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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.
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).
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.
Patterned tiles, or tiles making up patterns, were now mass-produced by machine and reliably level for floors and cheap to produce, especially for churches, schools and public buildings, but also for domestic hallways and bathrooms. For many uses the tougher encaustic tile was used.
Ceramic tiles are clay products that are formed into thin tiles and fired. Ceramic tiles are set in beds of mortar or mastic with the joints between tiles grouted. Varieties of ceramic tiles include quarry tile, porcelain, and terracotta. Many different natural stones are cut into a variety of sizes, shapes, and thicknesses for use as flooring.
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