By William M. Ivins

A hugely stimulating examine of the occasionally startling interrelationships among paintings and arithmetic all through heritage, as saw through the traditional Greeks, Renaissance theorists, artists akin to Albrecht D?rer, and so forth.

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Zong: The blocking numbers of convex bodies, Discrete Comput. Geom. 24 (2000) 267–277. [Da86] L. Danzer: Finite point-sets on S 2 with minimum distance as large as possible, Discrete Math. 60 (1986) 3–66. [FeT05] ´ th: Thinnest covering of a circle by eight, nine, G. E. , Cambridge Univ. Press, MSRI Publications 52 (2005), to appear. [FeT69] ´ th: Kreis¨ G. Fejes To uberdeckungen der Sph¨ are, Studia Sci. Math. Hung. 4 (1969) 225–247. [FeT72] ´ th: Lagerungen in der Ebene, auf der Kugel L. Fejes To und im Raum (2.

Meir (personal communication) has shown that k(4) = 17, and several others have proved that log k(d) = 1 + o (1) d log d, 2 as d → ∞. V. B´alint and V. B´ alint Jr. [B´ aB03] have supplied the ﬁrst written proofs for these results and made the conjectures k(5) = 34, k(6) = 76, k(7) = 152, and k(8) = 353. Makai Jr. 63901(1+o(1))d . 6 Packing Equal Circles into Squares, Circles, Spheres 35 There are very few nontrivial results concerning packings and coverings of higher-dimensional unit spheres Sd ⊂ IRd+1 (d ≥ 3) with spherical balls.

The easy proof of g (7) = 1/2 can be found in ˇ ˇ ˇ CJ74]. ˇ Skljarski˘ ı, Cencov, and Jaglom [Sk For n = 8, 9, and 10, the values g of (n) were determined by G. Fejes T´oth [FeT05]. For further construc¨ tions, conjectures, and results on g (n) up to n = 30, see [Nu99], [NuO00]. The thinnest coverings of a square by 1–5 and 7 circles The problem of thinnest covering of a unit equilateral triangle with n equal circles was solved by Melissen [Me97] for n ≤ 6 and for n = 9, 10. 34 1 Density Problems for Packings and Coverings This work as well as [Nu00] presented many conjecturally optimal coverings for n ≤ 36.