Tiling Project Dec 17, 2017

This project demonstrates some of the capabilities of Walter Trump’s 2017 tiling program. The above shape containing 36 triangles came from Walter’s symmetrical sphinx tiling of the T12 triangle.  This got posted to the OEIS and introduced the idea of hyper-packing as noted below …

 Maximum number of 6 sphinx tile shapes in a sphinx tiled hexagon of order n. +20 3
 30, 132, 306, 552, 870, 1260, 1722, 2256, 2862, 3540, 4290, 5112, 6006, 6972, 8010, 9120, 10302, 11556, 12882, 14280, 15750, 17292, 18906, 20592, 22350, 24180, 26082, 28056, 30102, 32220, 34410, 36672, 39006, 41412, 43890, 46440, 49062, 51756, 54522, 57360, 60270, 63252 (list; graph; refs; listen; ; internal format)
 OFFSET 1,1 COMMENTS The equilateral triangle composed of 144 smaller equilateral triangles is the smallest triangle that can be tiled with the sphinx.  This triangle is used to form all orders of the hexagon. Walter Trump enumerated all 830 sphinx tilings of this triangle and found six symmetrical examples one of which is used to produce this sequence. Hyper-packing is a term that describes the ability of a shape to contain a greater area of subshapes than its own area by overlapping the subshapes. There are 864 unit triangles in the order 1 hexagon. 30 of the subshapes hyper-packed into this hexagon would contain 30x6x6 or 1080 unit triangles if summed individually. LINKS Craig Knecht, Example for the sequence. Craig Knecht, Order 4 Hexagon with 246 subshapes. Wikipedia, Hyper-Packing the Sphinx Wikipedia, Symmetric sphinx tiled triangles Wikipedia, Walter Trump Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = 6*n*(6*n-1). - Walter Trump Cf.: 2*x*(15+21*x)/(1-x)^3. - Vincenzo Librandi, Sep 20 2017 MAPLE seq(6*n*(6*n-1), n=1..100); # Robert Israel, Sep 19 2017 MATHEMATICA Array[6 # (6 # - 1) &, 42] (* Michael De Vlieger, Sep 19 2017 *) CoefficientList[Series[2(15 + 21 x)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 20 2017 *) PROG (MAGMA) [6*n*(6*n-1): n in [1..50]]; // Vincenzo Librandi, Sep 20 2017 CROSSREFS Cf. A279887, A287999. KEYWORD nonn,easy AUTHOR Craig Knecht, Aug 30 2017 STATUS approved

Walter added the capability to his tiling program of being able to specify the number of shapes in the shape table that could be used in tiling an object.  He supplied 5 hexagons that contain the sum of all triangles in the shape table at 110 triangles.  It is easy to tile these 110 area hexagons with one each of each shape in the shape table. I tried to refine these “noble” hexagons by embedding two of these shapes in the hexagon that were made from all 12 examples of the shapes that use 6 triangles.  So now there is a separation of the smaller shapes in the table from the larger shapes in the table (placed in the macro shape) within the noble hexagon….    The above tiling exercise demonstrates some of the capabilities of Walter Trump’s 2017 tiling program.   