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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
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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

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