An adaptive method for polygonization of implicit surfaces is presented. The method insists on the shape of triangles and the accuracy of resulting approximation as well. The presented algorithm is based on the surface tracking scheme and it is compared with the other algorithms based on the similar principle, such as the Marching cubes and the Marching triangles methods. The main advantages of...

Dense packings of hard particles have important applications in many fields, including condensed matter physics, discrete geometry, and cell biology. In this paper, we employ a stochastic search implementation of the Torquato-Jiao adaptive-shrinking-cell (ASC) optimization scheme [Nature (London) 460, 876 (2009)] to find maximally dense particle packings in d-dimensional Euclidean space R(d). W...

The Pythagorean theorem can be proven geometrically through the use of dissections of squares and triangles. Four of these decompositions are paired into two compound dissections, which are used to create novel images. Pythagorean Theorem Many proofs of the Pythagorean theorem are given in the literature, for example, at the MathWorld web site [1]. Of particular interest to this work are dissec...

Monsky’s celebrated equidissection theorem follows from his more general proof of the existence a polynomial relation f among areas triangles in dissection unit square. More recently, authors studied different p, also such dissection, that is invariant under certain deformations dissection. In this paper we study relationship between these two polynomials. We first generalize notion allowing wh...

We describe computer algorithms that can enumerate and display, for a given n > 0 (in theory, of any size), all n-ominoes, niamonds, and n-hexes that can tile the plane using only rotations; these sets necessarily contain all such tiles that are fundamental domains for p4, p3, and p6 isohedral tilings. We display the outputs for small values of n. This expands on earlier work [3]. 1 Polyominoes...

We will make this more specific by specifying what sorts of measurements will be allowed. For example, in much of the paper, allowed measurements will include distances between pairs of vertices, angles between edges, angles between two intersecting face diagonals (possibly on different faces with a common vertex) or between a face diagonal and an edge, and dihedral angles (that is, angles betw...

This paper presents an adaptive modification of the Edge spinning method for polygonization of implicit surfaces. The method insists on the shape of triangles and the accuracy of resulting approximation as well. The main advantages of the triangulation presented are simplicity and the stable features that can be used for next expanding. The implementation is not complicated and only the standar...

The synthesis of holograms by computer requires the calculation of the complex amplitude emitted by an object in the hologram plane. Several approaches exist that decompose the object into primitives like points and lines. We assume that the object to be imaged can be approximately decomposed into congruent triangles. In a preprocessing step, we calculate wave elds for the triangle whose transf...

Here we discuss which triangles do, and which don’t, admit a tiling of H,E, and S. These notes are meant to pick up as “Regular Production Systems and Triangle Tilings” [16] leaves off. We pause for a conjecture and some additional nomenclature: Let T be any triangle in X = H,E, S. A configuration by T is a collection of congruent copies of T, for each pair of which meet edge-to-edge and vertex...

This article highlights interactions of diverse areas: the Heron formula for the area of a triangle, the Descartes circle equation, and right triangles with integer or rational sides. New and old results are synthesized. We first exploit elementary observations about circles to characterize an arbitrary triangle using three circles. A fourth circle brings a certain symmetry – the four radii are...