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31-Point Geometry

Published online by Cambridge University Press:  03 November 2016

W.L. Edge
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Extract

The following paragraphs have been assembled in consequence of my reading Dr. Cundy’s note on 25-point geometry. Towards the end of it, apparently mindful of the adjunction of a “ line at infinity ” to the Euclidean plane, he adjoins a line to the 25-point plane and so obtains a geometry of 31 points. Here I reverse this procedure : I start with the 31-point geometry and thereafter assign to one of its 31 lines the rôle of the “ line at infinity ”. This seems more in the spirit of Cayley's dictum at the end of his Sixth Memoir on Quantics, that “ descriptive geometry is all geometry ” and metrical geometry only a part thereof.

Type
Research Article
Information
The Mathematical Gazette , Volume 39 , Issue 328 , May 1955 , pp. 113 - 121
Copyright
Copyright © The Mathematical Association 1955

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References

1. Sylvester introduces synthemes of six objects in his paper “Elementary Researches in the Analysis of Combinatorial Aggregation”; this is on p. 91 of Vol. I of his Mathematical Papers and he gives an etymology for his neologism in a footnote on this page. On p. 92 he displays a synthematic total : i.e. a set of five synthemes of which the pairs, three in each syntheme, together exhaust all 6 C 2 = 15 pairs. Given any one syntheme two totals can be built to include it, wherefore the number of distinct totals is 15x2÷5 = 6.

2. When we adjoin to F either root of any one of those ten quadratics (f) whose discriminant is a non-square we thereby generate a larger field Φ of twenty-five marks. If the roots of x2 = 2 are ±j each of the ten quadratics has a pair of roots A ±Bj where A is a mark and Β a non-zero mark of F. These twenty roots, with the five marks of F, constitute Φ. Just as there are primitive marks, namely ±2, in F of which all non-zero marks are powers, so in Φ ; the first twenty-four powers of, for example, 2 -j yield all the non-zero marks of Φ. The one-dimensional geometry based on Φ consists of the six points of L and the pairs of foci of the ten involutions, 26 points in all.

3. Veronese’s long account of the Hexagrammum Mysticum of Pascal is in vol. 1 of the third series of Memoirs of the Atti della Reale Accademia dei Lincei (1877) ; pp. 649–702. The separation into six Desargues figures is on p. 661. It was on reading the manuscript of this memoir before its publication that Cremona realised how to obtain the whole Pascal figure by projection from a node of a cubic surface. Concerning these matters see Richmond, H.W.: Mathematische Annalen 53 (1899), 161–176.CrossRefGoogle Scholar

4. Cayley’s Sixth Memoir on Quanties was published in vol. 149 of the Philosophical Transactions in 1859 and occupies pp. 561–592 of vol. II of his Collected Mathematical Papers. It does not presume a knowledge of its five predecessors and should be read by every mathematician. It is a pleasure, as well as an education, to read it and Forsyth says, in his obituary notice of Cay ley, that it could not be presented in more attractive form.

The choice as absolute of a pair of points was made in § 15 in order to derive the geometry described by Dr. Cundy. But it is clear from the Memoir that we might equally well have chosen as absolute any of the 3100 conies and so obtained a geometry that is “non-euclidean” but in which the duality between points and lines is still symmetrical.

5. Given a quadrangle in Π there is a unique projectivity transforming its vertices into those of the same or any other quadrangle taken in any prescribed order. The group G of projectivities in Π is therefore of order 372000, 4! times the number of quadrangles. G is transitive on the lines and on the conies of Π.

Any non-singular three-rowed matrix M imposes, whenever its nine elements all belong to F, one of these projectivities, but the same projectivity is imposed by all four matrices ±M, ±2M. The number of non-singular matrices is therefore 1488000, four times the number of projectivities ; these matrices form the general linear homogeneous group (on three variables, over F) whose order is given in the classical treatises, for instance on p. 97 of Jordan’s, C. Traité des substitutions (Paris 1870) and on p. 77 Google Scholar of Dickson’s, L.E. Linear Groups (Leipzig 1901).Google Scholar From these sources we derive the order 1488000 in the form (53 - 1)(53 - 5)(53 - 52). Since the multiplication of M by a mark m of F causes the determinant | M | to be multiplied by m3 we can set up a(l, 1) correspondence between matrices and the projectivities which they impose by stipulating that | M | is always 1, and we thus obtain G as the special linear homogeneous group of order (53 - 1)(53 - 5)52

Any conic Σ is invariant for 372000 -f 3100= 120 projectivities of G. Since, by choice of the triangle of reference, Σ has the equation x2 + y2 + z2 = 0, the 120 unimodular matrices constitute the orthogonal group R (in three variables and over F). This is of the same order as the group, encountered in § 1, of projectivities on L ; indeed the two groups are not merely of the same order but are isomorphic, both of them being symmetric groups of degree 5. The projectivities on L permute five of six synthematic totals while R permutes those five self-polar triangles of Σ whose vertices are all D.