THE DENSITY OF ELLIPSES PACKINGS WITH SIX CONTACTING NEIGHBOURS. Takeo Matsumoto and Masaharu T anemura, Dept. Earth Sciences, Faculty of Science, Kanazawa University, Japan and Institute of Statistical Mathematics, Tokyo, Japan

In contrast with the closeset packing of circles, with plane group,p6mm,periodic close packings of identical ellipses do not always have the maximum density [[rho]]=[[pi]]_12 = 0.9069.....

Nowacki(1948) and Grünbaum & Shephard (1987) have shown altogether seven different close packings of ellipses, with six contacting neighbours, namely, c2mm, p2, p2gg, p31m, p3 and two p2gg's pasckings in plane groups.

Matsumoto(1968) and Matsumoto & Nowacki(1966) have shown that the first two of the above densest packings of ellipses, c2mm (2 a 2mm) and p2(1 a 2) always attain the above maximum density. That of the third, p2gg (2 a 2) cannot exceed this maximum density. The former two packings,c2mm and p2, are derived from the densest packing of circles,p6mm (1 a 6mm), by affine transformation,while the p2gg packing of ellipses can never attain the maximum density of [[rho]].

Tanemura & Matsumoto(1992) have recently indicated that the density of p31m(3 c m) packing of ellipses, the fourth one of the above list,never exceeds the maximum density, and shows a maximum only for the case of axial ratio = 1,where the packing is equivalent to the p6mm packing of circles.

Matsumoto & Tanemura(1995) have also shown that the density of p3(3 d 1) packing of ellipses, the fifth one of the above list, never exceeds the maximum density through numerical computations and series expansions. This maximum density is attained only by the closest packing of circles, p6mm.

We are now calculating the density of two p2gg's (both 4 c 1) packings, the last ones of the above list, by numerical calculations and by expanding forms in terms of [[epsilon]]-k-1 (k = axial ratio) and [[theta]] (tilting angle of ellipse). For these two cases, each density cannot exceed the maximum density.