We investigate the problem of cutting a triangle ABC into N congruent triangles (the “tiles”), which may or may not be similar to ABC. We wish to characterize the numbers N for which some triangle ABC can be tiled by N tiles, or more generally to characterize the triples (N,T ) such that ABC can be N-tiled using tile T . In the first part of the paper we exhibit certain families of tilings whic...

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 of these triangles; for this purpose we find for any positive integer n an explicit cubic number field Q(λ) (depending on n) and an explicit point Pλ of infi...

We investigate the problem of cutting a triangle ABC into N congruent triangles (the “tiles”), each of which is similar to ABC. The more general problem when the tile is not similar to ABC is not treated in this paper; see [1]. We give a complete characterization of the numbers N for which some triangle ABC can be tiled by N tiles similar to ABC, and also a complete characterization of the numb...

Given a simple polygon with rational coordinates having one vertex at the origin and an adjacent vertex on the x-axis, we look at the problem of the location of the vertices for a tiling of the polygon using lattice triangles (i.e., triangles which are congruent to a triangle with the coordinates of the vertices being integer). We show that the coordinate of the vertices in any tiling are ratio...

menelaus’ sphaerica can be considered as the most important classical text in the tradition of spherics books, written with the aim of the solution of problems arising in spherical astonomy. euclid’s elements is the the most important book on plane geometry. this article aims at a comparative study of menelaus’s sphaerica and euclid’s elements, to show that book i of sphaerica is an attempt to ...

Many years ago, Sommerville classified the tilings of the sphere with congruent spherical triangles – under some assumptions. Later, Davies did the same, under only the restriction that the triangles should meet edge-to-edge. This paper surveys a recently-discovered class of tilings with congruent spherical triangle tiles that do not meet edgeto-edge. Introduction. Any simple polyhedron can be ...

Learning takes time, namely, one needs to be exposed to contingency relations between stimulus dimensions in order to learn, whereas intentional control can be recruited through task demands. Therefore showing that control can be recruited as a function of experimental instructions alone, that is, adapting the processing according to the instructions before the exposure to the task, can be take...

A tiling of the plane by polygons is unilateral if each edge of the tiling is a side of at most one polygon of the tiling. A tiling is equitransitive if for any two congruent tiles in the tiling, there exists a symmetry of the tiling mapping one to the other. It is known that a unilateral and equitransitive (UE) tiling can be made with any finite number of congruence classes of squares. This ar...