An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T , overlapping only at their boundaries. The triangle T is the “tile”. The tile may or may not be similar to ABC. This paper is the first of four papers, which together seek a complete characterization of the triples (ABC,N, T ) such that ABC can be N-tiled by T . In this paper, we consider...

A geometric approach to the proof of Fermat’s last theorem is proposed. Instead integers a, b, c, considers a triangle with side lengths c. It proved that in case right-angled and obtuse-angled triangles Fermat's equation has no solutions. When considering when c are sides an acute triangle, it entire solutions for p>2. The numbers a=k, b=k+m, c=k+n, where k, m, n natural satisfying inequali...

We often need to refer to adjacent elements (e.g., vertices, edges and faces) in triangle meshes for rendering, mesh simplification and other processes. It is, however, sometimes impossible to prepare the enormous memory needed to represent element connectivity in gigantic triangle meshes. We proposed a memory efficient scheme for referring to adjacent faces around a vertex in non-manifold tria...

Isosceles trapezoids are dissected into three similar triangles and rearranged to form two additional isosceles trapezoids. Moreover, triangle centers, one from each similar triangle, form the vertices of a centric triangle which has special properties. For example, the centroidal triangles are congruent to each other and have an area one-ninth of the area of the trapezoids; whereas, the circum...

louis hubert farabeuf was born in bannost (sein-et-marne) on may 6th 1841 and died in beton-bazoche (aisne) in august 1910. he was professor and director of the anatomical department of the faculty of medicine in paris until 1878. in 1879, he founded the school of practical studies in paris, which contributed to anatomical studies. farabeuf designed many surgical procedures. his manual of opera...

We take a fresh look at the geometrization of logic using recently developed tools “quantum Riemannian geometry” applied in digital case over field $\mathbb{F}\_2={0,1}$, extending de Morgan duality to this context differential forms and connections. The $1$-forms correspond graphs exterior derivative subset amounts arrows that cross between set its complement. line graph $0-1-2$ has non-flat b...