M 



POLYGON AND POLYHEDRON. 



POLYGON AND POLTHEDEON. 



they outnumber the edges by 9)'; and since both side* of thin equation 

 receive the mine aoca*ion for erery new polyhedron, it remains always 

 th.it i. th tot.il number of corner* ami facet In Miy system <>f 

 I- -lj hnlr..ii!. each of which hu one ur more face* in common with 

 . r'.iVni, exceed* the total number of edge* and polyhedrons by 1. 



In .very face of a polyhedron take any point, which for abbreviation 

 we may call the centre of that face. .l..;n the . . ntre of each f 

 the centre* of the adjoining face*; we hare thus a new polyhedron, 

 awl tin- |.iuts may be so taken, that those lying in the fiu-m which 

 meat at any corner, shall all be in the same plane. The new polyhedron 

 ha* obriontly as many comer* a* the old one had face* ; and a* many 

 face* a* the old one baa corner* : the number of edge* being the lame 

 in both : and if we call a corner triangular, quadrangular, Ac., according 

 a* three, four, Ac., angle* meet there, the new solid has as many 

 triangular, Ac., face*, as the old solid ha* triangular, Ac., corner*, 

 and ri tvrwf. These polyhedron* may be called conjugate to one 

 another. 



Thu* there is a triangular tetrahedron (four-faced solid) with four 

 triangular corner* : consequently the conjugate solid in another tetra- 

 hedron of the name kind. The quadrangular hexahedron (of sit four- 

 Bided face*) baa 8 triangular corner* : the conjugate solid has therefore 

 8 triangular face*, and ix quadrangular corner* (the triangular 

 octahedron). The pentagonal dodecahedron (having 12 live .-id. -.1 

 face*) ha* 20 triangular corners: the conjugate solid has therefore 

 jn triangular faces and 13 pentagonal corners (the triangular icosa- 

 hedron). The *olids mentioned in this paragraph are those which 

 may be made of equilateral and equiangular face*. [BOBDLAft SOLIDS.] 



Again, a solid can be formed with 14 quadrangular faces, having 

 8 triangular comer* and 8 quadrangular ones ; its conjugate solid has 

 therefore 8 triangular and 8 quadrangular faces, with 14 quadrangular 

 corner*; the number of edge* in both being 8 + 8-1-14 2, or 28. 



Let r,, F,. r,, Ac., be the number of triangular, quadrangular, pent- 

 agonal, Ac., face* in a solid, and c,, i\, c s , Ac., the number of triangular, 

 ]ii:ulrangular, pentagonal, Ac., corners. Let F, o, V., be the total number 

 of face*, corner*, and edge* ; then we have 



F = F 5 + F,-rF s + ..... (1) 

 C = C,+C 4 +C,,-r ..... (2) 



Again, sinco 3 F s -f 4 F, + . . . . is the total number of sides of all the, 

 faces, before they are joined, and since the junction joins each with 

 another, we have half the preceding for the number of edges, or 



. ..(8) 

 . . . (4) 



But w + = + 2, whence wo deduce 



. . . 

 . . .(8) 



H- nee F, + F, 4- . . . . and c s 4 c, + . . . . must be even numbers ; for 

 if these be subtracted from the even numbers 2 c ami '2 r. it will In- 

 seen that even numbers are left : or the number of odd-sided figures 

 must lie even, and also the number of odd-angled corners. Moreover 

 the number of corners must be made up of (1) a couple ; (2) half a* 

 many a* there are odd-sided faces ; (3) 1 for every quadrangle and p. n- 

 i. '2 for every hexagon and heptagon, 3 for every octagon and 

 nonagon, Ac. ; and the same will be true if we write faces for corners, 

 and corners for face*. 



Since every face has at least three sides, and every corner at least 

 three angles, 2 K cannot fall short of 8 f, nor of 3 c. Hence, n.-il li.-r 

 4 R 6 F, nor 4 E can be negative, that is, neither of the following 

 can be negative : 



3c,42c,4c,-12 c, 2c,-8c, . ... (7) 



F,-r2F.+ F,-12-F,-2F.-2F,-. ... (8) 



Hence it appear* that there mu&t be either triangular, quadrangular, 

 r p nUgunal face*, and either three-angled, four-angled, or live-angled 

 corner*. Call these the ettcntiul faces and corners. Hence the fol- 

 lowing readily follow* : 



If the ooential face* be all triangle*, there must bo 4 at least ; if all 

 quadrangle*, 6 at least ; if all pentagons, 12 at least : and the 

 the corncm. If the non-essential face* be all hexagons, or the M<HI 

 reeential corner* six-angled, it would appear* that the minimum 

 number of emential face* and corners need not be increased, how ni.iny 

 hexagon* soever, or nix-angled corners, there may be. 



Wo easily show that the formula) in (7) and (8) are 2 F, -t 4 r, + 

 Or. -... and ao, + 4c,+ 6c,-t-. . . . from which, by addition, we 

 find 



That is, calling both triangle* and three-angled corner* by the name of 

 trij.lstt, quadrangle* and four-angled corner* by that of iyimr/m/./rfe. 

 Ac., we have the following theorem : Space cannot be inclosed without 



lUmembcr bowcrrr that lhl rabjrct It very Incompletely known ; and 

 thoujh MOM otcnsarv condition* ran to laid down, It hu notcr brrn found out 

 whtt condition* ire both twcc-urr and Hilllclcnt In nnlrr Hist ftlvrn 

 of fcc< nujr endow pcc. 



triplet* : and the triplets are In number 8, and one for each quintnpli t . 

 in. I two for each sextuplet, and three for oach septuple!, A 



If all the corner* be three-angled, we have il E = 8o, or (8) vanishes. 

 If then all the face* bo of aide* not exceeding six, we have 



Similarly, if all the faces be triangular, and the corner* nowhere 

 more than six-angled, we must hare 



Sc.-rSc. + c.^lZ. 



Hence it follows that when all the corners are three-angled. and all 



i-itli<-r i^ntagon* or hexagons, the numl-erof j -m 

 neither nion' nor less tlian 12 : also that when all the fa. -. -,. aietri 

 and all the corners five-angled or six-angled, the number of live-angled 

 comers can be neither more nor less than 12. 



If all the corner* be four-angled, we have 2 B= 4 c, or 



whence there must be at least 8 triangle*. And similarly, if all the 

 sides be quadrangular, there mutt be at least 8 three-angled corners. 

 If all the comer* be five-angled, we have 2 K= 5 c, or 



F,= 2042F 4 45F,-r8F,,-- ---- 



so that there must be at least 20 triangular faces. Similarly if all the 

 faces be pentagonal, there must be at least 20 three-angle'. I .. >i i 



Some of the most obvious ways in which figures may bo put t< . 

 so as to enclose space ore as follows : 



1. Two n-sidod faces, joined by n quadrangles. This includ. 

 prism and truncated pyramid, and also every quadrangular In \.i 

 Led run. 



2. The pyramid, with one n-sided face and n triangle*. 



8. The solid with n quadrangles, and '2 triangles, the symmetrical 

 case of which is a prism surmounted at each end by a pyramid. 



4. Two faces of n aides, and inn quadrangles, m being any whole 

 number. 



6. Twelve quadrangle* BO arranged that four of them are pl.n-.-d 

 corner to corner, the figure being finished by four others on eaeh side. 

 When the quadrangles ore all equilateral, this is the common rl* 

 dodecahedron. 



6. The pentagonal dodecahedron, in which there are two ]>ent 



each of which has another pentagon on every side, the two figures 

 being placed together so that the projecting angles of the one till up 

 the re-entering angles of the other. 



7. The triangular icosahedron, the conjugate solid of the lost, whie.h 

 may be thus imagined. Let a pentagonal prism bo surmount. .1 at 

 each extremity by a pyramid, and let the sides of the primn which join 

 the angles of the opposite pentagons, and also a diagonal in 



quadrangle, be supposed to be formed of extensible and contrail ille 

 threads. Turn one of the surmounting pyramids partly round : then 

 the sides and diagonals of the five quadrangles will no longer . ..ntinue 

 in the name plane, but will form ten triangles, whieh, with I 

 belonging to the pyramids, complete the number FMuimL 



Wli.-i) the sides of a polygon are given, the polygon itself is not 

 given, unless it be a triangle: thug there is .in infinite number of 

 quadrangles which have the same four sides. But it i.- > en- 

 able that when a solid is formed of given faces, in a giv. n 

 juxtaposition, those faces, if they form a solid at all, can only form 



