variomino

variomino, a MATLAB code which considers variominoes, which are polyominoes in which each square has been assigned a positive label or "variety", and the determination of tilings of a region using a specific set of variominoes.

Licensing:

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

Languages:

variomino is available in a MATLAB version.

Related Data and Programs:

variomino_test

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.

chrominoes, a MATLAB code which searches for tilings of a polygonal region using polyominoes, in which a coloring scheme is used to reduce the problem size and quickly eliminate certain arrangements.

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_hexity, a MATLAB code which evaluates and manipulates a six-fold parity quantity associated with grids and tiles used in the Eternity puzzle.

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, a MATLAB code which works with polyiamonds, simple shapes constructed by edgewise connections of congruent equilateral triangles.

polyomino_parity, a MATLAB code which uses parity considerations to determine whether a given set of polyominoes can tile a specified region.

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. 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.

Source code:

• cplex_solution_multiple_read.m, extracts solution data from a structure obtained from a cplex() solution file that contained multiple solutions.
• cplex_solution_read.m, reads an XML file, converts the data to a structure, and extracts the data representing solutions.
• cplex_solution_single_read.m, extracts solution data from a structure obtained from a cplex() solution file that contained a single solution.
• variomino_area.m, returns the area of a variomino, simply the number of nonzeros in the representation.
• variomino_condense.m, cleans up a matrix that represents a variomino by removing initial and final rows and columns of zeros.
• variomino_embed_list.m, for each possible embedding, lists the translation necessary to apply to the variomino.
• variomino_embed_number.m, reports the number of ways a variomino can be embedded in a region.
• variomino_equal.m, is true if two variominoes are equal.
• variomino_index.m, computes an index for each nonzero variomino entry.
• variomino_lp_write.m, writes an LP file describing a particular problem.
• variomino_matrix.m, determines the matrix and right hand side for a variomino problem.
• variomino_matrix_reid.m, returns the matrix associated with the Reid tiling problem.
• variomino_print.m, prints a variomino.
• variomino_tiling_print.m, is given matrices defining a region R and a set of variominoes P, and a solution vector X, which represents a tiling of R by the variominoes in P, and prints out a representation of that tiling.
• variomino_transform.m, carries out reflections and rotations of a variomino.
• variomino_variants.m, carries out reflections and rotations of a set of variominoes to determine which transformations yield distinct variants.
• variomino_variety.m, computes the variety of a variomino.
• variominoes_print.m, prints variominoes packed in an array.
• xml2struct.m, reads an XML file and converts the data to a structure.