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A topological basis is a subset B of a set T in which all other open sets can be written as unions or finite intersections of B. For the real numbers, the set of all open ...

A ring torus constructed out of a square of side length c can be dissected into two squares of arbitrary side lengths a and b (as long as they are consistent with the size of ...

Let CD be the altitude of a triangle DeltaABC and let E be its midpoint. Then area(DeltaABC)=1/2AB·CD=AB·DE, and ABFG can be squared by rectangle squaring. The general ...

The triangular cupola is Johnson solid J_3. Its faces consist of 4 equilateral triangles, 3 squares, and 1 hexagon. It is implemented in the Wolfram Language as ...

A series involving three sums. Examples of convergent triple series include sum_(i=1)^(infty)sum_(j=1)^(infty)sum_(k=1)^(infty)1/((ijk)^2) = 1/(216)pi^6 (1) ...

To truncate a real number is to discard its noninteger part. Truncation of a (positive) number x therefore corresponds to taking the floor function |_x_|. Truncation also ...

The Tucker-Brocard cubic is the triangle cubic with trilinear equation abcsum_(cyclic)aalpha(b^2beta^2+c^2gamma^2)=alphabetagammasum_(cyclic)a^2(b^4+c^4). It passes through ...

The uniform polychora are four-dimensional analogs of the uniform polyhedra. In fact, the uniform polyhedra are cells of the uniform polychora. There are more than 8000 known ...

A compact set W_infty with area mu(W_infty)=8/9(24)/(25)(48)/(49)...=pi/4 created by punching a square hole of length 1/3 in the center of a square. In each of the eight ...

The abundance of a number n, sometimes also called the abundancy (a term which in this work, is reserved for a different but related quantity), is the quantity ...