ON THE ENUMERATION AND CONSTRUCTION OF POLYEDRA WHOSE SUMMITS ARE ALL TRIEDRAL, AND WHICH HAVE NEITHER TRIANGLE NOR QUADRILATERAL.

BY THOMAS P. KIRKMAN, M.A., F.R.S.

THE theory of the Polyedra has long had a high place among the problems of mathematical science, and has figured among the unanswered prize-questions of the learned societies of Europe. The last time such question was proposed was in 1858, when the Imperial Institute of Paris offered, with three years' notice, for the Concours of 1861, their grand medal of 3,000 francs for any real information about it, in these words (vide Comptes Rendus, 1858):-" Perfectionner en quelque point important la théorie géométrique des polyèdres."

The reason why the question was so indefinite is a simple one. Former prize-questions had asked for all: it was desirable to ask for a part. But no learned body in 1858 could venture to frame out of previous knowledge a welldefined scientific question on the subject. For beyond the very ancient enumeration of the five regular solids, and the simple combinations of crystallography, no attempt that I have heard of had been made to scale its difficult and barren crags on any side, except that I had enumerated and constructed the fourteen 8-edra whose faces are all triangles; a contribution very minute to the whole theory of the polyedra. I gave the triplets of summits by which every solid can be drawn with ease upon any one of its triangles as base; but I gave no insight into the secret of symmetry.

E

The Academy's question, like all previous prize-questions on the subject, brought no answers for the competition of 1861 (vide Comtes Rendus). The same prize for the same work was offered again for one year more; but we heard of no candidates, and the glittering medal was at last taken down. Yet the subject must have been pondered by some good heads; for in the Manchester Memoirs for 1862, there appeared, from no less distinguished hand than Professor Cayley's, a paper on the triangular-faced polyacrons, in which he found and figured the fourteen solids which I had enumerated and constructed in the same Memoirs for 1854: of this, however, as he afterwards informed me, he was not aware when he communicated the paper. He did more: he gave in that memoir a method of enumerating and constructing such solids up to any number of faces: "stopping as to the examples at the same point with Mr. Kirkman ; for although perfectly practicable, it would be very tedious to carry them further, and there would be no commensurate advantage in doing so." He showed also that there is a mathematical elegance, and therefore greater ease, in this restricted enquiry, when further confined to triangular polyacrons which have no summit of fewer than five edges, and he gave a method of enumeration and construction for all cases so limited; yet without example of application or result of any kind.

The fault of his method is stated by himself: it is "very tedious." So far as I can see, it involves, as to the limited question, the enormous toil of comparing and identifying scores of repetitions in varying attitudes of by far the most of the polyacrons to be constructed. Professor Cayley touched no more than I had done in 1854, the grand puzzle of symmetry in his solids. (Vide for that Proceedings of the Royal Society, Jan., 1863, p. 375.)

The enumeration and construction of all the q-acral

p-edra which have only triangular faces is the simplest and easiest of all problems in the general theory of the polyedra. For any edge of such p-edron, across which lies no triangular section of the solid, being effaced, we have a q-acral (p-1)-edron of which all faces are triangles except one 4-gon. In the tables of the q-acral (p-1)-edra these 4-gons are given with their symmetry; and the number L of all their different diagonals is given. In the same tables is the entry (33)= M1+,N, the number of all the edges of these (p-1)-edra across which is a triangular section. Wherefore the entire number of edges of the q-acral p-edra to be entered is L+N1, the number N being unaltered, and the change of place of the suffix unit denoting that a delete is conceived to have become an effaceable, thus adding a face. (Vide page 182 of vol. xxxii of these Proceedings.) With this number L + N, in its specified parts with known symmetry all in the tables, is given everything required for the full and complete description and enumeration of these p-edra, by aid of the general theory. And after that, as the paper last referred to shows, their unrepeated construction is a bagatelle, compared with what precedes it. For these reasons, I took little interest at the time, and for nearly twenty years after, in Professor Cayley's short paper, except that it was a high gratification to see myself alone associated with a man of his power, to make up the complete list of writers on this difficult subject. Quite recently, however, I have amused myself with looking more closely into his judicious and pleasing limitation of this problem by exclusion of summits of three and of four edges; not without regret at having so long neglected the entire literature given to me in any language of this wide topic in Geometry.

I have studied with much enjoyment, useless as the whole theory is, the separate and independent construction of polyedra having only triad summits, which have neither

triangle nor quadrilateral; and this mathematically is one with Cayley's limited problem.

The following considerations prove that these solids can all be reduced by well-defined rules to fundamentals all of one simple form, on which, by the converse rules of construction, they can be built in order without risk of omission, and with repetitions few and all foreseen. Nothing is accomplished in this handling of Professor Cayley's "triangular-faced n-acrons of the third class," which is not done in a more powerful and compendious manner for every kind of polyedron by the processes of my complete theory of the Enumeration and Description of the Polyedra, which was presented to the Royal Society in 1861. But, alas! I can refer to no proof of this in print containing all the steps of the reasoning and the work even for solids of a small number n of faces. Indeed, I have too much cause to feel heartily ashamed of writing that theory; for it was easily proved, without perusal of six pages of it, twenty years ago, to have no scientific value that could entitle it to appear intelligibly in the Philosophical Transactions. Hence it has the honour to appear there unintelligibly. The case is very different with this limited problem. This has been selected from that huge heap of questions by a Cayley, and by him learnedly treated of in print; although from a point in my humble judgment far too remote for results. The problem has thus won front rank in science; so that I need not be ashamed of offering a complete analysis and workmanlike solution of it; and this by construction of the solids, not following description and enumeration, but handin-hand with them; which, however, makes the selected problem more difficult than it is when dealt with as a part of the great whole. When I say by construction of the solids, I mean that I give definite rules for that construction, at a cost of repetitions in the great majority of cases not

exceeding three or four, and in most cases not exceed ing two.

1. The n-edra II, which have only triad summits, and have neither triangle nor quadrilateral, are either solids P, which have a frame or frames, or solids Q, which have no summit 555.

Definition. A frame is a sheet of 5-gons, of which no two neighbours have not a common summit 555.

A frame has one or more summits 555, and has at least three 5-gons.

2. Frames are 3-4-5- or 6-walled. Every frame has a symbol, a contour, and a walling.

A least frame has the least number of walls. Of m-walled frames the least has the fewest pentagons.

3. If a P has no frame of fewer than six walls, the least

n

frame it can have has the symbol sp

summit, and p = 3 pentagons.

=

18, showing s = 1

The contour of 1, is 101010, and its walling is A,B,C,D,E,F, showing that the A-gonal wall has one raypoint or summit A55, as have C and E; while B D and F have none, but each a zero-edge of the contour, B5, D5, and F5, the 5's being all pentagons of the frame.

4. A, in this walling, is no 5-gon; for if A55 were 555, A would be no wall, but a pentagon of the frame. Neither can A, or A, carrying two or three ray-points be a 5-gon.

Ο

The walls B, D, F, may or may not be 5-gons, and they may be B = 0, &c., one or all of them, i.e., null walls, being simple edge or edges 55 in no summit 555, but common to pentagons in different frames.

5. Possible symbols with

=

wallings of three and of four

walls are 10, A,B,C,; 8, A,B,C,D,; 9, are―10,=

=

A,B,C,D,;

five and six walls are

20211; 7, 30220:

=