Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-09T00:07:02.898Z Has data issue: false hasContentIssue false
  • English
  • Français

Tetrads of Möbius tetrahedra

Published online by Cambridge University Press:  09 April 2009

Sahib Ram Mandan
Sahib Ram Mandan
Affiliation:
Indian Institute of Technology, Kharagpur
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A tetrad of Möbius tetrahedra consists of a set of 4 mutually inscribed and therefore circumscribed tetrahedra whose 16 vertices and 16 faces form a Kummer's 166 configuration (5; 11; 12; 21). As pointed out by the refere, fundamental to all work on the 166 figure are the 10 quadrics, called fundamental for the associated Kummer's quartic surface (13). To every quadric F correspond a matrix scheme of the 16 points or planes, arranged in 4 rows or columns, such that the 8 Rosenhain tetrahedra (7) formed of the rows and columns are all self-polar for F. The rows form one and the columns another tetrad of Mobius tetrahedra. Nine new schemes can be derived from one such scheme to make the total ten as explained by Baker (3, p. 133) leading to 80 Rosenhain tetrahedra in all. The 16 nodes (5; 8) or tropes of the Rummer's quartic are the 16 common elements of the 10 schemes such that the nodes and tropes are poles and polars for any one of the 10 quadrics. Each trope touches the quartic along a singular conic through the 6 points of the figure lying therein. The lines tangent to the surface at its each node N generate a quadric cone which is enveloped by the 6 tropes through N (12).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 3 , Issue 1 , February 1963 , pp. 68 - 78
Copyright
Copyright © Australian Mathematical Society 1963

References

[1]Baker, H. F., Principles of gemoetry 2 (Cambridge, 1930).Google Scholar
[2]Baker, H. F., Principles of geometry 3 (Cambridge, 1934).Google Scholar
[3]Baker, H. F., Principles of geometry 4 (Cambridge, 1940).Google Scholar
[4]Baker, H. F., A locus with 25920 linear transformations (Cambridge, 1946).Google Scholar
[5]Baker, H. F., Note to the preceding paper by C. V. H. Rao, Proc. Camb. Phil. Soc. 42 (1946), 226–229.CrossRefGoogle Scholar
[6]Court, N. A., Modern pure solid geometry (New York, 1935).Google Scholar
[7]Coxeter, H. S. M., The real projective plane (Cambridge, 1955).Google Scholar
[8]Edge, W. L., The net of quadrics associated with a piar of Möbius tetrads, Proc. Lond. Math. Soc. 41 (1936), 336–360.Google Scholar
[9]Edge, W. L., Geometry in three dimensions over GF(L3), Proc. Roy. Soc. A 222 (1954), 262–286.Google Scholar
[10]Hamil, C. M., On a finite group of order 576, Proc. Phil. Soc. 44 (1948), 26–36.CrossRefGoogle Scholar
[11]Hanumanta, C. V. and Baker, H. F., On the generation of sets of four tetrahedra of which any two are mutually inscribed, Proc. Camb. Phil. Soc. 20 (1921). 155–157.Google Scholar
[12]Hudson, R. W. H. T., Kummer's quartic surface (Cambridge, 1905).Google Scholar
[13]Mandan, S. R., Properties of mutually self-polar tetrahedra, Bull. Cal. Math. Soc. 33 (1941), 147–155.Google Scholar
[14]Mandan, S. R., Umbilical projection, Proc. Ind. Acad. Sc. A 15 (1942), 16.CrossRefGoogle Scholar
[15]Mandan, S. R., Möbius tetrads, Math. Mon. 64 (1957), 471–478.Google Scholar
[16]Mandan, S. R., Möbius tetrads, II, Math. Mon. 65 (1958), 247–251.Google Scholar
[17]Mandan, S. R., An S-configuration in Euclidean and elliptic n-space, Can. J. Math. 10 (1958), 489–501.CrossRefGoogle Scholar
[18]Mandan, S. R., Harmonic inversion, Math. Mag. 32 (1959), 71–78.CrossRefGoogle Scholar
[19]Mandan, S. R., On four intersecting spheres, Jour. Ind. Math. Soc. 23 (1959), 151–167.Google Scholar
[20]Mandan, S. R., Semi-isoldynamic and -isogonic tetrahedra, Rend. Mat. 19 (1960), 401–415.Google Scholar
[21]Rao, C. V. H., On the generation of sets of four tetrahedra of which any two are mutually inscribed, Proc. Camb. Phil. Soc. 42 (1946), 217–226.CrossRefGoogle Scholar
[22]Salmon, G.Analytical geometry of three dimensions 1 (New York, 1927).Google Scholar
[23]Smith, C., Solid geometry (London, 1951).Google Scholar
[24]Sommerville, D. M. Y., Analytical geometry of three dimensions (Camb., 1959).Google Scholar
[25]Vaidyanathswamy, R., A memoir on the cubic transformations associated with a desmic system, Jour. Ind. Math. Soc. Suppl. (1927).Google Scholar