Tiling a Room Irrationally

Numbers can be well approximated by numbers of the form , where and are integers and is irrational. This property can be used to tile a room almost perfectly, with just three tile sizes, and with no need to cut any tiles.

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E. Pegg Jr and A. Fisher, "Tessellation Set", US Patent 6309716.
The numbers are Euler's number , the golden ratio , Pythagoras's number , Khinchin's constant , Euler gamma , Glaisher's constant , Catalan's number , , and .
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