Page:EB1911 - Volume 22.djvu/42
| A. Area. |
Volume V. |
Radius of Circum-sphere. R. |
Radius of In-sphere. 𝑟. | |
| Tetrahedron | 𝑙2√3 (1·7321 𝑙2) |
𝑙3√2 (0·11785 𝑙3) |
𝑙.√6/4 | 𝑙.√6/12 |
| Cube | 6 𝑙2 | 𝑙3 | 𝑙.√3/2 | 1/2𝑙 |
| Octahedron | 𝑙2.2√3 (3·4642 𝑙2) |
𝑙3.√2/3 (0·47140 𝑙3) |
𝑙/√2 | 𝑙√6 |
| Dodecahedron | 𝑙2.15√(1+2√5) (20·64578 𝑙2) |
𝑙3.5√{(47+21√5)/40} (7·663119 𝑙3) |
𝑙.1/2√{(5+3√5)/2} |
𝑙.√{(20+11√5)/40} |
| Icosahedron | 𝑙2.5√3 (8·6605 𝑙2) |
𝑙3.5/6√{(7+3√5)/2} (2·11785 𝑙3) |
𝑙.1/2√{(5+√5)/2} |
𝑙.1/2√{(7+3√5)/6} |
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: (1) 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 polyédres” (Journ. École poly. [iv.] 1810), three of them having been previously considered by Kepler. They were afterwards treated by A. L. Cauchy (Journ. École 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.
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 /2π” 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 /2π.”
The following table gives these constants for the regular polyhedra; 𝑛 denotes the number of sides to a face, 𝑛1 the number of faces to a vertex:—
| F | V | E | 𝑛 | 𝑛1 | 𝑒 | 𝑒′ | D | 𝑘 | |
| Tetrahedron | 4 | 4 | 6 | 3 | 3 | 1 | 1 | 1 | 1 |
| Cube | 6 | 8 | 12 | 4 | 3 | 1 | 1 | 1 | 1 |
| Octahedron | 8 | 6 | 12 | 3 | 4 | 1 | 1 | 1 | 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 | 30 | 5 | 5 | 1 | 2 | 3 | 2 |
| Great dodecahedron | 12 | 12 | 30 | 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 |
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. τεσσαρες-και-δεκα, 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 semi-regular (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. δυο-καί-τριάκοντα, 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.
10. The snub cube is a 38-faced solid having at each corner 4 triangles and 1 square; 6 faces belong to a cube, 8 to the coaxial octahedron, and the remaining 24 to no regular solid.
11, 12. The rhombicosidodecahedra.—Two 62-faced solids are derived from the dodecahedron, icosahedron and the semi-regular