Page:EB1911 - Volume 22.djvu/41

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POLYGONAL NUMBERS—POLYHEDRON

I2 species; Rumex (fig. 8) (11 species) includes the various species of dock .(q.v.) and sorrel (R. Acelosa); and Oxyria digyna, an alpine plant (mountain sorrel), takes its generic name (Gr. éfus, sharp) from the acidity of its leaves. Rheum (Rhubarb, g.v.) is central Asiatic.


POLYGONAL NUMBERS, in mathematics. Suppose we have a number of equal circular counters, then the number of counters which can be placed on a regular polygon so that the tangents to the outer rows form the regular polygon and all the internal counters are in contact with its neighbours, is a “polygonal number ” of the order of the polygon. If the polygon be a triangle then it is readily seen that the numbers are 3, 6, IO, 1 5 and generally en (n + 1); if a square, 4, 9, 16, . and generally n2; if a pentagon, 5, 12, 22 and generally n(3n—1); if a. hexagon, 6, 15, 28, . and generally n(2n- 1); and similarly for a polygon of r sides, the general expression for the corresponding polygonal number is § n[(n- 1) (V-2)+2]

Algebraically, polygonal numbers may be regarded as the sums of consecutive terms of the arithmetical progressions having 1 for the first term and I, 2, 3, . . for the common differences. Taking unit common difference we have the series 1; 1+2=3; 1+2+3 =6; 1+2-+-3-1-4=1o; or generally I-l-2-f-3 -}- n§ n(n+I); these are triangular numbers. With a common difference 2 we have I: l+3=4; I+3+5=9; I +3+5+7= 16? or generally I-l-3-l-5+ . -l- (2n-1)=n2; and generally for the polygonal number of the rth order we take the sums of consecutive terms of the series

1, 1+(r-2), 1+2 (r-2), . 1+n-1 r-2;

and hence the nth polygonal number of the rth order is the sum of n terms of this series, i, e.,

x+1+(r-2)+1+2(r-2)+ -1-(1-}-n-1.r-2)

=n+§ n.n-1.r-2.

The series I, 2, 3, 4, ... or generally n, are the so-called"' linear numbers " (cf. FIGURATE NUMBERS).


POLYHEDRAL NUMBERS, in mathematics. These numbers are related to the polyhedral (see POLYHEDRON) in a manner similar to the relation between polygonal numbers (see above) and polygons. Take the case of tetrahedral numbers. Let AB, AC, AD be three co vertical edges of

a regular tetrahedron. Divide AB,

into parts each equal to AVI, so

that tetrahedral having the common

vertex A are obtained, whose linear

2 dimensions increase arithmetically.

3 Imagine that we have a number of

spheres (or shot) of a diameter equal

B D to the distance A1. It is seen that

4 shot having their centres at the

C vertices of the tetrahedron A1 will form a pyramid. In the case of the tetrahedron

of edge A2 we require 3 along each side of the base, i.e. 6, 3 along the base of AI, and I at A, making IO in all. To add a. third layer, we will require 4 along each base, i.e. 9, and 1 in the centre. Hence in the tetrahedron A3 we have 20 shot. The numbers 1, 4, 10, 20 are polyhedral numbers, and from their association with the tetrahedron are termed “ tetrahedral numbers.”

This illustration may serve for a definition of polyhedral numbers: a polyhedral number represents the number of equal spheres which can be placed within a polyhedron so that the spheres touch one another or the sides of the polyhedron. In the case of the tetrahedron we have seen the numbers to be I, 4, IO, 20; the general formula for the nth tetrahedral number is § n(n+1)(n-|-2). Cubic numbers are 1, 8, 27, 64, 125, &c; or generally na. Octahedral numbers are 1, 6, 19,44, &c., or generally § n(2n2+1). Dodecahedral numbers are 1, 20, 84, 220, &c.; or generally § n(9n2-9n-|-2). icosahedral numbers are 1, 12, 48, 124, &c., or generally § n(5n2-5n+2).


POLYHEDRON (Gr. 7l'0ll:)S, many, 35pa, 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 en wrapped by the faces. “ Regular polyhedral ” are such as have their faces all equal regular polygons, and all their solid angles equal; the term is usually restricted to the ive forms in which the centre is singly enclosed, viz. the Platonic solids, while the fourpolyhedra 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 polyhedral 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: (1) the tetrahedron, enclosed by four equilateral triangles; (2) the cube or hexahedron, enclosed by 6 squares; (5) the octahedron, enclosed by 8 equilateral triangles; (4) the dodecahedron, enclosed by 12 pentagons; (5) the icosahedron, enclosed by 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.1 If three equilateral triangles be placed at a common vertex with their co vertical 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 co vertical 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 co vertically 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 20 triangular faces, which met in pairs to form 30 edges, and in flves to form 12 vertices. This is the icosahedron. That the triangle could give rise to no other solid followed from the fact that six co vertically placed triangles formed a plane. The pentagon is the basis of the dodecahedron Three pentagons may be placed at a common vertex to forni 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 (10th century A.D.), who solved it with a single opening of the compasses. M ensuralion of the Platonic Solids.-The mensuration of the regular polyhedral 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. 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 1, the following general formulae hold, a being written for 271'/n, and B for 21r/m:-

Sin él =cos 13/sin a; tan él =cos /3/(sinz a-cos2 B)%. A=% l”nF cot a..

V=%rA= glllsn Frtan el cot' a.=91;l3n

F cot” u. cos B/(sin2 a-cos” /SH.

R=%l tan él tan B=§ l sin, B/(sinz a-cos' /3)fl. r= il tan él cot a.=%l cot a cos 13/(sinz a-cos2 /3)l. 1 In the language of Proclus, the commentator: “ The equilat;-al triangle is the proximate cause of the three elements, 'fire, ' 'air' and water '; but the square is annexed to the ' earth] "