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Take any triangle with polygon vertices A, B, and C. Pick a point A_1 on the side opposite A, and draw a line parallel to BC. Upon reaching the side AC at B_1, draw the line ...

Given a reference triangle DeltaABC and a point P, the triple (x,y,z), with x=PA, y=PB and z=PC representing the distances from P to the vertices of the reference triangle, ...

Just as the ratio of the arc length of a semicircle to its radius is always pi, the ratio P of the arc length of the parabolic segment formed by the latus rectum of any ...

The "witch of Agnesi" is a curve studied by Maria Agnesi in 1748 in her book Instituzioni analitiche ad uso della gioventù italiana (the first surviving mathematical work ...

In antiquity, geometric constructions of figures and lengths were restricted to the use of only a straightedge and compass (or in Plato's case, a compass only; a technique ...

In two dimensions, there are two periodic circle packings for identical circles: square lattice and hexagonal lattice. In 1940, Fejes Tóth proved that the hexagonal lattice ...

The Klein bottle is a closed nonorientable surface of Euler characteristic 0 (Dodson and Parker 1997, p. 125) that has no inside or outside, originally described by Felix ...

An (ordinary) torus is a surface having genus one, and therefore possessing a single "hole" (left figure). The single-holed "ring" torus is known in older literature as an ...

The cube is the Platonic solid composed of six square faces that meet each other at right angles and has eight vertices and 12 edges. It is also the uniform polyhedron with ...

Barycentric coordinates are triples of numbers (t_1,t_2,t_3) corresponding to masses placed at the vertices of a reference triangle DeltaA_1A_2A_3. These masses then ...