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BP1120 No same-sized copies of self overlap vs. distinct same-sized copies overlap.
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COMMENTS

With mathematical jargon:

No distinct same-sized copies of self overlap on a subset with positive measure in the Hausdorff measure using the Hausdorff dimension.


For a covering of a fractal by finitely many scaled down copies of itself, the condition of that no two have an intersection with positive measure is equivalent to the condition that the Hausdorff dimension coincides with the similarity dimension.

(There is another similar condition in this context called the "open set condition" which implies this but is not equivalent. The open set condition is equivalent to the condition that the Hausdorff measure using the similarity dimension is nonzero.)

REFERENCE

https://en.wikipedia.org/wiki/Hausdorff_dimension

https://en.wikipedia.org/wiki/Open_set_condition

CROSSREFS

Adjacent-numbered pages:
BP1115 BP1116 BP1117 BP1118 BP1119  *  BP1121 BP1122 BP1123 BP1124 BP1125

KEYWORD

challenge, perfect, infinitedetail

CONCEPT fractal (info | search),
recursion (info | search),
self-reference (info | search),
overlap (info | search)

WORLD

[smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

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