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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.
The six coordinates of each pentagon can be grouped into two triples of coordinates, in which each triple gives the coordinates of a hexagon in an analogous three-dimensional coordinate system for each of the two overlaid hexagon tilings. [10] The number of tiles that are steps away from any given tile, for =,,, …, is given by the ...
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *333333 with 6 order-3 mirror intersections. In Coxeter notation can be represented as [6 * ,6], removing two of three mirrors (passing through the hexagon center) in the [6,6] symmetry.
In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space.It is paracompact because it has cells with an infinite number of faces.
A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic. [3] The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are ...
The rhombille tiling is also used as a design for parquetry [12] and for floor or wall tiling, sometimes with variations in the shapes of its rhombi. [13] It appears in ancient Greek floor mosaics from Delos [ 14 ] and from Italian floor tilings from the 11th century, [ 15 ] although the tiles with this pattern in Siena Cathedral are of a more ...
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