Dimension by Box-Counting

Background - Dimensions of Euclidean Objects

Starting with a line segment of length one, we can scale the segment by 1/2 and observe 2 copies of length 1/2 which cover the original line segment.

We need three segments scaled by 1/3, four segments scaled by 1/4, etc.

Note that 2 = 1/(1/2), 3 = 1/(1/3), and 4 = 1/(1/4).

In general, representing the scale by r and the number of segments of scale r by N(r) we have the relationship

N(r) = 1/r.

Scaled by 1/2	Scaled by 1/3	Scaled by 1/4

To cover a square of side length 1 we need 4 copies of the square scaled by 1/2 along all its sides.

Similarly we need 9 squares scaled by 1/3, 16 squares scaled by 1/4, etc.

Note that 4 = (1/(1/2))², 9 = (1/(1/3))², and 16 = (1/(1/4))².

In general,

N(r) = (1/r)².

Scaled by 1/2	Scaled by 1/3	Scaled by 1/4

To cover a cube of side length 1 we need 8 copies of the cube scaled by 1/2 along all its sides.

Similarly we need 27 cubes scaled by 1/3, 64 cubes scaled by 1/4, etc.

Note that 8 = (1/(1/2))³, 27 = (1/(1/3))³, and 64 = (1/(1/4))³.

In general,

N(r) = (1/r)³.

Scaled by 1/2	Scaled by 1/3	Scaled by 1/4

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