boundary_word_drafter

boundary_word_drafter, a MATLAB code which describes the outline of an object using a string of symbols that represent the sequence of steps tracing out the boundary, on a grid of drafters, or 30-60-90 triangles. The objects include tiles for the eternity, serenity and trinity puzzles, and the hat and turtle aperiodic monotiles.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

boundary_word_drafter is available in a MATLAB version and an Octave version.

Related Data and Programs:

boundary_word_drafter_test

boat, a MATLAB code which considers the whale tiling puzzle, a smaller version of the eternity puzzle. The whale puzzle specifies a region R composed of 756 30-60-90 triangles, and a set of 21 "tiles", each consisting of 36 30-60-90 triangles, and seeks an arrangement of the tiles that exactly covers the region.

boundary_word_equilateral, a MATLAB code which describes the outline of an object on a grid of equilateral triangles, using a string of symbols that represent the sequence of steps tracing out the boundary.

boundary_word_hexagon, a MATLAB code which describes the outline of an object on a grid of hexagons, using a string of symbols that represent the sequence of steps tracing out the boundary.

boundary_word_right, a MATLAB code which describes the outline of an object on a grid of isoceles right triangles, using a string of symbols that represent the sequence of steps tracing out the boundary.

boundary_word_square, a MATLAB code which describes the outline of an object on a grid of squares, using a string of symbols that represent the sequence of steps tracing out the boundary.

eternity, a MATLAB code which considers the eternity puzzle, which considers an irregular dodecagon shape that is to be tiled by 209 distinct pieces, each formed by 36 contiguous 30-60-90 triangles, known as polydrafters.

eternity_tile, a MATLAB code which considers the individual tiles of the eternity puzzle, 209 distinct pieces, each formed by 36 contiguous 30-60-90 triangles, known as polydrafters.

pram, a MATLAB code which considers the pram puzzle, a smaller version of the eternity puzzle. The pram puzzle specifies a region R composed of 2304 30-60-90 triangles, and a set of 64 "tiles", consisting of 36 30-60-90 triangles, and seeks an arrangement of the tiles that exactly covers the region.

serenity, a MATLAB code which considers the serenity puzzle, a smaller version of the eternity puzzle. The serenity puzzle specifies a dodecagonal region R composed of 288 30-60-90 triangles, and a set of 8 "tiles", each consisting of 36 30-60-90 triangles, and seeks an arrangement of the tiles that exactly covers the region.

trinity, a MATLAB code which considers the trinity puzzle, a smaller version of the eternity puzzle. The trinity puzzle specifies a region R composed of 144 30-60-90 triangles, and a set of 4 "tiles", T1, T2, T3 and T4, each consisting of 36 30-60-90 triangles, and seeks an arrangement of the four tiles that exactly covers the region.

whale, a MATLAB code which considers the whale tiling puzzle, a smaller version of the eternity puzzle. The whale puzzle specifies a region R composed of 288 30-60-90 triangles, and a set of 8 "tiles", each consisting of 36 30-60-90 triangles, and seeks an arrangement of the tiles that exactly covers the region.

Reference:

1. Marcus Garvie, John Burkardt,
   A new mathematical model for tiling finite regions of the plane with polyominoes,
   Contributions to Discrete Mathematics,
   Volume 15, Number 2, July 2020.
2. Solomon Golomb,
   Polyominoes: Puzzles, Patterns, Problems, and Packings,
   Princeton University Press, 1996,
   ISBN: 9780691024448
3. Ed Pegg,
   Polyform Patterns,
   in Tribute to a Mathemagician,
   Barry Cipra, Erik Demaine, Martin Demaine, editors,
   pages 119-125, A K Peters, 2005.
4. Mark Wainwright,
   Prize specimens,
   Plus magazine,
   01 January 2001,
   https://plus.maths.org/content/prize-specimens

Source code:

• ch_wrap.m, forces a character to lie between given limits by wrapping.
• edge_plot.m, plots the edge nodes of an object.
• eternity_grid_word.m, returns the boundary word for the Eternity grid.
• eternity_tile_word.m, the boundary word for any tile, given its index number.
• hat_monotile_word.m, the boundary word for the hat monotile, which, along with its reflection, provides an aperiodic tiling of the plane.
• hexagon_word.m, boundary word for the hexagon grid.
• hexagon3_word.m, returns the boundary word for the hexagon3 grid.
• hexagon4_vertex_xy.m, returns (x,y) coordinates of the vertices for the hexagon4 grid.
• i4_wrap.m, forces an integer to lie between given limits by wrapping.
• is_octave.m, is TRUE if the function is called by Octave.
• polygon_contains_point.m, is TRUE if a polygon contains a point.
• rectangle_triangle_k.m, returns the indexes (k) of the three nodes forming each triangle for the NX by NY rectangle grid.
• rectangle_word.m, returns the boundary word for the NX by NY rectangle grid.
• s_substitute.m, substitutes characters in a string.
• triangle_k.m, returns the node indices of the vertices of the triangles contained in an object.
• turtle_monotile_word.m, the boundary word for the turtle monotile, which, without using reflection, provides an aperiodic tiling of the plane.
• vertex_plot.m, plots the vertices that define the polygon bounding an object.
• word_parity.m, determines the parity of a shape.
• word_print.m, prints a boundary word of a shape.
• word_range_ij.m, returns the horizontal and vertical limits of an object, given its boundary word, using the (i,j) coordinate system.
• word_range_xy.m, returns the horizontal and vertical limits of an object, given its boundary word, using the (x,y) coordinate system.
• word_reflect_ij.m, returns the boundary word for an object reflected across the 0, 30, 60, 90, 120, or 150 degree line through the base point, using the (i,j) coordinate system.
• word_reflect_xy.m, returns the boundary word for an object reflected across the 0, 30, 60, 90, 120, or 150 degree line through the base point, using the (x,y) coordinate system.
• word_representative.m, returns the representative for a boundary word, the lexically first member of the equivalence class.
• word_reverse.m, returns the boundary word for an object when it is traversed in the reverse direction.
• word_rotate_xy.m, returns the boundary word for a rotated object, using the (i,j) coordinate system.
• word_rotate_xy.m, returns the boundary word for a rotated object, using the (x,y) coordinate system.
• word_to_edge_ij.m, determines the (i,j) coordinates of edge nodes from a boundary word.
• word_to_edge_xy.m, determines the (x,y) coordinates of edge nodes from a boundary word.
• word_to_vertex_ij.m, determines the (i,j) coordinates of vertices from a boundary word.
• word_to_vertex_xy.m, determines the (x,y) coordinates of vertices from a boundary word.
• word_translate.m, returns the boundary word for a translated object.