Polyhedron

Polyhedron (Gr. iroXus, many, g 5pa, a base), in geometry, a solid figure contained by plane faces. If the figure be entirely to one side of any face the polyhedron is said to be " convex, " and it is obvious that the faces enwrap the centre once; if, on the other hand, the figure is to both sides of every face it is said to be concave, " and the centre is multiply enwrapped by the faces. " Regular polyhedra " are such as have their faces all equal regular polygons, and all their solid angles equal; the term is usually restricted to the five forms in which the centre is singly enclosed, viz. the Platonic solids, while the four polyhedra in which the centre is multiply enclosed are referred to as the Kepler-Poinsot solids, Kepler having discovered three, while Poinsot discovered the fourth. Another group of polyhedra are termed the " Archimedean solids," named after Archimedes, who, according to Pappus, invented them. These have faces which are all regular polygons, but not all of the same kind, while all their solid angles are equal. These figures are often termed " semi-regular solids," but it is more convenient to restrict this term to solids having all their angles, edges and faces equal, the latter, however, not being regular polygons.

Platonic Solids. The names of these five solids are: (r) the tetrahedron, enclosed by four equilateral triangles; (2) the cube or hexahedron, enclosed by 6 squares; (3) the octahedron, enclosed by 8 equilateral triangles; (4) the dodecahedron, enclosed by 12 pentagons; (5) the icosahedron, enclosedby 20 equilateral triangles.

The first three were certainly known to the Egyptians; and it is probable that the icosahedron and dodecahedron were added by the Greeks. The cube may have originated by placing three equal squares at a common vertex, so as to form a trihedral angle. Two such sets can be placed so that the free edges are brought into coincidence while the vertices are kept distinct. This solid has therefore 6 faces, 8 vertices and 12 edges. The equilateral triangle is the basis of the tetrahedron, octahedron and icosahedron.' If three equilateral triangles be placed at a common vertex with their covertical sides coincident in pairs, it is seen that the base is an equal equilateral triangle; hence four equal equilateral triangles enclose a space. This solid has 4 faces, 4 vertices and 6 edges. In a similar manner, four covertical equilateral triangles stand on a square base. Two such sets placed base to base form the octahedron, which consequently has 8 faces, 6 vertices and 12 edges. Five equilateral triangles covertically placed would stand on a pentagonal base, and it was found that, by forming several sets of such pyramids, a solid could be obtained which had zo triangular faces, which met in pairs to form 30 edges, and in fives to form 12 vertices. This is the icosahedron. That the triangle could give rise to no other solid followed from the fact that six covertically placed triangles formed a plane. The pentagon is the basis of the dodecahedron. Three pentagons may be placed at a common vertex to form a solid angle, and by forming several such sets and placing them in juxtaposition .a solid is obtained having 12 pentagonal faces, 30 edges, and 20 vertices.

These solids played an important part in the geometry of the Pythagoreans, and in their cosmology symbolized the five elements: fire (tetrahedron), air (octahedron), water (icosahedron), earth (cube), universe or ether (dodecahedron). They were also discussed by the Platonists, so much so that they became known as the " Platonic solids." Euclid discusses them in the thirteenth book of his Elements, where he proves that no more regular bodies are possible, and shows how to inscribe them in a sphere. This latter problem received the attention of the Arabian astronomer Abul Wefa (loth century A.D.), who solved it with a single opening of the compasses.

Mensuration of the Platonic Solids.-The mensuration of the regular polyhedra is readily investigated by the methods of elementary geometry, the property that these solids may be inscribed in and circumscribed to concentric spheres being especially useful.

| 

A.

Area.

| 

Volume

V.

| 

Radius of Circum- sphere.

R.

| 

Radius of In-sphere.  
  
---|---|---|---|---  
  
Tetrahedron. ... .

| 

l2.3/ 3

(1.7321 12)

| 

l3/63/ 2

(0.11785 l3)

| 

-V6/4

| 

l. 3/ 6/12  
  
Cube. .. ... .

| 

6 2

| | 

3/ 2

| 

Zl  
  
Octahedron. .

| 

12.2," 3

| 

\- 2

| |   
| 

(3464 212)

| 

(04714-0313)

| 

l;'3/ 2

| 

l 3/ 6

/  
  
Dodecahedron.. .

| 

1 2.153/ (1+23/ 5)

| 

l3. 5A, 1 {(47+213/ 5)/40}

| |   
| 

(zo6 457 8 / 2)

| 

(7.663119 l 3) 

| | 

I. A l {(20+1111 5)1401  
  
Icosahedro

| 

l 253/3

(8.6605 2)

| 

l3 s3/ {(7 +33/5)/ 2)

(2.18169

| 

1.2,/{(5-1-?15)/2} 

| 

Lill   
  
If F be the number of faces, n the number of edges per face, m the number of faces per vertex, and l the length of an edge, and if we denote the angle between two adjacent faces by I, the area by A, the volume by V, the radius of the circum-sphere by R, and of the in-sphere by r, the following general formulae hold, a being written for 21r/n, and a for 27r/m:- Sin z I =cos 1 3/sin a; tan II =cos l3/ (sin'- a -cos t R) 2. A =1 l 2 nF cot a.  V = 3rA = 2 1 4 / 3 n F tan -II cot e a = 2 I 1 3 n F cot e a cos a/ (sin' a -cos t 13) 2 R =1-/ tan IT tan 0=1/ sin 13/(sin e a-cost r =Zl tan 21 cot a= Il cot a cos 13/(sin" a -cos' (3)L  1 In the language of Proclus, the commentator: " The equilateral triangle is the proximate cause of the three elements, ` fire,' ` air ' and ` water '; but the square is annexed to the ` earth.' " A The following Table gives the values of A, V, R, r for the five Polyhedra: - Kepler-Poinsot Polyhedra. - These solids have all their faces equal regular polygons, and the angles at the vertices all equal. They bear a relation to the Platonic solids similar to the relation of " star polygons " to ordinary regular polygons, inasmuch as the centre is multiply enclosed in the former and singly in the latter. Four such solids exist: (I) small stellated dodecahedron; (2) great dodecahedron; (3) great stellated dodecahedron; (4) great icosahedron. Louis Poinsot discussed these solids in his memoir, " Sur les polygones et les polyedres " (Journ. Ecole poly. [iv.] 1810), three of them having been previously considered by Kepler. They were afterwards treated by A. L. Cauchy (Journ. Ecole poly. [ix.] 1813), who showed that they were derived from the Platonic solids, and that no more than four were possible. A. Cayley treated them in several papers (e.g. Phil. Mag., 1859, 17, p. 123 seq.), considering them by means of their projections on the circumscribing sphere and not, as Cauchy, in solido.  The small stellated dodecahedron is formed by stellating the Platonic dodecahedron (by "stellating " is meant developing the faces contiguous to a specified base so as to form a regular pyramid). It has 12 pentagonal faces, and 30 edges, which intersect in fives to form 12 vertices. Each vertex is singly enclosed by the five faces; the centre of each face is doubly enclosed by the succession of faces about the face; and the centre of the solid is doubly enclosed by the faces. The great dodecahedron is determined by the intersections of the twelve planes which intersect the Platonic icosahedron in five of its edges; or each face has the same boundaries as the basal sides of five covertical faces of the icosahedron. It is the reciprocal (see below) of the small stellated dodecahedron. Each vertex is doubly enclosed by the succession of covertical faces, while the centre of the solid is triply enclosed by the faces. The great stellated dodecahedron is formed by stellating the faces of a great dodecahedron. It has 12 faces, which meet in 30 edges; these intersect in threes to form 20 vertices. Each vertex is singly enclosed by the succession of faces about it; and the centre of the solid is quadruply enclosed by the faces. The great icosahedron is the reciprocal of the great stellated dodecahedron. Each of the twenty triangular faces subtend at the centre the same angle as is subtended by four whole and six half faces of the Platonic icosahedron; in other words, the solid is determined by the twenty planes which can be drawn through the vertices of the three faces contiguous to any face of a Platonic icosahedron. The centre of the solid is septuply enclosed by the faces.

| 

F

| 

V

| 

En

| | 

n i

| 

e

| 

e' 

| 

D 

| 

k   
  
---|---|---|---|---|---|---|---|---|---  
  
Tetrahedron

| 

4

| 

4

| 

6

| 

3

| 

3

| 

1

| 

1

| 

I

| 

I  
  
Cube

| 

6

| 

8

| 

12

| 

4

| 

3

| 

I

| 

I

| 

I

| 

1  
  
Octahedron

| 

8

| 

6

| 

12

| 

3

| 

4

| 

1

| 

1

| 

I

| 

1  
  
Dodecahedron

| 

12

| 

20

| 

30

| 

5

| 

3

| 

1

| 

1

| 

1

| 

1  
  
Icosahedron. .. .

| 

20

| 

12

| 

30

| 

3

| 

5

| 

1

| 

1

| 

1

| 

1  
  
Small stellated dodecahedron .

| 

12

| 

12

| 

3 0

| 

5

| 

5

| 

I

| 

2 

| 

3

| 

2  
  
Great dodecahedron. .

| 

12

| 

12

| 

3 0

| 

5

| 

5

| 

2

| 

1

| 

3

| 

3  
  
Great stellated dodecahedron .

| 

12

| 

20

| 

30

| 

5

| 

3

| 

1

| 

2

| 

7

| 

4  
  
Great icosahedron. .. .

| 

20

| 

12

| 

30

| 

3

| 

' 5

| 

2 

| 

1

| 

7

| 

7  
  
A connexion between the number of faces, vertices and edges of regular polyhedra was discovered by Euler, and the result, which assumes the form E + 2' = F ± V, where E, F, V are the number of edges, faces and vertices, is known as Euler's theorem on polyhedra. This formula only holds for the Platonic solids. Poinsot gave the formula E 2k = eV + F, in which k is the number of times the projections of the faces from the centre on to the surface of the circumscribing sphere make up the spherical surface, the area of a stellated face being reckoned once, and e is the ratio " angles at a vertex /21r" as projected on the sphere, E, V, F being the same as before. Cayley gave the formula E + 2D = eV + e'F, where e, E, V, F are the same as before, D is the same as Poinsot's k with the distinction that the area of a stellated face is reckoned as the sum of the triangles having their vertices at the centre of the face and standing on the sides, and e' is the ratio: " the angles subtended at the centre of a face by its sides /2rr."  The following table gives these constants for the regular polyhedra; n denotes the number of sides to a face, n 1 the number of faces to a vertex: - Archimedean Solids. - These solids are characterized by having all their angles equal and all their faces regular polygons, which are not all of the same species. Thirteen such solids exist.

1\. The truncated tetrahedron is formed by truncating the vertices of a regular tetrahedron so as to leave the original faces hexagons. (By the truncation of a vertex or edge we mean the cutting away of the vertex or edge by a plane making equal angles with all the faces composing the vertex or with the two faces forming the edge.) It is bounded by 4 triangular and 4 hexagonal faces; there are 18 edges, and 12 vertices, at each of which two hexagons and one triangle are covertical.

2\. The cuboctahedron is a tesserescae-decahedron (Gr. rEoo-apes-KaiSEKa, fourteen) formed by truncating the vertices of a cube so as to leave the original faces squares. It is enclosed by 6 square and 8 triangular faces, the latter belonging to a coaxial octahedron. It is a common crystal form.

3\. The truncated cube is formed in the same manner as the cuboctahedron, but the truncation is only carried far enough to leave the original faces octagons. It has 6 octagonal faces (belonging to the original cube), and 8 triangular ones (belonging to the coaxial octahedron).

4\. The truncated octahedron is formed by truncating the vertices of an octahedron so as to leave the original faces hexagons; consequently it is bounded by 8 hexagonal and 6 square faces. 5, 6. Rhombicuboctahedra. - Two Archimedean solids of 26 faces are derived from the coaxial cube, octahedron and semiregular (rhombic) dodecahedron (see below). The " small rhombicuboctahedron " is bounded by 12 pentagonal, 8 triangular and 6 square faces; the " great rhombicuboctahedra " by 12 decagonal, 8 triangular and 6 square faces.

7\. The icosidodecahedron or dyocaetriacontahedron (Gr. Svo - Kat- rpoieKovra, thirty-two), is a 32-faced solid, formed by truncating the vertices of an icosahedron so that the original faces become triangles. It is enclosed by 20 triangular faces belonging to the original icosahedron, and 12 pentagonal faces belonging to the coaxial dodecahedron.

8\. The truncated icosahedron is formed similarly to the icosidodecahedron, but the truncation is only carried far enough to leave the original faces hexagons. It is therefore enclosed by 20 hexagonal faces belonging to the icosahedron, and 12 pentagonal faces belonging to the coaxial dodecahedron.

9\. The truncated dodecahedron is formed by truncating the vertices of a dodecahedron parallel to the faces of the coaxial icosahedron so as to leave the former decagons. It is enclosed by 20 triangular faces belonging to the icosahedron and 12 decagons belonging to the dodecahedron.

to. The snub cube is a 38-faced solid having at each corner 4 triangles and I square; 6 faces belong to a cube, 8 to the coaxial octahedron, and the remaining 24 to no regular solid.

II, 12. The rhombicosidodecahedra. - Two 62-faced solids are derived from the dodecahedron, icosahedron and the semi-regular triacontahedron. In the " small rhombicosidodecahedron " there are 12 pentagonal faces belonging to the dodecahedron, 20 triangular faces belonging to the icosahedron and 30 square faces belonging to the triacontahedron. In the " great rhombicosidodecahedron " the dodecahedral faces are decagons, the icosahedral hexagons and the triacontahedral squares; this solid is sometimes called the " truncated icosidodecahedron." 13. The snub dodecahedron is a 92-faced solid having 4 triangles and a pentagon at each corner. The pentagons belong to a dodecahedron, and 20 triangles to an icosahedron; the remaining 60 triangles belong to no regular solid.

## Semi-regular Polyhedra

Although this term is frequently given to the Archimedean solids, yet it is a convenient denotation for solids which have all their angles, faces, and edges equal, the faces not being regular polygons. Two such solids exist: (1) the " rhombic dodecahedron, " formed by truncating the edges of a cube, is bounded by 12 equal rhombs; it is a common crystal form (see Crystallography); and (2) the " semi-regular triacontahedron," which is enclosed by 30 equal rhombs.

The interrelations of the polyhedra enumerated above are considerably elucidated by the introduction of the following terms: (1) Correspondence. Two polyhedra correspond when the radii vectores from their centres to the mid-point of the edges, centre of the faces, and to the vertices, can be brought into coincidence. (2) Reciprocal. Two polyhedra are reciprocal when the faces and vertices of one correspond to the vertices and faces of the other. (3) Summital or facial. A polyhedron (A) is said to be the summital or facial holohedron of another (B) when the faces or vertices of A  correspond to the edges of B, and the vertices or faces of A correspond to the vertices and faces together of B. (4) Hemihedral.  A polyhedron is said to be the hemihedral form of another polyhedron when its faces correspond to the alternate faces of the latter or holohedral form; consequently a hemihedral form has half the number of faces of the holohedral form. Hemihedral forms are of special importance in crystallography, to which article the reader is referred for a fuller explanation of these and other modifications of polyhedra (tetartohedral, enantiotropic, &c.).

It is readily seen that the tetrahedron is its own reciprocal, i.e.  it is self-reciprocal; the cube and octahedron, the dodecahedron and icosahedron, the small stellated dodecahedron and great dodecahedron, and the great stellated dodecahedron and great icosahedron are examples of reciprocals. We may also note that of the Archimedean solids: the truncated tetrahedron, truncated cube, and truncated dodecahedron, are the reciprocals of the crystal forms triakistetrahedron, triakisoctahedron and triakisicosahedron. Since the tetrahedron is the hemihedral form of the octahedron, and the octahedron and cube are reciprocal, we may term these two latter solids " reciprocal holohedra " of the tetrahedron. Other examples of reciprocal holohedra are: the rhombic dodecahedron and cuboctahedron, with regard to the cube and octahedron; and the semiregular triacontahedron and icosidodecahedron, with regard to the dodecahedron and icosahedron. As examples of facial holohedra we may notice the small rhombicuboctahedron and rhombic dodecahedron, and the small rhombicosidodecahedron and the semiregular triacontahedron. The correspondence of the faces of polyhedra is also of importance, as may be seen from the manner in which one polyhedron may be derived from another. Thus the faces of the cuboctahedron, the truncated cube, and truncated octahedron, correspond; likewise with the truncated dodecahedron, truncated icosahedron, and icosidodecahedron; and with the small and great rhombicosidodecahedra.

The general theory of polyhedra properly belongs to combinatorial analysis. The determination of the number of different polyhedra of n faces, i.e. n-hedrons, is reducible to the problem: In how many ways can multiplets, i.e. triplets, quadruplets, &c., be made with n symbols, so that (1) every contiguous pair of symbols in one multiplet are a contiguous pair in some other, the first and last of any multiplet being considered contiguous, and (2) no three symbols in any multiplet shall occur in any other. This problem is treated by the Rev T. P. Kirkman in the Manchester Memoirs (18 55, 18 571860); and in the Phil. Trans. (1857).

See Max Bruckner, Vielecke and Vielflache (1900); V. Eberhard, Zur Morphologie der Polyeder (1891).