Related papers: Parallel packing a square with isosceles right tri…
A geometric inequality among three triangles, originating in circle packing problems, is introduced. In order to prove it, we reduce the original formulation to the nonnegativity of a particular polynomial in four real indeterminates.…
A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$…
Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction…
For a positive integer $n\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded $n$-sided polygons with unit length…
In this paper we continue the investigation of finding the max/min polygons which can be inscribed in a given triangle. Here we are concerned with equilateral triangles. This may seem uninteresting or benign at first, but there are some…
Every regular map on a closed surface gives rise to generally six regular maps, its "Petrie relatives", that are obtained through iteration of the duality and Petrie operations (taking duals and Petrie-duals). It is shown that the skeletal…
The problem widely known as Moser's Square Packing Problem asks for the smallest area $A$ such that for any set $S$ of squares of total area $1$, there exists a rectangle $R$ of area $A$ into which the squares in $S$ permit an…
In this paper we prove that if $P_1, P_2$ are isogonal points in the triangle $ABC$, and if $A_1B_1C_1$ and $A_2B_2C_2$ are their corresponding pedal triangles such that the triangles $ABC$ and $A_1B_1C_1$ are homological (the lines $AA_1,…
An upward equilateral triangle of side $n$ can be partitioned into $n$ unit upward equilateral triangles and $\frac{n(n-1)}{2}$ unit rhombi with $60^{\circ}$ and $120^{\circ}$ angles. In this paper, we focus on understanding such partitions…
Answering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene…
We prove a theorem on the relationships between the lengths of sides of a spherical quadrilateral with three right angles. They are analogous to the relationships in the Lambert quadrilateral in the hyperbolic plane. We apply this theorem…
In this pedagogical note we present a short proof of the following main result of arxiv.org/abs/0911.5319, and clarify its relation to the isoperimetric problem. On the hyperbolic plane consider triangles ABC with fixed lengths of AB and…
We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in…
We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not…
In this paper, we investigate the homothetic point enclosure problem: given a set $S$ of $n$ triangles with sides parallel to three fixed directions, find a data structure for $S$ that can report all the triangles of $S$ that contain a…
We obtain a rigidity result for compact three-dimensional Heterotic solitons with parallel non-trivial torsion. We show that they are either hyperbolic three-manifolds or compact quotients of the Heisenberg group equipped with a…
We determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, we establish maximality by extending the Lov\'asz theta number to Cayley graphs on the special orthogonal group…
Let D = {D_{1},...,D_{l}} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let \Omega^{1}_{P^n}(log D) be the logarithmic bundle attached to it. Following [1], we show that…
In this paper we consider the problem of packing a fixed number of identical circles inside the unit circle container, where the packing is complicated by the presence of fixed size circular prohibited areas. Here the objective is to…
The intersection matrix of a simplicial complex has entries equal to the rank of the intersection of its facets. In [1] the authors prove the intersection matrix is enough to determine a triangulation of a surface up to isomorphism. In this…