File:Square-InfinitelyQuartered.svg

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.

code

max = 8;
scale = 2^max;

min[n_Integer] := scale/2^n

square[x_String, y_String, w_String, fill_String] := StringJoin[
	"<rect x=\"",
	x,
	"\" y=\"",
	y,
	"\" width=\"",
	w,
	"\" height=\"",
	w,
	"\" fill=\"",
	fill,
	"\" stroke=\"#000\" />"
]

whitesquare[n_] := With[
	{w = min[n]},
	square[ToString[scale-w], "0", ToString[w], "#fff"]
]

yellowsquare[n_] := With[
	{w = min[n]},
	square[ToString[scale - w], ToString[w/2], ToString[w/2], "#ff6"]
]

diagramCode=StringJoin[
	"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"256\" height=\"256\" viewBox=\"0 0 256 256\" version=\"1.1\">\n\t",
	Riffle[Join[whitesquare /@ Range[0, max], yellowsquare /@ Range[0, max - 1]], "\n\t"],
	"\n</svg>"
];

Export[FileNameJoin[{Directory[], "Downloads", "Square-InfinitelyQuartered.svg"}], ExportForm[diagramCode,"HTML"]]