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Tiled by finitely many smaller copies of itself (different sizes allowed) vs. not so.

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	COMMENTS	`These are sometimes called "irreptiles".`
	CROSSREFS	`See BP532 for the version with only one size of tile allowed.` `Adjacent-numbered pages: BP1114 BP1115 BP1116 BP1117 BP1118 * BP1120 BP1121 BP1122 BP1123 BP1124`
	KEYWORD	`hardsort, proofsrequired, perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search),` `tiling (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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`

Connected fractal vs. disconnected fractal.

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	CROSSREFS	`Adjacent-numbered pages: BP1232 BP1233 BP1234 BP1235 BP1236 * BP1238 BP1239 BP1240 BP1241 BP1242`
	KEYWORD	`notso, perfect, infinitedetail`
	CONCEPT	`connected_component (info \| search),` `fractal (info \| search)`
	WORLD	`fractal_self_tile [smaller \| same \| bigger] zoom in left (connected_fractal_self_tile)`
	AUTHOR	`Jago Collins`

Fractal with hole vs. fractal with no hole (simply connected).

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	CROSSREFS	`Adjacent-numbered pages: BP1233 BP1234 BP1235 BP1236 BP1237 * BP1239 BP1240 BP1241 BP1242 BP1243`
	KEYWORD	`notso, perfect, infinitedetail`
	CONCEPT	`closed_open (info \| search),` `existence (info \| search),` `fractal (info \| search),` `hole (info \| search),` `loop (info \| search)`
	WORLD	`connected_fractal_self_tile [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

Any point contained in (arbitrarily) smaller version of self vs. not so.

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	COMMENTS	`Note if any point is contained in some smaller version of the whole, then any point is contained in arbitrarily smaller versions of the whole.` `It isn't possible to unambiguously communicate in a picture whether or not a few specific points are included in the fractal. The pictures are interpreted as what is intuitively simplest. To make matters less ambiguous, all the fractals here contain all points arbitrarily close to points in them. (They are topologically closed. See also BP1239.)` `The left hand side of this is a stronger condition than the left hand side of BP1116.`
	CROSSREFS	`Adjacent-numbered pages: BP1236 BP1237 BP1238 BP1239 BP1240 * BP1242 BP1243 BP1244 BP1245 BP1246`
	KEYWORD	`notso, perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	WORLD	`connected_fractal [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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