Neal, here's a proposed text. Rewrite welcome. I'm torn between precision and length. I added a new term, which differs from A001004, so I removed the cross reference. The counts really need independent confirmation: It took me fifteen minutes to locate the 24th heptiamond, and that's with knowing what the count should be. Dan, I think heteromino is a superior word to polyanygon, but the poly-xxx usage is already established, and I think descriptiveness trumps sound quality. I've thought a little about tiling with polyanygons. There are obvious tilings with impure combinations like a row of squares terminated by triangles: 344443 A random polyanygon will usually have impossible angles, with no combinations even tiling around a vertex. Two interesting cases turned up: triamond on domino, __ / \ |__| and the brother with the same polygons arranged around the central vertex in the order 33434. Do these tile the plane? Are there any tilings with pieces containing pentagons or heptagons? Another theoretical complication is tilings with non- registered vertices, where the vertices from one anygon don't coincide with the vertices of the adjacent anygon. This is even possible when tiling with squares or some polyominos, but in these cases there's always a vertex-aligned tiling too. Can there be polyanygon tilings where there's no vertex- aligned tiling? Rich 1,2,3,9,20,75,255 Number of polyanygons with score N, with all symmetries removed. Polyanygons are like polyominos, but they are built by gluing together arbitrary regular polygons of side = 1. The 3 polyanygons of score 3 are the triamond (three equilateral triangles), a square stuck to a triangle (a house), and the regular pentagon. The score for a polyanygon is the total of its components' scores. Triangle = 1, square = 2, N-gon = N-2. [Fine print: Holes are allowed. Non-zero area overlap of polygons is forbidden. Glued edges must line up vertex to vertex. Non-glued edges may occasionally have a slide offset, and vertices may touch edges internally or contact other vertices. Higher scoring polyanygons can have partially offset edges, so the perimeter is not always an integer. A regular hexagon is regarded as different from the same shape made out of six triangles, and a dodecagon is different from a hexagon plus an alternating triangle/square fringe. Two polyanygons made of the same regular polygons in the same relative positions and orientations, but differing in order of assembly or which edges are glued, are considered the same.] Richard Schroeppel (schroeppel(AT)alum.mit.edu), Oct 23 2005 ________________________________ From: N. J. A. Sloane [mailto:njas@research.att.com] Sent: Sun 10/23/2005 1:23 PM To: Schroeppel, Richard; math-fun@mailman.xmission.com Subject: Re polyanygons I'm adding this to the OEIS, OK? %I A111189 %S A111189 1,2,3,9,20,75 %N A111189 Number of polyanygons with score n. %C A111189 These are similar to polyominos, except you can use any regular polygons of side = 1, as long as there's no overlapping. %C A111189 A regular polygon of N sides has a score of N-2: triangles score 1, squares 2, etc. The score for a polyanygon is the total of its components. %C A111189 These are with all symmetries (rotations, reflections) removed. %C A111189 Note that higher scoring polyanygons can have partially offset edges, so the perimeter might not always be an integer. %C A111189 The regular hexagon is regarded as different from the same shape made out of six triangles. It's also possible to make a dodecagon by adding a fringe or alternating triangles and squares to a regular hexagon. I think these are the only ambiguous cases. %e A111189 The 3 polyanygons of score 3 are the triamond (three regular triangles), a square stuck to a triangle (a house), and the pentagon. %O A111189 1,2 %K A111189 nonn,more %Y A111189 Initially this agrees with A001004 but will eventually diverge from it. %A A111189 Richard Schroeppel (rschroe(AT)sandia.gov), Oct 23 2005 NJAS