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Three-dimensional generalization of the polyominoes to n dimensions. The number of polycubes N(n) composed of n cubes are 1, 1, 2, 8, 29, 166, 1023, ... (OEIS A000162; Ball ...

Let two spheres of radii R and r be located along the x-axis centered at (0,0,0) and (d,0,0), respectively. Not surprisingly, the analysis is very similar to the case of the ...

Cantellation, also known as (polyhedron) expansion (Stott 1910, not to be confused with general geometric expansion) is the process of radially displacing the edges or faces ...

A cube can be divided into n subcubes for only n=1, 8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46, and n>=48 (OEIS A014544; Hadwiger 1946; Scott 1947; Gardner 1992, ...

Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A ...

What space-filling arrangement of similar cells of equal volume has minimal surface area? This questions arises naturally in the theory of foams when the liquid content is ...

A formula relating the number of polyhedron vertices V, faces F, and polyhedron edges E of a simply connected (i.e., genus 0) polyhedron (or polygon). It was discovered ...

Let two points x and y be picked randomly from a unit n-dimensional hypercube. The expected distance between the points Delta(n), i.e., the mean line segment length, is then ...

The first few terms in the continued fraction of the Champernowne constant are [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 45754...10987, 6, 1, 1, 21, ...] ...

A generalization of calculus of variations which draws the relationship between the stationary points of a smooth real-valued function on a manifold and the global topology ...