Inter-Kissing Number for Non-Spheres - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:14:43Z http://mathoverflow.net/feeds/question/106163 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106163/inter-kissing-number-for-non-spheres Inter-Kissing Number for Non-Spheres bobuhito 2012-09-02T07:33:50Z 2012-09-02T09:08:34Z <p>In 3D, the maximum number of spheres which can inter-touch is 5 (mathoverflow.net/questions/106120). This maximum reduces to 4 for unit spheres.</p> <p>Is there a different shape (e.g., an egg, or a pyramid) for which these maximums are not 5 and 4? If so, what shape has the highest maximum? To avoid "corner touching" (e.g., 8 cubes could all touch at one corner), please additionally require that every "touch-point" have only 1 "official connection" (e.g., only 2 of the 8 cubes can be declared as touching at the corner).</p> http://mathoverflow.net/questions/106163/inter-kissing-number-for-non-spheres/106168#106168 Answer by Douglas Zare for Inter-Kissing Number for Non-Spheres Douglas Zare 2012-09-02T08:53:41Z 2012-09-02T09:08:34Z <p>There is no upper bound. There can be arbitrarily many congruent convex solids which pairwise touch face-to-face. See Erickson, J. Kim, S. <a href="http://compgeom.cs.uiuc.edu/~jeffe/pubs/crum.html" rel="nofollow">"Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes,"</a> for many references.</p> <p><img src="http://compgeom.cs.uiuc.edu/~jeffe/pubs/pix/crum3.gif" alt="Voronoi cells from helix"></p>