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The Kummer quartic and the tetrahedroids based on the Maschke forms

Published online by Cambridge University Press:  24 October 2008

W. L. Edge
W. L. Edge
Affiliation:
Mathematical Institute16 Chambers StreetEdinburgh 1
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Extract

1. The quadrics of space are linearly dependent on ten among them; any ten linearly independent quadrics may be chosen to constitute the base, but it is customary in analytical work to select the four squares and the six products of pairs of four planes which are the faces of some tetrahedron of reference. This choice is adequate enough for many purposes, but it gives an unsymmetrical twist to the work; whereas four quadrics of the base are repeated planes the other six are plane pairs, and the curve common to two of the ten base quadrics can be a skew quadrilateral, a repeated line-pair, or a line counted four times. This radical defect can be mitigated by taking as base a certain set of ten non-singular quadrics, namely, the set of ten fundamental quadrics which is so prominent a feature of Klein's figure of six mutually apolar linear complexes. Every pair of such a set of ten quadrics has its two members related to one another in precisely the same way; their common curve is a skew quadrilateral and they are their own polar reciprocals with respect to each other.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 45 , Issue 4 , October 1949 , pp. 519 - 535
Copyright
Copyright © Cambridge Philosophical Society 1949

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References

* Cayley, A.: J. Math. 83 (1877), 215Google Scholar and Mathematical Papers, 10, 161;Google ScholarBaker, H. F., Abelian Functions (Cambridge, 1897), 487Google Scholar and Principles of Geometry, 4 (Cambridge, 1925 and 1940), 216;Google ScholarHudson, R. W. H. T., Kummer's Quartic Surface (Cambridge, 1905), 81.Google Scholar

* Math. Ann. 30 (1887), 504.Google Scholar

† Inscribed, be it noted, in the sense that two pairs of opposite edges lie on a Quadric.

* Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, 44 (1892), 156.Google Scholar This volume is mistakenly cited by Hudson, on p. 83 of his book, as 48.

* Quart. J. Pure Appl. Math. 34 (1902), 128.Google Scholar

† G is indeed the direct product of the symmetric group of permutations of the six Maschke surfaces and a group of order 16 which leaves each individual M i as a whole unchanged but permutes its points in Kummer sets.

* E. Study, loc. cit. 143; Baker, H. F., Principles of Geometry, 4 (Cambridge, 1925 and 1940), 208.Google Scholar

† Math. Ann. 18 (1881), 138.Google Scholar

* J. Math. Pures Appl. 11 (1846), 291;Google ScholarMathematical Papers, 1, 302.Google Scholar

† Berichte…zu Leipzig, 36 (1884), 10.Google Scholar

‡ Math. Ann. 2 (1870), 211 and 217;Google ScholarGes. Math. Abh. 1, 66, 71–2.Google Scholar

§ Berichte…zu Leipzig, 36 (1884), 132.Google Scholar