The On-Line Encyclopedia of Bongard Problems

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Fractal tiles itself with smaller non-rotated (nor reflected) copies of itself vs. fractal requires turning to tile itself.

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	COMMENTS	`No included examples involve reflection.`
	CROSSREFS	`Adjacent-numbered pages: BP524 BP525 BP526 BP527 BP528 * BP530 BP531 BP532 BP533 BP534`
	KEYWORD	`perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `rotation_required (info \| search),` `self-reference (info \| search),` `tiling (info \| search)`
	WORLD	`fractal_self_tile [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

Self-tiling fractal using one size of tile vs. does not tile itself with a single size of itself.

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	CROSSREFS	`This is BP344 ("rep-tiles") but for fractals.` `See BP1119 for the version with multiple different sizes of tile allowed.` `Adjacent-numbered pages: BP527 BP528 BP529 BP530 BP531 * BP533 BP534 BP535 BP536 BP537`
	KEYWORD	`hardsort, proofsrequired, perfect, infinitedetail, contributepairs`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search),` `tiling (info \| search)`
	WORLD	`[smaller \| same \| bigger] zoom in left (fractal_self_tile)`
	AUTHOR	`Aaron David Fairbanks`

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`

Fractal topologically closed (each white point has a neighborhood of pure white surrounding it) vs. not

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	COMMENTS	`It isn't possible to unambiguously communicate whether or not a few specific points are missing.`
	CROSSREFS	`Adjacent-numbered pages: BP1234 BP1235 BP1236 BP1237 BP1238 * BP1240 BP1241 BP1242 BP1243 BP1244`
	KEYWORD	`notso, perfect, infinitedetail`
	CONCEPT	`fractal (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

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