CERTAIN FORMS OF THE ICOSAHEDRON AND A METHOD FOR DERIVING AND DESIGNATING HIGHER POLYHEDRA BY ALBERT HARRY WHEELER, North High School, Worcester, Massachusetts, U.S.A. The Five Regular Solids have afforded material for profound mathematical investigation from the days of Pythagoras to modern times. They were treated in Book XIII of Euclid's Elements and it is questionable to whom to ascribe the first knowledge of their existence. The tetrahedron, cube and octahedron were probably known earlier than the two more complicated bodies. It is probable that Pythagoras was acquainted with the five bodies. Certainly the Pythagoreans were, since Philolaus speaks of the five bodies in the sphere. These are often referred to as the Pythagorean solids and also as the Platonic bodies, since Plato concerned himself with them. It is not known in what connection Archimedes undertook his investigations of the half-regular solids. He knew thirteen half-regular polyhedra but did not refer to the "Archimedean Prism", so-called, nor to the antiprism. The only point I wish to make in this connection is that Archimedes derived new solids from the five regular bodies by a systematic removal of various portions. The existence of the inscribed and the circumscribed spheres of the five regular solids and of the thirteen Archimedean solids was discovered and many of the geometrical relations of the bodies among themselves as to inscription and circumscription were developed, but centuries passed until Kepler, by applying the method of extending the faces of a solid, obtained two new star dodecahedra. He also inscribed two tetrahedra in a cube and formed the stella octangula or group of two orthogonally intersecting concentric tetrahedra. Whether or not Wenzel Jamitzer of Nuremberg, 1508-1586, anticipated the discovery by Kepler, 1571-1630, of regular star polyhedra does not concern us at present, for little attention was given to these new forms until Poinsot in 1809 disclosed his discovery of the four regular star polyhedra derived from the regular convex polyhedra by extending their faces until they intersect. Mathematicians including Cauchy 1811, Bertrand 1858, Cayley 1859, Wiener 1867, Hess 1872 and others occupied themselves with these new higher polyhedra and established many theorems relating to them. The usual "Netze" or "Blanks" for forming the five regular solids from suitable material by folding together configurations of their faces were drawn by Dürer 1525.
702 ALBERT HARRY WHEELER I find an advantage in departing from these common forms and adopting those shown in the accompanying Fig. 1. These "Blanks" are constructed by selecting a face of a given regular solid as a first face and then joining to it a succession of adjacent faces, each connected to the next following by a common edge, and these faces are chosen to follow preferably a positive direction of rotation about the line joining the centre of the first selected face with the centre of the diametrically opposite face of the solid as an axis. They are roulettes of the regular solids. Fig. 1 Fig. 2 These faces are designated by assigning symbols to them, such as the letters of the alphabet a, b, c,... or by the numerals as 1, 2, 3,.... In the present discussion of a Method for Deriving and Designating Polyhedra, I shall limit myself to a consideration of the regular convex icosahedron and some of the forms derived from it, and shall apply certain elementary
A METHOD FOR DERIVING AND DESIGNATING POLYHEDRA 703 principles of group theory to the derivation and designation of a few forms and also to their actual construction from certain "elements". The numbering of the faces of an icosahedron which I shall adopt is shown in Figs. 1 and 2. The common numbering as shown by Brückner and others is shown in Fig. 3. We will designate the faces of the regular icosahedron taken in the order shown in Fig. 1, by two sets of marks, a, b, c,..., r, s, t, and 1, 2, 3,..., 18, 19, 20. Furthermore, a and 1 designate the first face, b and 2 the second face, c and 3 the third face, and so on in the order chosen, as shown in Fig. 1. It will be observed by referring to Fig. 1, that this blank or group of faces, 1, 2, 3,..., 18, 19, 20, consists of two sub-groups, 1, 2, 3,..., 10, and 11, 12, 13,..., 20 which possess symmetry with reference to the mid-point of the edges separating Fig. 4 the faces 10 and 11, and either group may be transformed into the other by a rotation through an angle IT about this point. The first of these sub-groups 1, 2, 3,..., 10, consists of the " visible" or "upper half " of the convex icosahedron when viewed from points on the axis Oi O20 perpendicular to thé faces 1 and 20, Fig. 2, and the second sub-group 11, 12, 13,..., 20, constitutes the "lower half". Each of these sub-groups in position on the icosahedron can be transformed into the other by rotating the solid through an angle w about an axis passing through the points designated as Q and JR, Fig. 2, which are the midpoints of the edges formed by the intersections of the faces 4 and 14, and by 9 and 18 respectively. It should be observed that an icosahedron coincides with itself when rotated about an axis passing through the geometric centres of two diametrically opposite
704 ALBERT HARRY WHEELER faces through angles of, and 27r. By referring to Fig. 2, it may be seen Ó Ó that the sub-group 1, 2, 3,..., 10, can therefore be separated into the cyclic subgroups, 1; and 2, 5, 8; 3, 6, 9; and 4, 7, 10. It also appears that the subgroup 11, 12, 13,..., 20, can be separated into the cyclic sub-groups 11, 14, 17; 12, 15, 18; 13, 16, 19, and 20, respectively. Omitting 1 and 20, it may be seen that the original group 1,2,3,..., 18,19,20, can be separated into the following cyclic sub-groups: 2, 5, 8, 11, 14, 17, 3, 6, 9, 12, 15, 18, 4, 7, 10, 13, 16, 19. Now, returning to the icosahedron, let some face as a, be selected as a plane of reference, and denote the intersections or traces of the planes of the remaining faces 2, 3, 4,..., with a, by the notation a 2, #3, #4,..., #is, #19 as shown in Fig. 4. It should be remembered that since the faces 1 and 20 are parallel, there will be no finite trace of 20 on 1, and hence a 20 will not appear in the figure. Similarly for other faces as, b, we shall have the traces b\, b 2, b s,, b ì5, bn, bis, big, &2o, and in this case there will be no trace b i6 in the group. Fig. 4 which shows the traces of the faces on the plane of the face a or 1, is readily constructed by observing the grouping of the faces of the icosahedron relatively to face 1 and to each other, and in particular, because of the pentagonal arrangement of groups of isosceles triangular faces about single vertices, it develops that the sides of the triangle V\ V\ V\, Fig. 4, are divided at the six points Vl in extreme and mean ratio. It may be seen furthermore, that the 18 lines in Fig. 4 are grouped form groups of equilateral triangles, as: i. «2, #5, a 8. II. Ö3, Ô6> a ) #4, #7, OK). III. #12, #15, #18,' oil» #14, #17- IV. #2, #10, #13; #5, #13, #16j #8, #16, #19- V. an, aie, #19. Vi. a 2, #5, #13; #5, #8, #16; #8, #2, #19. The triangles of these groups may be made to coincide either with themselves or with each other by rotating the entire configuration about the centre 0\ to through angles of,, and 27r. It follows that any configuration of lines selected from among the totality of lines a 2, #3, #4,, #19, may be transformed into a congruent configuration by rotations through angles of, and 2r. O ó We shall now define Fa x a y a z... to denote the plane figure formed by the lines a x a y a z.... In particular, we shall fix our attention on any portion of the plane, of which the lines a x a y a z... taken successively in the cyclic order given,
A METHOD FOR DERIVING AND DESIGNATING POLYHEDRA 705 form a closed boundary. Thus, referring to Fig. 5, the figure F a 2 #5 # 8 defines the triangle whose vertices are V 2 V 3 Vi. Also Fa 3 #9#4#io defines the "kite" with one vertex at V\. We shall call any enclosed space of the plane thus defined, an element, and shall denote it by Ea x a y a z.... Thus, in Fig. 5 we have the elements Ea 2 a b a % and Ea3#9#4#10- Fig. 5 The symbols in the Table are elements of faces of icosahedra, and the entire "visible" parts of these faces may be obtained by applying the substitutions where 1, S, S 2, 5s2, 5, 8-3, 6, 94, 7, 10-11, 14, 17-12, 15, 18-13, 16, 19. It should be understood that the classification of elements in the following table is not exhaustive, but it shows certain varieties and points out relations with reference to the complete figure from which they are obtained by selection. On transforming any symbol as Ea P a Q a r...a w by a substitution, S, we shall obtain a second element, and obviously on applying a cyclic substitution repeatedly we shall eventually reproduce the original element. The totality of such "elements" thus derived and lying in the same plane form a group and constitute the visible parts of one face of a polyhedron. Hence we may regard the Table as designating certain Icosahedra by means of Elements from which their faces may be constructed. As Archimedes, Kepler and Poinsot derived new forms by removing portions of others or by adding to them by extending faces and edges, so we may proceed to derive additional forms by selecting groups of elements from among the lines of some complete figure, as Fig. 4.
706 ALBERT HARRY WHEELER TABLE OF ICOSAHEDRA No. Vertices Elements Varieties 1 V 1 i= Ea2#5#8, Regular Convex Triangular. 2 V 2 JEi == a2#io#3, Hexagonal. 3 F 3 i = #9#3#4#io> Five Intersecting Octahedra. 4 I/ 4 i= 3= jea 8 #n#9, Easanai 2 aio, <X4#12#5, Nonagonal. 5 F 5 Ei jea2#i8#n#8, 2 = #3#9#12#11#4#10, Archimedean Variety (A.V.) of No. 13. 6 F«1= Eana8#2#i8#i7, Five Intersecting Tetrahedra (Right). 7 F6 1= jeana-8#2#i8#i2, Five Intersecting Tetrahedra (Left). V 6 Er 2= Ez- Ea^au, #17#15#5#8, ai 8 #6#9, Ten Intersecting Tetrahedra,. No. 6+N0. 7. 9 V 6 Ei= ai6#5#8, Möbius (Concave). 10 V 6 Ei= Ea 8 a b agaea 1 a i, Twenty Pointed, Six-Edged. 11 V 7 Er 2= 3= #16#14#7, #16#15#17, ai6#6#18, Regular Star. Poinsot. 12 F 8 1= 2 = 3= #17#16#13, ai8#16#14, ai90i6#15, Complete. 13 F 7 l~~ #9#4#10#3, Kite, Archimedean Variety of No. 11. 14 F l == öi6#4#7, 2 = #16#6#9, Archimedean Variety of No. 11. 15 F 6, F 7 l = ö9(x4#ll#18#12, Double. No. 13 +No. 7. 16 V», yv l = 0-904011014012, Double. No. 13 + No. 6.
A METHOD FOR DERIVING AND DESIGNATING POLYHEDRA 707 TABLE OF ICOSAHEDRA (continued) 17 F 6, V 7 1= ai2#9#4#n, F 6, V 7 2= 0-i6#5#17#15#8, 1= ai2#io#5, 18 2= #12#11#9#4, 3= #n#8#3, 19 P 5, V 7 1= #i8#8#n, 2 = #3#12#9, 3 = #9#4#11#12, 4 == Ea^anaio, 5= ai2#5#14, Double. No. 13 + No. 9. Double. No. 13 + No. 10. Triple. No. 13 + No. 6 + N0. 7. 20 F 6 1= + ai6#5#i7#i5#8, Hollow, Labyrinth. 2 a6#18#17#10#7#9, 21 P 5 1= ~ a 4 a 7 #i4, 2 = + ai6#15#17, 3 = ai8#6#9, 22 I/ 7, F 5 1= + a 9 #4#n#i2, 2 = 0-2#5#12#14, Discrete, Skeleton. Discrete. Twelve Pointed, Crown Rimmed Group. NOTE The notation +, - (see Nos. 20, 21, 22) denotes that the elements are on "opposite sides" of a given plane. We will apply the method of derivation and designation under consideration to the triple icosahedron composed of the group of five intersecting regular tetrahedra capable of inscription in a regular convex pentagonal dodecahedron, its enantomorph of five other intersecting regular tetrahedra, and the twelve pointed star "kite" icosahedron, the elements of a face of which are listed as No. 19 in the Table. By letting S represent the substitution above given and applying the substitutions 1, S, S 2 to the elements 1, 2, 3, E± and 5 we may obtain the group G a of elements in the plane a formed by their configuration and constituting the visible portions of the face a of the polygon considered. Hence we have ( 1, 2, 3, -Ë4, 5, G = a -l SEi, SE 2, SEs, SE4 SE5, [S 2 Ei, S 2 E 2, S 2 E 3, S 2 E 4, S 2 E 5. With reference to the face, a, of the polyhedron, the remaining faces are given by groups of simply isomorphic substitutions.
708 ALBERT HARRY WHEELER It will be convenient to follow the method of Klein and others and at the same time to designate or name each domain or element of a face of a polyhedron after the operation through which it is derived by a suitable substitution from one amongst a given group of elements. Thus, the elements of the group G a of the triple icosahedron are represented in Fig. 6. Fig. 6 After having determined the elements, G a, of a particular polyhedron it is a simple matter to select from among them those elements which when properly grouped together about a point form the face angles of some polyhedral angle whose vertex is a point of the complete figure. By making a sufficient number of duplicates of the group of elements thus selected and arranged out of suitable material and properly assembling these groups together, models of polyhedra may be readily constructed.