
 
Caption: Mathematical Models by Arnold Emch  Series 3 (1925) This is a reducedresolution page image for fast online browsing.
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30. Quintic cyclide.1 (By Dr. H . P. Pettit) T h e surface is one of the few anallagmatic quintic cyclides, this one being anallagmatic with respect to eight spheres. It is projectively generated by the tangent planes of a cone of second order and a pencil of concentric spheres xX2 + yX + z = 0, where p _ XQ = 0, P = x2 + y2 + z2 — r2, Q = x2 + y2 + z2 + r2, and has the form x(x2 + y2 + z2  r2)2 + y(x2 + y2 + z2  r2) (x2 + y2 + z2 + r2) + z (x2 + y2 + z2 + r2)2 = 0. The vertex of the double tangent cone and the center of the pencil of spheres coincide at the origin. T h e surface has one infinite set of circular generators and is symmetrical with respect to the origin. T h e section by the xyplane is a quintic composed of a cubic and a circle of radius r. T h e section by the xzplane is a proper quintic. T h e section by the yzplane is a degenerate quintic composed of a cubic and an imaginary circle. T h e topography in the neighborhood of the xyplane shows the change from the degenerate quintic to the proper quintic with the loop and serpentine, Fig. 5. 31. Model of the symmetric substitution group of order 24 (G24). T h e symmetric substitution group of n elements of order nl m a y be represented geometrically in a projective space of n — \ dimensions and thus lends to important properties of certain geometric forms. T h e author has recently2 considered some of these applications, principally with reference to the G6 and G24 T h e four elements, numbers, X\> x2y xs, #4 of the G24 are chosen as projective coordinates of a point in 3space. T h e model illustrates the theorem: Any set of 24 points of the G24 lies on 16 conies of a quadric whose planes by four pass through the four lines S{ cut out from the unitplane e by the coordinateplanes X{ = 0. The points of the group lie two by two on 72 lines which in sets of 12 pass through the six vertices E{k of *A general cyclide with special reference to the quintic cyclide. The Tohoku Mathematical Journal, Vol. 23 (1923), pp. 125. 2The American Journal of Mathematics, Vol. X L V (1923), pp. 192207. ni]
 