A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Gr\"unbaum , who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Gr\"unbaum conjectured that the number of triangular cells p_3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least 2n-4. We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family with p_3(A)/n \to 16/11 = 1.45. We expect that the lower bound p_3(A) \geq 4n/3 is tight for infinitely many simple arrangements. It may however be that digon-free arrangements of n pairwise intersecting circles indeed have at least 2n-4 triangles. For pairwise intersecting arrangements with digons we have a lower bound of p_3 \geq 2n/3, and conjecture that p_3 \geq n-1. Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that p_3 \le 2n^2/3 +O(n). This is essentially best possible because families of pairwise intersecting arrangements of n pseudocircles with p_3/n^2 \to 2/3 as n \to \infty are known. The paper contains many drawings of arrangements of pseudocircles and a good fraction of these drawings was produced automatically from the combinatorial data produced by the generation algorithm. In the final section we describe some aspects of the drawing algorithm.
A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Gr\"{u , who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Gr\"{u conjectured that the number of triangular cells p_3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least 2n-4. We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family with p_3(A)/n \to 16/11 = 1.45. We expect that the lower bound p_3(A) \geq 4n/3 is tight for infinitely many simple arrangements. It may however be that digon-free arrangements of n pairwise intersecting circles indeed have at least 2n-4 triangles. For pairwise intersecting arrangements with digons we have a lower bound of p_3 \geq 2n/3, and conjecture that p_3 \geq n-1. Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that p_3 \le 2n^2/3 +O(n). This is essentially best possible because families of pairwise intersecting arrangements of n pseudocircles with p_3/n^2 \to 2/3 as n \to \infty are known. The paper contains many drawings of arrangements of pseudocircles and a good fraction of these drawings was produced automatically from the combinatorial data produced by the generation algorithm. In the final section we describe some aspects of the drawing algorithm.
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Gr\"unbaum
Gr\"{u
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Gr\"unbaum
Gr\"{u