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The shortest circular sequence of length sigma^n such that every string of length n on the alphabet a of size sigma occurs as a contiguous subrange of the sequence described ...

There are a number of attractive polyhedron compounds of two cubes. The first (left figures) is obtained by allowing two cubes to share opposite polyhedron vertices then ...

A 3-cusped hypocycloid, also called a tricuspoid. The deltoid was first considered by Euler in 1745 in connection with an optical problem. It was also investigated by Steiner ...

Given a unit disk, find the smallest radius r(n) required for n equal disks to completely cover the unit disk. The first few such values are r(1) = 1 (1) r(2) = 1 (2) r(3) = ...

The Fuhrmann triangle of a reference triangle DeltaABC is the triangle DeltaF_CF_BF_A formed by reflecting the mid-arc points arcM_A, arcM_B, arcM_C about the lines AB, AC, ...

For an integer n>=2, let gpf(x) denote the greatest prime factor of n, i.e., the number p_k in the factorization n=p_1^(a_1)...p_k^(a_k), with p_i<p_j for i<j. For n=2, 3, ...

Consider two mutually tangent (externally) spheres A and B together with a larger sphere C inside which A and B are internally tangent. Then construct a chain of spheres each ...

The Kakeya needle problems asks for the plane figure of least area in which a line segment of width 1 can be freely rotated (where translation of the segment is also ...

The plastic constant P, sometimes also called le nombre radiant, the minimal Pisot number, plastic number, plastic ratio, platin number, Siegel's number, or silver number, is ...

Hardy and Littlewood (1914) proved that the sequence {frac(x^n)}, where frac(x) is the fractional part, is equidistributed for almost all real numbers x>1 (i.e., the ...