English: A unit square that is infinitely quartered with one corner highlighted from each quartering is a geometric representation of the infinite series:

${\displaystyle {\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,{\frac {1}{64}}\,+\,{\frac {1}{256}}\,+\,\cdots \;=\;{\frac {1}{3}}.}$

The area summation of all highlighted regions adds up to exactly 1/3. While not necessarily an intuitive result, this can be easily verified as the 'L' shape made by the largest three squares (the three untouched after the first quartering) has exactly 1/3 of its area highlighted, and this shape and proportion repeats infinitely unchanged throughout subsequent quarterings.

This square serves as a verification that the numbers 0.999... and 1.000... are exactly equal. While the one-third of the square that is highlighted has the decimal notation of 0.333..., filling in the other two-thirds of the 'L' shape results in 3 x 0.333... = 0.999... which results in the entire unit square being filled in, so 0.999... = 1 exactly.

An alternative way to view this image is in Base 4, in which the comparable equality becomes 0.333...₄ = 1. (Note that in Base 4, the simple summation: 3 + 1 = 10₄) The infinite series representing the area of the highlighted squares becomes:

0.1₄ + 0.01₄ + 0.001₄ + 0.0001₄ + ...

And the summation of this series equals 1/3:

0.1111...₄ = 1/3

Filling in the other two-thirds of the 'L' shape gives:

3 x 0.1111...₄ = 0.3333...₄

This results in the entire unit square being filled in, so 0.3333...₄ = 1 exactly.

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See also GeometricSquares.png, similar image uploaded in 2008.