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In geometry, the Waterman polyhedra are a family of polyhedra invented around 1990 by the mathematician Steve Waterman. A Waterman polyhedron is created by packing spheres according to the cubic close(st) packing (CCP), then sweeping away the spheres that are farther from the center than a defined radius, then creating the convex hull of the resulting pack of spheres. Image:Waterman_Packed_Spheres_0024.1.png|Cubic Close(st) Packed spheres with radius Image:Waterman_0024.1.png|Corresponding Waterman polyhedron W24 Origin 1 Waterman polyhedra form a vast family of polyhedra. Some of them have a number of nice properties such as multiple symmetries, or interesting and regular shapes. Others are just a collection of faces formed from irregular convex polygons. The most popular Waterman polyhedra are those with centers at the point (0,0,0) and built out of hundreds of polygons. Such polyhedra resemble spheres. In fact, the more faces a Waterman polyhedron has, the more it resembles its circumscribed sphere in volume and total area. With each point of 3D space we can associate a family of Waterman polyhedra with different values of radii of the circumscribed spheres. Therefore, from a mathematical point of view we can consider Waterman polyhedra as a 4D space W(x, y, z, r), where x, y, z are coordinates of a point in 3D, and r is a positive number greater than 1.〔(Visualizing Waterman Polyhedra with MuPAD ) by M. Majewski〕 ==Seven origins of cubic close(st) packing (CCP)== There can be seven origins defined in CCP,〔(7 Origins of CCP Waterman polyhedra ) by Mark Newbold〕 where n = : * Origin 1: offset 0,0,0, radius sqrt(2n) * Origin 2: offset 1/2,1/2,0, radius sqrt(2+4n)/2 * Origin 3: offset 1/3,1/3,2/3, radius sqrt(6(n+1))/3 * Origin 3 *: offset 1/3,1/3,1/3, radius sqrt(3+6n)/3 * Origin 4: offset 1/2,1/2,1/2, radius sqrt(3+8(n-1))/2 * Origin 5: offset 0,0,1/2, radius sqrt(1+4n)/2 * Origin 6: offset 1,0,0, radius sqrt(1+2(n-1)) Depending on the origin of the sweeping, a different shape and resulting polyhedron are obtained. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Waterman polyhedron」の詳細全文を読む スポンサード リンク
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