----- Original Message ----- From To: <math-fun@mailman.xmission.com> Sent: Sunday, February 01, 2004 2:24 PM Subject: Re: [math-fun] Two questions in geometric combinatorics ------------------------------------------------------------------------- Dylan asked: << On Fri, Jan 30, 2004 at 05:17:01PM -0500, asimovd@aol.com wrote: << Let A_n denote a regular tetrahedral arrangement of points in 3-space, n to an edge. (The layers successively contain 1, 3, 6,... points.)
Can you be more explicit what this arrangement is? I am completely baffled.
The arrangement A_n of points in 3-space is just a regular tetrahedron's worth of the face-centered cubic lattice (from the triangular viewpoint). I.e., starting from the top, the (k)th layer is an k-sided equilateral triangle of k(k+1)/2 points arranged as in the triangular lattice. The layers are spaced so each point's nearest neighbors are (let's say) exactly one unit apart from it. The problem (to repeat) is, For which n can A_n be partitioned into disjoint "tetrads", where a tetrad here means 4 mutually nearest points) ? An obvious necessary condition is that 4 divides #(A_n), which happens exactly when n = 0,2,4,6, or 7 (mod 8). --Dan