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Register a new userOptimal packings of congruent circles on a square flat torus
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We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason - the problem of super resolution of images. We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.
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A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in $mathbb{R}^3$ is not greater than $13.92$. We also find new upper bounds for the average degree of contact graphs in $mathbb{R}^4$ and $mathbb{R}^5$.
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Viera v{C}erv{n}anova (Department of Mathematics
, Computern Science
, Faculty of Education
2020
Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. The configuration and its components, which are two orthogonal frames and two orthogonal families of chains, are in some way connected to classical geometric configurations such as the arbelos or the Pappus chain, or the Apollonian packing from the 20th century. In this paper, we build the fabric and list some of the obvious properties that result from this construction. Next, we focus on the curvature inside the individual components: we show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures of the circles of each chain are arranged in a quadratic bi-sequence. Because solving geometric sangaku problems was a gateway to our discovery of the fabric, we conclude this paper with two sangaku problems and their solutions using our results on curvatures.
Define the augmented square twist origami crease pattern to be the classic square twist crease pattern with one crease added along a diagonal of the twisted square. In this paper we fully describe the rigid foldability of this new crease pattern. Specifically, the extra crease allows the square twist to rigidly fold in ways the original cannot. We prove that there are exactly four non-degenerate rigid foldings of this crease pattern from the unfolded state.
Monskys theorem from 1970 states that a square cannot be dissected into an odd number of triangles of the same area, but it does not give a lower bound for the area differences that must occur. We extend Monskys theorem to constrained framed maps; based on this we can apply a gap theorem from semi-algebraic geometry to a polynomial area difference measure and thus get a lower bound for the area differences that decreases doubly-exponentially with the number of triangles. On the other hand, we obtain the first superpolynomial upper bounds for this problem, derived from an explicit construction that uses the Thue-Morse sequence.
Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz square peg problem. We prove Hadwigers 1971 conjecture that any simple loop in $3$-space inscribes a parallelogram. We show that any simple planar loop inscribes sufficiently many rectangles that their vertices are dense in the loop (independently due to Schwartz). If the loop is rectifiable, there is a rectangle that cuts the loop into four pieces that can be rearranged to form two loops of equal length. A rectifiable loop in $d$-space can be cut into $(r-1)(d+1)+1$ pieces that can be rearranged by translations to form $r$ loops of equal length. We relate our results to fair divisions of necklaces in the sense of Alon and to Tverberg-type results. This provides a new approach and a common framework to obtain variants of Toeplitz square peg problem for the class of all continuous curves.
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