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The matching rules force a particular substitution: the two A L tiles in a φ A L tile must form a kite, and thus a kite decomposes into two kites and a two half-darts, and a dart decomposes into a kite and two half-darts. [45] [46] Enlarged φ B-tiles decompose into B-tiles in a similar way (via φ A-tiles).
From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal and order-8 triangular tilings.. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.
Subtle alternating colors may be used to create a distinctive floor pattern, or the materials used may be the same, causing the floor to look uniform from a distance. Laying a herringbone floor is very challenging, since the multitude of small rows must be made to line up smoothly, which can be difficult in a room which is not perfectly flat.
The Penrose tiles, and shortly thereafter Amman's several different sets of tiles, [21] were the first example based on explicitly forcing a substitution tiling structure to emerge. Joshua Socolar , [ 22 ] [ 23 ] Roger Penrose , [ 24 ] Ludwig Danzer , [ 25 ] and Chaim Goodman-Strauss [ 20 ] have found several subsequent sets.
Hexagonal tiling is the densest way to arrange circles in two dimensions. The honeycomb conjecture states that hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter.
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