Related papers: A characterization of regular tetrahedra in Z^3
In this note we describe a procedure of calculating the number all regular tetrahedra that have coordinates in the set {0,1,...,n}. We develop a few results that may help in finding good estimates for this sequence which is twice A103158 in…
We study the existence of equilateral triangles of given side lengths and with integer coordinates in dimension three. We show that such a triangle exists if and only if their side lengths are of the form $\sqrt{2(m^2-mn+n^2)}$ for some…
This paper is a continuation of previous work of the authors. We extend one of the theorems that gave a way to construct equilateral triangles whose vertices have integer coordinates to the general situation. An approximate extrapolation…
In this paper we describe a procedure for calculating the number of regular octahedrons that have vertices with coordinates in the set {0,1,...,n}. As a result, we introduce a new sequence in ``The Online Encyclopedia of Integer Sequences"…
Extending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates ("in $\mathbb Z^3$"), we look at the problem of characterizing all regular polyhedra (Platonic Solids) with the…
In this paper we describe a procedure of calculating the number cubes that have coordinates in the set {0,1,...,n}. We adapt the code that appeared in [11] developed to calculate the number of regular tetrahedra with coordinates in the set…
We consider the problem of enumerating integer tetrahedra of fixed perimeter (sum of side-lengths) and/or diameter (maximum side-length), up to congruence. As we will see, this problem is considerably more difficult than the corresponding…
We propose a classification of polyhedra (planar, $3$-connected graphs) according to their type i.e., their set of quantities of common neighbours for each pair of distinct vertices. For every (finite) set of non-negative integers, we…
We give a characterization of all three points in $\mathbb R^4$ with integer coordinates which are at the same Euclidean distance apart. In three dimension the problem is characterized in terms of solutions of the Diophantine equations…
We prove inequalities involving intrinsic and extrinsic radii and diameters of tetrahedra.
We completely describe the structure of the connected components of transversals to a collection of n line segments in R^3. We show that n>2 arbitrary line segments in R^3 admit 0, 1, ..., n or infinitely many line transversals. In the…
Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic…
Integer geometry on a plane deals with objects whose vertices are points in $\mathbb Z^2$. The congruence relation is provided by all affine transformations preserving the lattice $\mathbb Z^2$. In this paper we study circumscribed circles…
We prove that every tetrahedron T has a simple, closed quasigeodesic that passes through three vertices of T. Equivalently, every T has a face whose "exterior angles" are at most pi.
We describe a procedure of counting all equilateral triangles in the three dimensional space whose coordinates are allowed only in the set $\{0,1,...,n\}$. This sequence is denoted here by ET(n) and it has the entry A102698 in "The On-Line…
We analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call "trihexes". Trihexes are analogous to fullerenes, which are 3-regular planar…
In this paper we discuss the classification rank $3$ lattices preserved by finite orthogonal groups of isometries and derive from it the classification of regular polyhedra in the $3$-dimensional torus. This classification is highly related…
In general graph theory, the only relationship between vertices are expressed via the edges. When the vertices are embedded in an Euclidean space, the geometric relationships between vertices and edges can be interesting objects of study.…
We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of…
Three--dimensional colored triangulations are gluings of tetrahedra whose faces carry the colors 0, 1, 2, 3 and in which the attaching maps between tetrahedra are defined using the colors. This framework makes it possible to generalize the…