polyiamonds

polyiamonds, a MATLAB code which considers polyiamonds, simple connected shapes constructed from equilateral triangles connected edgewise.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

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

Related Data and Programs:

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.

boundary_word_triangle, a MATLAB code which describes the outline of an object on a grid of triangles, 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.

pariomino, a MATLAB code which considers pariominoes, which are polyominoes with a checkerboard parity, and the determination of tilings of a region using a specific set of pariominoes.

polyiamonds_test

polyominoes, a MATLAB code which defines, solves, and plots a variety of polyomino tiling problems, which are solved by a direct algebraic approach involving the reduced row echelon form (RREF) of a specific matrix, instead of the more typical brute-force or backtracking methods.

Reference:

1. Martin Gardner,
   Mathematical Games: On Polyiamonds: Shapes That are Made Out of Equilateral Triangles, Scientific American,
   Volume 211, December 1964.
2. 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.
3. Solomon Golomb,
   Polyominoes: Puzzles, Patterns, Problems, and Packings,
   Princeton University Press, 1996,
   ISBN: 9780691024448
4. T H O'Beirne,
   Pentominoes and Hexiamonds,
   New Scientist,
   Volume 12, pages 379-380, 1961.
5. Torbijn,
   Polyiamonds,
   Journal of Recreational Mathematics,
   Volume 2, pages 216-227, 1969.

Source code:

• boolean_to_string.m, given a boolean value, returns the character vector 'True' or 'False'.
• gort_boundary.m, returns the boundary word of the Gort polyiamond.
• boundary_hexiamond.m, returns the boundary word for a given hexiamond.
• boundary_is_legal.m, given the boundary word of a polyiamond, determines if it is legal.
• boundary_plot.m, given the boundary word of a polyiamond, creates a plot.
• boundary_print.m, given the boundary word of a polyiamond, prints it.
• boundary_range.m, given the boundary word of a polyiamond, returns its range in the I and J directions.
• boundary_reflect.m, given the boundary word of a polyiamond, returns the boundary word of the reflected polyiamond.
• boundary_representative.m, returns the representative for a boundary word, the lexically first member of the equivalence class.
• boundary_rotate.m, given the boundary word of a polyiamond, rotates it through a multiple of 60 degrees.
• boundary_snap.m, given the boundary word of a polyiamond, "snaps" it so its minimum I and J node coordinates are 0.
• boundary_sort.m, given the boundary word of a polyiamond, returns a sorted version of the boundary word.
• boundary_tiny.m, returns the boundary word of the tiny polyiamond.
• boundary_to_edge.m, given the boundary word of a polyiamond, converts it to a sequence of nodes that bound the polyiamond.
• boundary_to_triangle.m, given the boundary word of a polyiamond, converts it to a triangle list.
• boundary_to_vertex.m, given the boundary word of a polyiamond, converts it to a vertex list.
• boundary_translate.m, given the boundary word of a polyiamond, returns the boundary word of the translated polyiamond.
• chvec_lt.m, is TRUE if c1
• edge_gort.m, returns the edge list of the Gort polyiamond.
• edge_hexiamond.m, returns the edge list of a given hexiamond.
• edge_plot.m, given the edge list of a polyiamond, plots the polyiamond.
• edge_print.m, given the edge list of a polyiamond, prints it.
• edge_range.m, given the edge list of a polyiamond, returns its range in the I and J directions.
• edge_reflect.m, given the edge list of a polyiamond, returns the edge list of a reflected polyiamond.
• edge_snap.m, given the edge list of a polyiamond, "snaps" it so its minimum I and J node coordinates are 0.
• edge_sort.m, given the edge list of a polyiamond, returns a sorted version of the edge list.
• edge_tiny.m, returns the edge list of the tiny polyiamond.
• edge_to_boundary.m, given the edge list of a polyiamond, returns the corresponding boundary word.
• edge_to_triangle.m, given the edge list of a polyiamond, returns the corresponding triangle list.
• edge_to_vertex.m, given the edge list of a polyiamond, returns the corresponding vertex list.
• edge_translate.m, given the edge list of a polyiamond, returns the edge list of a translation of the polyiamond.
• grid_plot.m, plots a parallelogram grid divided into equilateral triangles.
• i4_wrap.m, forces an I4 to lie between given limits, using wrapping.
• i4vec_lt.m, returns TRUE if vector 1 is less than vector 2, using the lexical ordering.
• ij_neighbors.m, given the (i,j) parallelogram coordinates of a node, returns the (i,j) coordinates of the six neighboring nodes.
• ij_reflect.m, given the (i,j) parallelogram coordinates of a node, returns the (i,j) coordinates after reflection.
• ij_rotate.m, given the (i,j) parallelogram coordinates of a node, returns the (i,j) coordinates after R 60 degree rotations.
• ij_to_xy.m, given the (i,j) parallelogram coordinates of a node, returns the (x,y) Cartesian coordinates.
• ij_translate.m, given the (i,j) parallelogram coordinates of a node, returns the (i,j) coordinates after displacement by a vector D.
• ijk_fill.m, given (i,j,k) parallelogram triangle coordinates, plots a filled triangle on the current figure.
• ijk_neighbors.m, given (i,j,k) parallelogram triangle coordinates, returns the IJK coordinates of 3 neighbor triangles.
• ijk_to_ij.m, given (i,j,k) parallelogram triangle coordinates, returns the (i,j) parallelogram coordinates of the nodes that are vertices.
• ijk_to_xy.m, given (i,j,k) parallelogram triangle coordinates, returns the (x,y) Cartesian point coordinates of the nodes that are vertices.
• is_octave.m, is TRUE if Octave is executing.
• name_hexiamond.m, given an index between 1 and 12 of a hexiamond, returns the corresponding name.
• polyiamond_free_enumerate.m, enumerates the free polyiamonds.
• s_escape_tex.m, de-escapes TeX escape sequences.
• triangle_contains_ijk.m, given the triangle list of a polyiamond, determines if an IJK triangle is inside the polyiamond.
• triangle_gort.m, returns the triangle list of the Gort polyiamond.
• triangle_hexiamond.m, returns the triangle list of a given hexiamond.
• triangle_ijk_order.m, given the triangle list of a polyiamond, returns the row in which a given IJK triangle occurs.
• triangle_parity.m, given the triangle list of a polyiamond, computes the parity (k=1 minus k=0).
• triangle_plot.m, given the triangle list of a polyiamond, plots the corresponding polyiamond.
• triangle_print.m, given the triangle list of a polyiamond, prints it.
• triangle_range.m, given the triangle list of a polyiamond, returns its range in the I and J directions.
• triangle_reflect.m, given the triangle list of a polyiamond, returns the triangle list of the reflected polyiamond.
• triangle_snap.m, given the triangle list of a polyiamond, "snaps" it so its minimum I and J node coordinates are 0.
• triangle_sort.m, given the triangle list of a polyiamond, sorts the triangle list.
• triangle_tiny.m, returns the triangle list of the tiny polyiamond.
• triangle_to_boundary.m, given the triangle list of a polyiamond, and returns the boundary word of the polyiamond.
• triangle_to_edge.m, given the triangle list of a polyiamond, returns the boundary node list of the polyiamond.
• triangle_to_vertex.m, given the triangle list of a polyiamond, returns the vertex list of the polyiamond.
• triangle_translate.m, given the triangle list of a polyiamond, returns the triangle list of a translation of the polyiamond.
• vertex_contains_ijk.m, given the vertex list of a polyiamond, determines if an IJK triangle is contained inside the polyiamond.
• vertex_contains_point.m, given the vertex list of a polyiamond, determines if a point is contained inside the polyiamond.
• vertex_gort.m, returns the vertex list of the Gort polyiamond.
• vertex_hexiamond.m, returns the vertex list of a given hexiamond.
• vertex_plot.m, given the vertex list of a polyiamond, plots the polyiamond.
• vertex_print.m, given the vertex list of a polyiamond, prints it.
• vertex_range.m, given the vertex list of a polyiamond, returns its range in the I and J directions.
• vertex_reflect.m, given the vertex list of a polyiamond, returns the vertex list of the reflected polyiamond.
• vertex_snap.m, given the vertex list of a polyiamond, "snaps" it so its minimum I and J node coordinates are 0.
• vertex_sort.m, given the vertex list of a polyiamond, returns a sorted version of the vertex list.
• vertex_tiny.m, returns the vertex list of the tiny polyiamond.
• vertex_to_boundary.m, given the vertex list of a polyiamond, produces the corresponding boundary word.
• vertex_to_edge.m, given the vertex list of a polyiamond, produces the corresponding edge list.
• vertex_to_triangle.m, given the vertex list of a polyiamond, produces the corresponding triangle list.
• vertex_translate.m, given the vertex list of a polyiamond, returns the vertex list of a translation of the polyiamond.
• xy_to_ij.m, given the (x,y) Cartesian coordinates of a point, returns the corresponding (i,j) parallelogram coordinates.