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"God is or He is not...Let us weigh the gain and the loss in choosing...'God is.' If you gain, you gain all, if you lose, you lose nothing. Wager, then, unhesitatingly, that ...

The "perp dot product" a^_|_·b for a and b vectors in the plane is a modification of the two-dimensional dot product in which a is replaced by the perpendicular vector ...

A 1-form omega=sum_(i=1)^na_i(x)dx_i such that omega=0.

An algebra in which the associator (x,x,x)=0. The subalgebra generated by one element is associative.

An epicycloid with n=5 cusps, named after the buttercup genus Ranunculus (Madachy 1979). Its parametric equations are x = a[6cost-cos(6t)] (1) y = a[6sint-sin(6t)]. (2) Its ...

A polynomial having only real numbers as coefficients. A polynomial with real coefficients is a product of irreducible polynomials of first and second degrees.

Given a rectangle BCDE, draw EF=DE on an extension of BE. Bisect BF and call the midpoint G. Now draw a semicircle centered at G, and construct the extension of ED which ...

Specifying two sides and the angle between them uniquely (up to geometric congruence) determines a triangle. Let c be the base length and h be the height. Then the area is ...

In any triangle, the locus of a point whose pedal triangle has a constant Brocard angle and is described in a given direction is a circle of the Schoute coaxal system.

For p(z)=a_nz^n+a_(n-1)z^(n-1)+...+a_0, (1) polynomial of degree n>=1, the Schur transform is defined by the (n-1)-degree polynomial Tp(z) = a^__0p(z)-a_np^*(z) (2) = ...