The n-bone conjecture: Torsten Sillke, Nov. 1992 ====================== The n-bone are n spheres in a row glued together. So the 5-bone looks like: ooooo John H. Conway solved the 3-bone conjecture: It is impossible to tile with 3-bones a triangular arangement of spheres. See: W. P. Thurston, Conway's Tiling Groups, AMM 97 (1990) 757-773 You see easily that with 2-bones there is a tiling of the triangle or tetrahedron iff the number of spheres is odd. On the right is a 2 example tiling of 2 1 a triangle with 2-bones 0 0 1 My extended conjecture is: (the n-bone conjecture) There is no tiling of the d-simplex with n-bones, where d>=2 and n>=3. The simplex had to be finite of course. Known Cases: For d=3 layer coloring with n colors exclude all Tetrahedra(k) except: k = -2, -1, 0 (modulo n*n) The smallest open cases are therefore: 7, 8, 9, 16, 17, 18, 25, 26, 27, ... for n=3 14, 15, 16, ... for n=4 23, 24, 25, ... for n=5. The 3-bone case can be strengthend see below. All open cases are too big as one can exclude by computer. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 3-Bones: Torsten Sillke, 1994-08-21 (Coloring found) ======== A short note for impossible tilings of the tetrahedron with 3-bones. Coloration of the Tetrahedron with colors {a,b,c}. Example n=7 a b b b b a b a a b a b b a b a b b b a b b a b b b c c b a c a b b b b a b b a b a a c a a b c c b b a b a a b a b b a b b a b a b b a b b b a b b a b b a b a b b b a This coloration has the property, that a 3-bone has 3 possebilities (a,b,c) is (1,2,0), (0,1,2), or (2,0,1). Let x, y, z the number of 3-bones of the 3 types. If there were a tiling of the tetrahedron, we had an integral non-negative solution of the linear system: [ 1 0 2 ] [x] [30] [ 2 1 0 ] * [y] = [48] [ 0 2 1 ] [z] [ 6] [ 1 0 2 ]-1 [ 1 4 -2 ] [ 2 1 0 ] = 1/9[-2 1 4 ] [ 0 2 1 ] [ 4 -2 1 ] As the unique solution is not integral, there is no tiling. This coloration excludes all tetrahedra with n = 7, 8, 9, 16, 17, 18 (mod 27). The remaining ones are n = 25, 26, 27 (mod 27). In these cases the solutions are integral. Are there tilings for n=25, 26, 27 ? ------------------------------------ Tiling cubes with 3-bones: Torsten Sillke, 1996-05-11 -------------------------- Cubes are not excluded by the coloration given above. The 7-cube_- (spheres = (7*7*7 - 1)/2 = 171) is tilable. Coloration of the 7-cube_+ (spheres = (7*7*7 + 1)/2 = 172): a c c a b c b c b b c c a c b c b b a a b a b a c b b c c b b c a b a b a a b b c b c a c c b b c b c b a c c a c b b c a b a b a a b b c b b c b b a a b a b a c b b c a c c a b c b c b b c c a c layers 1,7 layers 2,6 layers 3,5 layer 4 The 7-cube_- is the first figure I found that was tilable by 3-bones. Tiling pyramids with 3-bones: ----------------------------- No pyramid of size 8 is possible. [ 1 0 2 ] [x] [ 66] [ 2 1 0 ] * [y] = [108] => (x,y,z) = (146,32,26)/3 [ 0 2 1 ] [z] [ 30] . . . . . . . a . . . . . . . b . . . . . . b . b . . . . . . . b . . . . . . . b . . . . . . a . a . . . . . b . c . b . . . . . . a . a . . . . . . . b . . . . . . . a . . . . . . b . b . . . . . b . c . b . . . . a . c . c . a . . . . . b . c . b . . . . . . b . b . . . . . . . a . . . . . . . b . . . . . . b . b . . . . . a . b . a . . . . b . a . a . b . . . b . b . c . b . b . . . . b . a . a . b . . . . . a . b . a . . . . . . b . b . . . . . . . b . . . . . . . b . . . . . . a . a . . . . . b . c . b . . . . b . a . a . b . . . a . a . b . a . a . . b . c . b . b . c . b . . . a . a . b . a . a . . . . b . a . a . b . . . . . b . c . b . . . . . . a . a . . . . . . . b . . . . . . . a . . . . . . b . b . . . . . b . c . b . . . . a . c . c . a . . . b . b . c . b . b . . b . c . b . b . c . b . a . c . c . a . c . c . a . . b . c . b . b . c . b . . . b . b . c . b . b . . . . a . c . c . a . . . . . b . c . b . . . . . . b . b . . . . . . . a . . . . . . . b . . . . . . b . b . . . . . a . b . a . . . . b . a . a . b . . . b . b . c . b . b . . a . b . a . a . b . a . b . a . a . b . a . a . b b . b . c . b . b . c . b . b . b . a . a . b . a . a . b . . a . b . a . a . b . a . . . b . b . c . b . b . . . . b . a . a . b . . . . . a . b . a . . . . . . b . b . . . . . . . b -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/