Mathematics

First described in 2014, Roli's cube R is a chiral 4 -polytope, faithfully realized in Euclidean 4 -space (a situation earlier thought to be impossible). Here we describe R in a new way, determine its minimal regular cover, and reveal connections to the Möbius-Kantor configuration.

We investigate several topics of triangle geometry in the elliptic and in the extended hyperbolic plane, such as: centers based on orthogonality, centers related to circumcircles and incircles, radical centers and centers of similitude, orthology, Kiepert perspectors and related objects, Tucker circles, isoptics, substitutes for the Euler line. For both, the elliptic and the extended hyperbolic plane, a uniform metric is used.

Let A= 3 – √ . There are two main conjectures about paper Moebius bands. First, a smooth embedded paper Moebius band must have aspect ratio at least A. Second, any sequence of smooth embedded paper Moebius bands having aspect ratio converging to A converges, in the Hausdorff topology and up to isometries, to an equilateral triangle of semiperimeter A. We will reduce these conjectures to 10 statements about the positivity of certain explicit piecewise algebraic expressions.

This note presents simple linear algebraic proofs of theorems due to Sinajova, Rankin and Kuperberg concerning spherical point configurations. The common ingredient in these proofs is the use of spherical Euclidean distance matrices and the Perron-Frobenius theorem.

A remarkably simple Diophantine quadratic equation is known to generate all Apollonian integral gaskets (disk packings). A new derivation of this formula is presented here based on inversive geometry. Also occurrences of Pythagorean triples in such gaskets is discussed.

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.

We investigate the uniqueness for the Monge-Ampère type equation det( u ij + δ ij u ) n−1 i,j=1 =G(u), (∗) on S n−1 , where u is the restriction of the support function on the sphere S n−1 of a convex body that contains the origin in its interior and G:(0,∞)→(0,∞) is a continuous function. The problem was initiated by Firey (1974) who, in the case G(θ)= θ −1 , asked if u≡1 is the unique solution to (*). Recently, Brendle, Choi and Daskalopoulos [9] proved that if G(θ)= θ −p , p>−n−1 , then u has to be constant, providing in particular a complete solution to Firey's problem. Our primary goal is to obtain uniqueness (or nearly uniqueness) results for (*) for a broader family of functions G . Our approach is very different than the techniques developed in [9] .

Let d≥2 and let K and L be two convex bodies in R d . We say that K and L satisfy the (d+1) -equichordal property if L⊂intK , the boundary of L does not contain a segment, and for any line ℓ supporting the boundary of L and for two points { ζ ± } of an intersection of the boundary of K with ℓ one has dist d+1 (L∩ℓ, ζ + )+ dist d+1 (L∩ℓ, ζ − )=c, where a constant c is independent of ℓ . It was shown in \cite{R} that if d≥3 and a smooth K is such that its Dupin floating body coincides with the Bárány-Larman-Schütt-Werner convex floating body K δ , 0<δ< vol d (K) , then K floats in equilibrium in every direction, provided K and K δ satisfy the (d+1) -equichordal property. We prove that if K and L have C 3 -smooth boundaries and L is a body of revolution around an axis l such that L∩l is symmetric with respect to the origin, then K and L are concentric Euclidean balls.

Ulam's problem 19 from the Scottish Book asks: {\it is a solid of uniform density which floats in water in every position necessarily a sphere?} We obtain several results related to this problem.

Following the train of thought from our previous paper we revisit the theorems of Pongsriiam and Termwuttipong by further developing their characterization of certain property-preserving functions using the so-called triangle triplets. We develop more general analogues of disjoint sum lemmas for broader classes of metric-type spaces and we apply these to extend results of Borsìk an Doboš as well as those obtained by Khemaratchatakumthorn and Pongsriiam. As a byproduct we obtain methods of generating non-trivial and infinite strong b -metric spaces which are not metric.