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The symmetric group S_n of degree n is the group of all permutations on n symbols. S_n is therefore a permutation group of order n! and contains as subgroups every group of ...

Two points z and z^S in C^* are symmetric with respect to a circle or straight line L if all circles and straight lines passing through z and z^S are orthogonal to L. Möbius ...

The trace of a second-tensor rank tensor T is a scalar given by the contracted mixed tensor equal to T_i^i. The trace satisfies ...

A number of attractive 18-compounds of the regular tetrahedron can be constructed. The compound illustrated above will be implemented in a future version of the Wolfram ...

Given a regular tetrahedron of unit volume, the mean triangle area of a triangle picked at random inside it is approximately A=0.1811+/-0.0012, and the variance is ...

A 30-sided polygon. The regular triacontagon with side length 1 has inradius r, circumradius R, and area A given by r = 1/4(sqrt(15)+3sqrt(3)+sqrt(2)sqrt(25+11sqrt(5))) (1) R ...

Construction of the angle pi/4=45 degrees produces an isosceles right triangle. Since the sides are equal, sin^2theta+cos^2theta=2sin^2theta=1, (1) so solving for ...

An error in a statistical test which occurs when a false hypothesis is accepted (a false positive in terms of the null hypothesis).

A cube whose edge lengths are 1. The unit cube therefore has unit volume.

Given two intersecting lines, the two nonadjacent angles with the same vertex are said to be vertical angles. One can easily prove that vertical angles are congruent. Some ...