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Two figures are said to be similar when all corresponding angles are equal, and are directly similar when all corresponding angles are equal and described in the same ...

The envelope of a one-parameter family of curves given implicitly by U(x,y,c)=0, (1) or in parametric form by (f(t,c),g(t,c)), is a curve that touches every member of the ...

1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line ...

The study of how the intrinsic structure of graphs ensures certain types of properties (e.g., clique-formation and graph colorings) under appropriate conditions.

A fallacy is an incorrect result arrived at by apparently correct, though actually specious reasoning. The great Greek geometer Euclid wrote an entire book on geometric ...

An origami configuration that can be pressed to a plane figure without crumpling it or adding new creases.

A differential k-form can be integrated on an n-dimensional manifold. The basic example is an n-form alpha in the open unit ball in R^n. Since alpha is a top-dimensional ...

Let (Omega)_(ij) be the resistance distance matrix of a connected graph G on n nodes. Then Foster's theorems state that sum_((i,j) in E(G)))Omega_(ij)=n-1, where E(g) is the ...

Given three coins of possibly different sizes which are arranged so that each is tangent to the other two, find the coin which is tangent to the other three coins. The ...

Because the Legendre polynomials form a complete orthogonal system over the interval [-1,1] with respect to the weighting function w(x)=1, any function f(x) may be expanded ...