File:GeometricSquares.svg

An illustration of geometric series. Each of the purple squares is 1/4 of the area of the previous square (1/2×1/2=1/4, 1/4×1/4=1/16), with the total area being

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

The area summation of all highlighted regions adds up to exactly 1/3. While not necessarily an intuitive result, this can be easily verified by examining the 'L' shape made by the largest three squares (the three untouched after the first quartering). Exactly 1/3 of the area of these three squares is highlighted. This shape and proportion repeats infinitely unchanged throughout subsequent quarterings. So by inspection of the symmetry, the infinite series maintains the exact total of 1/3.

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.