BeschreibungSquare-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:
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.
Mathematica code that generated this image:
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"]]
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