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  "body": "\n## Example\n\nA *Penrose tiling* is a [[tiling|GKJK293DXS]] of the plane by copies of two prototiles that obey matching rules.\nRoger Penrose introduced these tilings in the 1970s.\nIn the rhombus version, the prototiles are two rhombi with sides of length $1$.\nThe thick rhombus has angles $72^\\circ$ and $108^\\circ$, and the thin rhombus has angles $36^\\circ$ and $144^\\circ$.\nThe matching rules mark the edges of the rhombi and allow two tiles to meet along an edge only when the marks agree.\nIn the kite and dart version, the two prototiles are a convex kite and a non-convex dart, again with matching rules.\n\nThe two prototiles with their matching rules admit tilings of the whole plane, and none of these tilings is [[periodic|WFRGSBFZGV]].\nSo each version is an [[aperiodic set of prototiles|Z3FCJZF6WK]].\nIn every Penrose tiling the ratio of the number of thick to thin rhombi, counted in large disks, tends to the [[golden ratio|0SC8T7548D]] $\\varphi = (1 + \\sqrt{5})/2$.\nSince $\\varphi$ is irrational, this ratio cannot come from a periodic tiling, whose ratio would be the rational ratio within one period.\n\n## Remarks\n\nThe matching rules are needed.\nWithout them, a thick rhombus is already a parallelogram that tiles the plane periodically, and a kite and a dart fit together into a rhombus that tiles periodically.\nThere are uncountably many different Penrose tilings, and every finite patch of one of them appears in every other one.\nSome Penrose tilings have a global fivefold rotational symmetry about one point, which no periodic tiling of the plane can have.\n\n## Explanations\n\n### L1 human\n\nA Penrose tiling covers the plane with just two shapes, yet the pattern never repeats.\nYou can slide it in any direction, and it never lands exactly on itself.\nStill, it is highly ordered: patches with fivefold symmetry, like small stars, appear again and again.\nThe two kinds of tiles occur in the proportion of the golden ratio, about $1.618$ to $1$.\n",
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