.

    A regular polygon is easily seen to have a centre, from which all the vertices are at the same distance 0R, while all the sides are at the same distance 1R. This means that there are two concentric circles, the circum-circle and in-circle, which pass through the vertices and touch the sides, respectively.

    It is sometimes helpful to think of the sides of a p-gon as representing p vectors whose sum is zero. They may then be compared with p segments issuing from one point, the angle between two consecutive segments being equal to an exterior angle of the p-gon. It follows that the sum of the exterior angles of a plane polygon is a complete turn, or 2π. Hence each exterior angle of {p} is 2π/p, and the interior angle is the supplement,

    1·11

    This may alternatively be seen from the right-angled triangle O2 O1 O0 of Fig. 1.1A, where O2 is the centre, O1 is the mid-point of a side, and O0 is one end of that side. The right angle occurs at O1, and the angle at O2 is evidently π/p. If 2l is the length of the side, we have

    O0 O1 = l, O0 O2 = 0R, O1 O2 = 1R ;

    therefore

    1·12

    The area of {p}, being made up of 2p such triangles, is

    1·13

    (in terms of the half-side l). The perimeter is, of course,

    1·14

    FIG. 1.1A

    As p increases without limit, the ratios S/0R and S/1R both tend to 2π, as we would expect. (This is how Archimedes estimated π, taking p=96.)

    We may take the Cartesian coordinates of the vertices to be

    Then, in the Argand diagram, the vertices of a {p} of circum-radius 0R=1 represent the complex numbers e²kπi/p, which are the roots of the cyclotomic equation

    1·15

    It is sometimes desirable to extend our definition of a p-gon by allowing the sides to be curved ; e.g., we shall have occasion to consider spherical polygons, whose sides are arcs of great circles on a sphere. This extension makes it possible to have p=2 : a digon has two vertices, joined by two distinct (curved) sides.

    1·2. Polyhedra. A polyhedron may be defined as a finite, connected set of plane polygons, such that every side of each polygon belongs also to just one other polygon, with the proviso that the polygons surrounding each vertex form a single circuit (to exclude anomalies such as two pyramids with a common apex). The polygons are called faces, and their sides edges. Until Chapter VI we insist that the faces do not cross one another. Thus the polyhedron forms a single closed surface, and decomposes space into two regions, one of which, called the interior, is finite. We shall often find it convenient to regard the polyhedron as consisting of its interior as well as its N2 faces, N1 edges, and N0 vertices.

    The most important case is when none of the bounding planes penetrate the interior. We then have a convex polyhedron, which may be described (in terms of Cartesian coordinates) by a system of inequalities

    These inequalities must be consistent but not redundant, and must provide the range for a finite integral

    (which measures the volume).

    Certain polyhedra are almost as familiar as the polygons that bound them. We all know how a point and a p-gon can be joined by p triangles to form a pyramid, and how two equal p-gons can be joined by p rectangles to form a right prism. After turning one of the two p-gons in its own plane so as to make its vertices (and sides) correspond to the sides (and vertices) of the other, we can just as easily join them by 2p triangles to form an antiprism, whose 2p lateral edges make a kind of zigzag.

    A tetrahedron is a pyramid based on a triangle. Its faces consist of four triangles, any one of which may be regarded as the base. If all four are equilateral, we have a regular tetrahedron. This is the simplest of the five Platonic solids. The others are the octahedron, cube, icosahedron, and (pentagonal) dodecahedron. (See Plate I, Figs. 1-5.)

    PLATE I

    REGULAR, QUASI-REGULAR AND RHOMBIC SOLIDS

    1·3. The five Platonic solids. A convex polyhedron is said to be regular if its faces are regular and equal, while its vertices are all surrounded alike. (We shall see in § 1•7 that the regularity of faces may be waived without causing anything worse than a simple distortion. A more economical definition will be given in § 2·1.) If its faces are {p}’s, q surrounding each vertex, the polyhedron is denoted by {p, q}.

    The possible values for p and q may be enumerated as follows. The solid angle at a vertex has q face-angles, each (1–2/p)π, by 1•11. A familiar theorem states that these q angles must total less than 2π. Hence 1–2/p<2/q ; i.e.,

    1·31

    or (p–2)(q–2)<4. Thus {p, q} cannot have any other values than

    {3,3}, {3, 4}, {4,3}, {3,5}, {5,3}.

    The tetrahedron {3, 3} has already been mentioned. To show that the remaining four possibilities actually occur, we construct the rest of the Platonic solids, as follows.

    By placing two equal pyramids base to base, we obtain a dipyramid bounded by 2p triangles. If the common base is a {p} with p<6, the altitude of the pyramids can be adjusted so as to make all the triangles equilateral. If p=4, every vertex is surrounded by four triangles, and any two opposite vertices can be regarded as apices of the dipyramid. This is the octahedron, {3, 4}.

    By adjusting the altitude of a right prism on a regular base, we may take its lateral faces to be squares. If the base also is a square, we have a cube {4, 3}, and any face may be regarded as the base.

    Similarly, by adjusting the altitude of an antiprism, we may take its 2p lateral triangles to be equilateral. If p=3, we have the octahedron (again). If p=4 or 5, we can place pyramids on the two bases, making 4p equilateral triangles altogether. If p=5, every vertex is then surrounded by five triangles, and we have the icosahedron, {3, 5}.

    There is no such simple way to construct the fifth Platonic solid. But if we fit six pentagons together so that one is entirely surrounded by the other five, making a kind of bowl, we observe that the free edges are the sides of a skew decagon. Two such bowls can then be fitted together, decagon to decagon, to form the dodecahedron,⁴ {5, 3}.

    1·4. Graphs and maps. The edges and vertices of a polyhedron constitute a special case of a graph, which is a set of N0 points or nodes, joined in pairs by N1 segments or branches (which need not be straight). If a node belongs to q branches, we have evidently

    1·41

    where the summation is taken over the N0 nodes. For a connected graph (all in one piece) we must have

    1·42

    One graph is said to contain another if it can be derived from the other by adding extra branches, or both branches and nodes. A graph may contain a circuit of p branches and p nodes, i.e., a p-gon (p≥2). A graph which contains no circuit is called a forest, or, if connected, a tree. In the case of a tree, the inequality 1•42 is replaced by the equation

    1·43

    for a tree may be built up from any one node by adding successive branches, each leading to a new node.

    The theory of graphs belongs to topology ( rubber sheet geometry ), which deals with the way figures are connected, without regard to straightness or measurement. In this spirit, the essential property of a polyhedron is that its faces together form a single unbounded surface. The edges are merely curves drawn on the surface, which come together in sets of three or more at the vertices.

    In other words, a polyhedron with N2 faces, N1 edges, and N0 vertices may be regarded as a map, i.e., as the partition of an unbounded surface into N2 polygonal regions by means of N1 simple curves joining pairs of N0 points. One such map may be seen by projecting the edges of a cube radially onto its circum-sphere ; in this case N0=8, N1=12, N2= 6, and the regions are spherical quadrangles.

    From a given map we may derive a second, called the dual map, on the same surface. This second map has N2 vertices, one in the interior of each face of the given map ; N1 edges, one crossing each edge of the given map ; and N0 faces, one surrounding each vertex of the given map. Corresponding to a p-gonal face of the given map, the dual map will have a vertex where p edges (and p faces) come together. (See, for instance, the maps formed by the broken and unbroken lines in Fig. 1.4A·.) Duality is a symmetric relation : a map is the dual of its dual.

    FIG. 1.4A·

    By counting the sides of all the faces (of a polyhedron or map), we obtain the formula

    1·44

    where the summation is taken over the N2 faces. Dually, by counting the edges that emanate from all the vertices, we obtain 1·41. It follows from 1·44 that the number of odd faces (i.e., p-gonal faces with p odd) must be even. In particular, if all but one of the faces are even, the last face must be even too.

    1·5. A voyage round the world. Hamilton proposed the following diversion.⁵ Suppose that the vertices of a polyhedron (or of a map) represent places that we wish to visit, while the edges represent the only possible routes. Then we have the problem of visiting all the places, without repetition, on a single journey.

    FIG. 1.5A

    FIG. 1.5B

    Fig. 1.5A shows a solution of this problem in a special caste ⁶ which is of interest as being the simplest instance where the journey cannot possibly be a round trip. Fig. 1.5B shows a map for which the problem is insoluble even if we are allowed to start from any one vertex and finish at any other.⁷

    Although it is not always possible to include all the vertices of a polyhedron in a single chain of edges, it certainly is possible to include them all as nodes of a tree (whose N0–1 branches occur among the N1 edges). This merely requires repeated application of the principle that any two vertices may be connected by a chain of edges. In fact, every connected graph has a tree for its scaffolding (Gerüst ⁸), and the connectivity of the graph is defined as the number of its branches that have to be removed to produce the tree, namely 1–N0+N1.

    1·6. Euler’s Formula. In defining a polyhedron, we did not exclude the possibility of its being multiply-connected (i.e., ring-shaped, pretzel-shaped, or still more complicated). The special feature which distinguishes a simply-connected polyhedron is that every simple closed curve drawn on the surface can be shrunk, or that every circuit of edges bounds a region (consisting of one face or more). For such a polyhedron, the numbers of elements satisfy Euler’s Formula

    1·61

    which can be proved in a great variety of ways.⁹ The following proof is due to von Staudt.

    Consider a tree whose nodes are the N0 vertices, and whose branches are N0–1 of the N1 edges (i.e., a scaffolding of the graph of vertices and edges). Instead of the remaining edges, take the corresponding edges of the dual map (as in Fig. 1.4A, where the selected edges are drawn in heavy lines). These edges of the dual map form a graph with N2 nodes, one inside each face of the polyhedron. Its branches are entirely separate from those of the tree. It is connected, since the only way in which one of its nodes could be inaccessible from another would be if a circuit of the tree came between, but a tree has no circuits. On the other hand, a circuit of the graph would decompose the surface into two separate parts, each containing some nodes of the tree, which is impossible. So in fact the graph is a second tree, and has N2–1 branches. But every edge of the polyhedron corresponds to a branch of one tree or the other. Hence

    (N0–1) + (N2–1) = N1.

    This argument breaks down for a multiply-connected surface, because there the graph of edges of the dual map does contain circuits (although these do not decompose the surface). For instance, the unbroken lines in Fig. 1.6A form the unfolded net of a map of sixteen quadrangles on a ring-shaped surface; the heavy lines form a scaffolding, and the broken lines cross the remaining edges. Two circuits of broken lines can be seen: one through the mid-point of AD, and another through the mid-point of AE.

    Any orientable unbounded surface (e.g., any closed surface in ordinary space that does not cross itself) can be regarded as a sphere with p handles . (Thus p=0 for a sphere or any simply-connected surface, p= 1 for a ring, and p=2 for the surface of a solid figure-of-eight.) The number p is called the genus of the surface. It can be shown¹⁰ that the appropriate generalization of 1•61 is

    1·62

    FIG. 1.6A

    The unbroken lines in Fig. 1.4A· form a Schlegel diagram for the dodecahedron : one face is specialized, and the rest of the surface is represented in the interior of that face (as if we projected the polyhedron onto the plane of that face from a point just outside). Such a diagram can be made for any simply-connected polyhedron. ¹¹We may regard the whole plane as representing the whole surface, by letting the exterior region of the plane represent the interior of the special face.

    If a simply-connected map has only even faces (like Fig. 1.5A or B), we can show that every circuit of edges consists of an even number of edges. For, such a circuit, of (say) N edges, decomposes the map into two regions which have the circuit as their common boundary. If we modify the map, replacing one of the two regions by a single N-sided face, then the rest of the faces (belonging to the other region) are all even. Hence, by the remark at the end of § 1·4 N is even.

    It follows that alternate vertices of any even-faced simply-connected map can be picked out in a consistent manner (so that every edge joins two vertices of opposite types). For instance, alternate vertices of a cube belong to two inscribed tetrahedra (Plate I, Fig. 6).

    1·7. Regular maps. A map is said to be regular, of type {p, q}, if there are p vertices and p edges for each face, q edges and q faces at each vertex, arranged symmetrically in a sense that can be made precise.¹² Thus a regular polyhedron (§ 1·3) is a special case of a regular map. By 1·41 and 1·44, we have

    1·71

    For each map of type {p, q} there is a dual map of type {q, p} ; e.g., a self-dual map of type {4, 4} is produced if we divide a torus or ring-surface into n² squares by drawing n circles round the ring and n other circles threading the ring. (Fig. 1.6A shows the case when n=4. The surface has been cut along the circles ABCD and AEFG, one of each type.)

    This example is ruled out if we restrict consideration to simply-connected polyhedra. Then the possible values of p and q are limited by the inequality 1•31, and for each admissible pair of values there is essentially only one polyhedron {p, q}. In fact, the relations 1•61 and 1•71 yield

    1·72

    which expresses N1 in terms of p and q. The inequality 1·31 is an obvious consequence of 1·72. The solutions

    {3, 3}, {3, 4}, {4, 3}, {3, 5}, {5, 3}

    give the tetrahedron, octahedron, cube, icosahedron, dodecahedron. As maps we have also the dihedron {p, 2} and the hosohedron ¹³{2, p}. The latter is formed by p digons or lunes.

    1·8. Configurations. A configuration in the plane is a set of N0 points and N1 lines, with N01 of the lines passing through each of the points, and N10 of the points lying on each of the lines. Clearly

    N0 N01 = N1 N10.

    For instance, Npoints of intersection, form a configuration in which N01= 2 and N10 = N1−1. Again, a p-gon is a configuration in which N0 =N1=p, N01=N10=2. (Further points of intersection, of sides produced, are not counted.)

    Analogously, a configuration in space is a set of N0 points, N1 lines, and N2 planes, or let us say briefly Nj j-spaces (j=0, 1, 2), where each j-space is incident with Njk of the k-spaces (j ≠ k).¹⁴ Clearly

    1·81

    These configurational numbers are conveniently tabulated as a matrix

    where Njj is the number previously called Nj.

    The subject of configurations belongs essentially to projective geometry, in which the principle of duality enables us to preserve the relations of incidence after interchanging points and planes. Thus, for any configuration there is a dual configuration, whose matrix is derived from that of the given configuration by a central inversion (replacing Njk by Nj′k′ where j + j′=k + k′= 2).

    In particular, for each Platonic solid {p, q} we have a configuration

    Here the relations 1.71 or 1.81 determine the ratios N0 : N1 : N2, and then 1.61 fixes the precise values

    1·82

    (See Table I, on page 292.)

    1·9. Historical remarks. Sir D’Arcy W. Thompson once remarked to me that Euclid never dreamed of writing an Elementary Geometry : what Euclid really did was to write a very excellent (but somewhat long-winded) account of the Five Regular Solids, for the use of Initiates. However, this idea, first propounded by Proclus, is denied by Heath.

    The early history of these polyhedra is lost in the shadows of antiquity. To ask who first constructed them is almost as futile as to ask who first used fire. The tetrahedron, cube and octahedron occur in nature as crystals¹⁵ (of various substances, such as sodium sulphantimoniate, common salt, and chrome alum, respectively). The two more complicated regular solids cannot form crystals, but need the spark of life for their natural occurrence. Haeckel observed them as skeletons of microscopic sea animals called radiolaria, the most perfect examples being Circogonia icosahedra and Circorrhegma dodecahedra.¹⁶ Turning now to mankind, excavations on Monte Loffa, near Padua, have revealed an Etruscan dodecahedron which shows that this figure was enjoyed as a toy at least 2500 years ago. So also to-day, an intelligent child who plays with regular polygons (cut out of paper or thin cardboard, with adhesive flaps to stick them together) can hardly fail to rediscover the Platonic solids. They were built up that childish way by Plato himself (about 400 B.C.) and probably before him by the earliest Pythagoreans, ¹⁷ one of whom, Timaeus of Locri, invented a mystical correspondence between the four easily constructed. solids (tetrahedron, octahedron, icosahedron, cube) and the four natural elements (fire, air, water, earth). Undeterred by the occurrence of a fifth solid, he regarded the dodecahedron as a shape that envelops the whole universe.

    All five were treated mathematically by Theaetetus of Athens, and in Books XIII-XV of Euclid’s Elements ; e.g., 1.71 is Euclid XV, 6. (Books XIV and XV were not written by Euclid himself, but by several later authors.) The pyramids and prisms are much older, of course ; but antiprisms do not seem to have been recognized before Kepler (A.D. 1571-1630).¹⁸

    The Greeks understood that some regular polygons can be constructed with ruler and compasses, while others cannot. This question was not cleared up until 1796, when Gauss, investigating the cyclotomic equation 1.15, concluded that the only {p}’s capable of such Euclidean construction are those for which the odd prime factors of p . This practically¹⁹ means that p must be a divisor of

    multiplied by any power of 2. The simplest rules for constructing {5} and {17} have been given by Dudeney and Richmond. Richelot and Schwendenwein constructed {257} about 1832, and J. Hermes wasted ten years of his life on {65537}. His manuscript is preserved in the University of Göttingen.

    The theory of graphs (so named by Sylvester) began with Euler’s problem of the Bridges in Königsberg, and was developed by Cayley, Hamilton, Petersen, and others. Euler discovered his formula 1.61 in 1752. Sixty years later, Lhuilier noticed its failure when applied to multiply-connected polyhedra. The subject of Topology (or Analysis situs) was then pursued by Listing, Möbius, Riemann, Poincaré, and has accumulated a vast literature.

    The theory of maps received a powerful stimulus from Guthrie’s problem of finding the smallest number of colours that will suffice for the colouring of every possible map. The question whether this number (for a simply-connected surface) is 4 or 5, has been investigated by Cayley, Kempe, Tait, Heawood, and others, but still remains unanswered. Evidently two colours suffice for the octahedron, three for the cube or the icosahedron, four for the tetrahedron or the dodecahedron.

    The well-known figure of two perspective triangles with their centre and axis of perspective is a configurations (as defined in §1.8) with N0=N1=10 and N01=N10=3, first considered by Desargues in 1636. The use of a symbol such as {p, q} (for a regular polyhedron with p-gonal faces, q at each vertex) is due to Schläfli (4, p. 44), so we shall call it a Schläfli symbol . The formulae 1.82 are his also.

    CHAPTER II

    REGULAR AND QUASI-REGULAR SOLIDS

    THIS chapter opens with a new economical definition for regularity : a polyhedron is regular if its faces and vertex angles are all regular. In § 2·2 we see how {q, p} can be derived from {p, q} by reciprocation. Much use is made later of the self-reciprocal property

    which is the number of sides of the skew polygon formed by certain edges (see § 2.6). The computation of metrical properties (in § 2.4) is facilitated by considering some auxiliary polyhedra which are not quite regular, but more than semi-regular, so it is natural to call them quasi-regular. §§ 2.7 and 2·8 deal with solids bounded by rhombs or other parallelograms ; these are described in such detail, not only for their intrinsic interest, but for use in Chapter XIII.

    2·1. Regular polyhedra. The definition of regularity in § 1·3 involves three statements : regular faces, equal faces, equal solid angles. (Regular solid angles can then be deduced as a consequence.) All three statements are necessary. For : the triangular dipyramid formed by fusing two regular tetrahedra has equal, regular faces ; prisms and antiprisms of suitable altitude have regular faces and equal solid angles; and certain irregular tetrahedra, called disphenoids, have equal faces and equal solid angles. (To make a model of a disphenoid, cut out an acute-angled triangle and fold it along the joins of the mid-points of its sides. The disphenoid is said to be tetragonal or rhombic according as the triangle is isosceles or scalene.)

    It is interesting to find that another definition, involving only two statements, is powerful enough to have the same effect : we shall see that regular faces and regular solid angles suffice. For simplicity, we replace the consideration of solid angles (which are rather troublesome) by that of vertex figures.²⁰

    The vertex figure at the vertex O of a polygon is the segment joining the mid-points of the two sides through O ; for a {p} of side 2l, this is a segment of length

    (See the broken line in Fig. 1.1A on page 3.) The vertex figure at the vertex O of a polyhedron is the polygon whose sides are the vertex figures of all the faces that surround O ; thus its vertices are the mid-points of all the edges through O. For instance, the vertex figure of the cube (at any vertex) is a triangle.

    Now, according to our revised definition, a polyhedron is regular if its faces and vertex figures are all regular.

    Since the faces are regular, the edges must be all equal, of length 2l, say. Similarly, since the vertex figures are regular, the faces must be all equal ; for otherwise some pair of different faces would occur with a common vertex O, at which the vertex figure would have unequal sides, namely 2l cos π/p for two different values of p. Moreover, the dihedral angles (between pairs of adjacent faces) are all equal ; for, those occurring at any one vertex belong to a right pyramid whose base is the vertex figure. Each lateral face of this pyramid is an isosceles triangle with sides l, l, 2l cos π/p. The number of sides of the base cannot vary without altering the dihedral angle. Hence this number, say q, is the same for all vertices, and the vertex figures must be all equal.

    We thus have the regular polyhedron {p, q}. Its face is a {p} of side 2l, and its vertex figure is a {q} of side 2l cos π/p.

    We easily see that the perpendicular to the plane of a face at its centre will meet the perpendicular to the plane of a vertex figure at its centre in a point O3 which is the centre of three important spheres : the circum-sphere which passes through all the vertices (and the circum-circles of the faces), the mid-sphere which touches all the edges (and contains the in-circles of the faces), and the in-sphere which touches all the faces.²¹ Their respective radii ²² will be denoted by 0R, 1R, and 2R.

    Let O2, be the centre of a face, O1 the mid-point of a side of this

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