Academ Periodic tiling where eighteen triangles encircle each hexagon

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Baelde

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https://czwiki.cz/b/1f4ef

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Wikimedia Commons

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600 x 600 Pixel (1037 Bytes)

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Wheher triangular or hexagonal,  all tiles of this tessellation are convex regular polygons edge‑to‑edge.  Out of the three colours of surfaces,  one is specific to the hexagons,  whereas one or the other of the lightest two colours fills each triangular tile.  Every hexagon adjoins six triangles,  three of one colour,  three of the other.  Every triangle adjoins three tiles:  either three triangles or two triangles and one hexagon.  A common edge of two adjacent tiles always separates two different colours.
 
This tiling is periodic.  All its tiles have the same side length which,  multiplied by 7,  becomes the side length of a rhombic repetitive pattern.  Its sides are parallel to two sides of a triangular tile.  Its area is the one of  7 × 7 × 2 = 98  triangles.  Such a pattern can contains  5  entire hexagons,  four halves of hexagons,  plus  7 × 8 = 56  equilateral triangles.  So its area can be seen as the area of the following number of triangles:  (5 + ⁴/₂ ) 6 + 7 × 8  =  7 (6 + 8).
 
Such a pattern,  formed by two equilateral triangles edge‑to‑edge,  can be transformed into a regular hexagonal pattern of same area,  of which two sides are parallel to the large diagonal of the repetitive rhombus.  So this image is classified within  Category:Honeycombs (geometry).

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