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  "body": "\n## Example\n\nConsider [[tilings|GKJK293DXS]] of the plane whose tiles are convex [[regular polygons|01KJJ9A400]].\nThe tiling is edge-to-edge if two tiles that meet share either one full edge or one vertex.\nThe interior angle of a regular $n$-gon is $(n-2)\\pi/n$.\nA tiling by copies of a single regular $n$-gon is called *regular*.\nAt each vertex of such a tiling some number $k \\ge 3$ of tiles meet, so $k(n-2)\\pi/n = 2\\pi$, which gives $(n-2)(k-2) = 4$.\nThe only solutions are $(n, k) = (3, 6), (4, 4), (6, 3)$.\nSo there are exactly three regular tilings.\nThey use equilateral triangles, six at each vertex; squares, four at each vertex; and regular hexagons, three at each vertex.\nIn particular, regular pentagons do not tile the plane edge-to-edge, since $3 \\cdot 3\\pi/5 \u003c 2\\pi \u003c 4 \\cdot 3\\pi/5$.\n\n## Remarks\n\nAn edge-to-edge tiling by regular polygons is *semiregular*, or Archimedean, if it uses more than one kind of polygon and its symmetries map any vertex to any other vertex.\nThere are exactly eight semiregular tilings.\nOne of them has at each vertex a triangle, a square, a hexagon and a square, in this cyclic order; the angles add up to $\\pi/3 + \\pi/2 + 2\\pi/3 + \\pi/2 = 2\\pi$.\nThe three regular and the eight semiregular tilings are all [[periodic|WFRGSBFZGV]].\n\n## Explanations\n\n### L1 human\n\nWhich regular shapes can cover a floor with no gaps, all meeting edge to edge?\nThe corners meeting at a point must add up to a full turn of $360^\\circ$.\nTriangles have $60^\\circ$ corners, squares $90^\\circ$ and hexagons $120^\\circ$, and these divide $360^\\circ$ evenly.\nNo other regular polygon works on its own, which is why floors use these three shapes so often.\nMixing several regular shapes gives eight more patterns in which every corner looks the same.\n",
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