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As defined by Gray (1997, p. 201), Viviani's curve, sometimes also called Viviani's window, is the space curve giving the intersection of the cylinder of radius a and center ...

For a power function f(x)=x^k with k>=0 on the interval [0,2L] and periodic with period 2L, the coefficients of the Fourier series are given by a_0 = (2^(k+1)L^k)/(k+1) (1) ...

For even h, (1) (Nagell 1951, p. 176). Writing out symbolically, sum_(n=0)^h((-1)^nproduct_(k=0)^(n-1)(1-x^(h-k)))/(product_(k=1)^(n)(1-x^k))=product_(k=0)^(h/2-1)1-x^(2k+1), ...

A d-hyperoctant is one of the 2^d regions of space defined by the 2^d possible combinations of signs (+/-,+/-,...,+/-). The 2-hyperoctant is known as a quadrant and the ...

Given a metric g_(alphabeta), the discriminant is defined by g = det(g_(alphabeta)) (1) = |g_(11) g_(12); g_(21) g_(22)| (2) = g_(11)g_(22)-(g_(12))^2. (3) Let g be the ...

A sequence of polynomials p_i(x), for i=0, 1, 2, ..., where p_i(x) is exactly of degree i for all i.

A number of attractive 60-compounds of the regular tetrahedron can be constructed. The compound illustrated above will be implemented in a future version of the Wolfram ...

A polygon which has both a circumcircle (which touches each vertex) and an incircle (which is tangent to each side). All triangles are bicentric with R^2-x^2=2Rr, (1) where R ...

If the square is instead erected internally, their centers form a triangle DeltaI_AI_BI_C that has (exact) trilinear vertex matrix given by (1) (E. Weisstein, Apr. 25, 2004). ...

The chromatic number of a graph G is the smallest number of colors needed to color the vertices of G so that no two adjacent vertices share the same color (Skiena 1990, p. ...