Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-04T21:24:18.575Z Has data issue: false hasContentIssue false
  • English
  • Français

On a conjectured analogue to Clifford's chain in [4]

Published online by Cambridge University Press:  24 October 2008

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Trinity College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

Clifford's well-known chain of theorems concerning circles in a plane may be stated as follows (the first two links being trivial):

(C2) Given any point A, and through it two circles C1 and C2, these meet again in a point A12, say.

(C3) Given three circles C1, C2, C3 through A, the three points A23, A13, An lie on a circle C123, say.

(C4) Given four circles C1,…,C4 through A, the four circles C234,…,C123 meet in a point A1234 (Wallace(1)).

(C5) Given five circles C1,…,C5 through A, the five points A2345,…,A1234 lie on a circle C12345 (Miquel (2)).

And so on (Clifford(3)).

Grace (4) showed that in three dimensions [3], there is no direct analogue to Clifford's chain, but in [4] he noted the following:

(G4) Given any point A in [4], and through it four hyperspheres S1,…, S4, these meet again in a point A1234, say.

(G5) Given five hyperspheres Sl,…,S5 through a point A, the five points A2345, A1234 lie on a hypersphere S12345, say.

(G6) Given six hyperspheres Sl,…, S7 through A, the six hyperspheres S23456,…, A12345 have in common a point A123456.

This leads naturally to the conjecture

(G7) Given seven hyperspheres Sl,…, S7 through A, the seven points A234567,…, A123456 lie on a hypersphere S1234567.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 73 , Issue 1 , January 1973 , pp. 37 - 43
Copyright
Copyright © Cambridge Philosophical Society 1973

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Wallace, W.Question No. 86 The Mathematical Repository, ed. Leybourn, T., vol. 1 (New Series) (1804) Part C, p. 22.Google Scholar
(2)Miquel, A.Mémoire de géometrie (2me partie) Angles curvilignes, intersections des cercles et des sphères. Liouville J. de Math. 10 (1845), 347–350.Google Scholar
(3)Clifford, W. K.Synthetic proof of Miquel's theorem. Oxford, Cambridge and Dublin Messenger of Math. 5 (1870), 124–141.Google Scholar
(4)Grace, J. H.Circles, spheres and linear complexes. Proc. Cambridge Philos. Soc. 16 (1898), 153–190.Google Scholar
(5)Brown, L. M.A configuration of points and spheres in four-dimensional space. Proc. Roy. Soc. Edinburgh Sect. A 64, 145–149.Google Scholar
(6)Hirschfeld, J. W. P.A projective configuration. Computational problems in abstract algebra, ed. Leech, J., pp. 321–323. (Oxford, Pergamon Press, 1970.)Google Scholar
(7)Longuet-Higgins, M. S.A numerical disproof of a conjecture in projective geometry. Proc. Roy. Soc. Ser. A 323 (1971), 443–453.Google Scholar
(8)Wren, T. L.Some applications of the two-three birational space transformation. Proc. London Math. Soc. 15 (1916), 144–165.Google Scholar
(9)Telephone Directory, Section 20 (Cambridge Area), p. 224. (H.M. Stationery Office, 1969.)Google Scholar
(10)Todd, J. A.Configurations defined by six lines in space of three dimensions. Proc. Cambridge Philos. Soc. 29 (1933), 52–68.Google Scholar
(11)Hirschfeld, J. W. P.A configuration of lines in three dimensions. Proc. Edinburgh Math. Soc. (in Press).Google Scholar
(12)Longuet-Higgins, M. S.Clifford's chain and its analogues in relation to the higher polytopes. Proc. Roy. Soc. London A 330 (1972), 443–466,Google Scholar