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Tiling In everyday life, geometric patterns are ubiquitous: for example, in parquet flooring, bathroom tiles, asphalt slabs in pedestrian zones, graph paper in school, and so on. The tiles or slabs are usually made up of simple regular geometric shapes, such as rectangles or squares. By arranging them edge to edge in all directions, they can cover an area of any size without gaps. In mathematics, tiling (also known as tessellation, paving, or covering) refers to the seamless and non-overlapping covering of a plane by uniform tiles. The concept can also be extended to higher dimensions. It involves the optimal utilization of a given area or space.
If the shapes are of the same form, same size, and also regular/equilateral polygons, there are only three possible shapes for a seamless tiling: squares, triangles, or hexagons. Beyond seven sides, a non-overlapping tiling is no longer possible, and with equilateral pentagons, there remains a triangular gap between the tiles. The shapes that allow for an infinite pattern in the plane are distinguished by the number of their symmetry properties, i.e., how many times they can be rotated or shifted without changing the appearance of the pattern. In total, there are 17 symmetry groups, each with different types of infinite pattern repetition. These symmetry groups are of great importance in the study of crystal structures.
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Tessellation Pattern (Colored sand on lightweight board, 70 x 70 cm). The shapes were constructed based on 8x8 squares and colored accordingly. |
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Triangle Tile Composition (Wooden sticks on Kappa foam board, 70x70 cm). |
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Truchet Tiles The first person to systematically delve into the seemingly trivial topic of dividing a surface with uniform patterns was Jean Truchet (1657-1729), a French Dominican born in Lyon who lived under Louis XIV. He worked in fields such as mathematics, hydraulics, graphics, and typography. Inspired by decorations he saw on tiles, Truchet studied decorative patterns on ceramic tiles. One particular pattern he examined involved square tiles divided by a diagonal line into two triangles and decorated in contrasting colors. By placing these tiles in different orientations relative to each other as part of a square tiling, Truchet found that many different patterns could be formed. This model of pattern formation is now known to mathematicians and designers as Truchet tiles. Truchet's pattern investigations were later taken up by the French typographer Pierre Simon Fournier (1712-1768), who introduced the point as a unit for font size and developed the typographic measurement system. As a wood engraver, Fournier had an interest in ornaments for the design of printed works. Inspired by Truchet, Fournier divided each element of a large ornament into small elements called "combinatorial" ornaments or vignettes. His work "Manuel Typographique," published in 2 volumes in 1764/68, was of great significance to the art of bookmaking and the development of various typefaces.
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