Related papers: Optimizing for the Rupert property
We prove results that imply, under various hypotheses, that every elliptic curve over a number field $k$ corresponding to a point on a modular curve has bad reduction at a certain prime $p$ of $\mathcal{O}_k$. For example, every elliptic…
Let $P$ be a polygon with $r>0$ reflex vertices and possibly with holes and islands. A subsuming polygon of $P$ is a polygon $P'$ such that $P \subseteq P'$, each connected component $R$ of $P$ is a subset of a distinct connected component…
We consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations. For classical finite elements, such methods are well known to be fast solvers…
The landscapes of a polyhedron are subsets of its nets one must consider to identify all shortest paths. Landscapes of cubes and tetrahedra have been used to identify coordinate based formulas for the lengths of the shortest paths between…
Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. We consider this property for regular polytopes in arbitrary dimensions, notably the simplex, cube, and…
The equality between the number of odd spin structures on a Riemann surface of genus g, with $2^g - 1$ being a Mersenne prime, and the even perfect numbers, is an indication that the action of the modular group on the set of spin structures…
Let $p$ be an odd natural number $\ge 3$. Inspired by results from Euclid's {\em Elements}, we express the irrational $$y=\sqrt[p]{d+\sqrt R}, $$ whose degree is $2p$, as a polynomial function of irrationals of degrees $\le p$. In certain…
A rational triangle is a triangle with rational sides and rational area. A Heron triangle is a triangle with integral sides and integral area. In this article we will show that there exist infinitely many rational parametrizations, in terms…
Octupolar tensors are third order, completely symmetric and traceless tensors. Whereas in 2D an octupolar tensor has the same symmetries as an equilateral triangle and can ultimately be identified with a vector in the plane, the symmetries…
A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2 if it is combinatorially regular but "fails geometric regularity by a factor of 2"; its combinatorial automorphism group is flag-transitive but its geometric…
There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is…
There is a highly special point configuration in $\mathbb{P}^2$ of 31 points, naturally arising from the geometry of the icosahedron. The 15 planes of symmetry of the icosahedron projectivize to 15 lines in $\mathbb{P}^2$, whose points of…
Not only is the geometry of rock fragments often well approximated by ideal convex polyhedra having few faces and vertices, but these numbers carry vital geophysical information on the fragmentation process. Despite their significance, the…
We study the bit complexity of two methods, related to the Euclidean algorithm, for computing cubic and quartic analogs of the Jacobi symbol. The main bottleneck in such procedures is computation of a quotient for long division. We give…
We develop the basic tools for classifying edge-to-edge tilings of the sphere by congruent pentagons. Then we prove that, for the edge combination $a^2b^2c$, such tilings are three two-parameter families of pentagonal subdivisions of the…
In this article, we prove that every integer can be written as an integer combination of exactly 4 tetrahedral numbers. Moreover, we compute the modular periodicity of platonic numbers.
The pooling problem has applications, e.g., in petrochemical refining, water networks, and supply chains and is widely studied in global optimization. To date, it has largely been treated deterministically, neglecting the influence of…
This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere $\mathbb{S}^2$ so that the surface area of the convex hull of the points is maximized. It is shown that the optimal…
We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard…
It is shown that there exists a charge five monopole with octahedral symmetry and a charge seven monopole with icosahedral symmetry. A numerical implementation of the ADHMN construction is used to calculate the energy density of these…