boundary_word_drafter

boundary_word_drafter, an Octave 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, an Octave 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, an Octave 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, an Octave 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, an Octave 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, an Octave 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, an Octave 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, an Octave 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, an Octave 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, an Octave 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, an Octave 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, an Octave 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.