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`

BP530

Fractal tiles itself with uniformly scaled-down copies of itself vs. fractal tiles itself with stretched copies of itself.

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	COMMENTS	`"Self-similar" vs. "self-affine."`
	CROSSREFS	`Adjacent-numbered pages: BP525 BP526 BP527 BP528 BP529 * BP531 BP532 BP533 BP534 BP535`
	KEYWORD	`perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `self-reference (info \| search),` `tiling (info \| search)`
	WORLD	`fractal_self_tile_affine_allowed [smaller \| same \| bigger] zoom in left (fractal_self_tile)`
	AUTHOR	`Aaron David Fairbanks`

BP531

Fractal is tiled by three smaller copies of itself vs. fractal is tiled by five smaller copies of itself.

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	COMMENTS	`More specifically, all left examples shown in this Problem have Hausdorff dimension log2(3) while all right examples have Hausdorff dimension log3(5).` `Left examples can tile themselves by any power of 3 smaller same-sized copies of themselves while right examples can tile themselves by any power of 5 smaller same-sized copies of themselves.` `Homage to Bongard's original three versus five Problems.`
	CROSSREFS	`Adjacent-numbered pages: BP526 BP527 BP528 BP529 BP530 * BP532 BP533 BP534 BP535 BP536`
	KEYWORD	`perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search),` `tiling (info \| search),` `three (info \| search),` `five (info \| search)`
	WORLD	`fractal_self_tile [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP532

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`

BP533

Contains smaller copy of itself vs. doesn't.

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	CROSSREFS	`Adjacent-numbered pages: BP528 BP529 BP530 BP531 BP532 * BP534 BP535 BP536 BP537 BP538`
	EXAMPLE	`A smaller copy of EX6409 (the black area) can be located within itself, but some of the white space inside it is not retained in this smaller copy.`
	KEYWORD	`perfect, infinitedetail, contributepairs`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	WORLD	`connected_built_from_self_tile_fractals [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP543

Image depicting infinitely many objects vs. image depicting finitely many objects.

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	COMMENTS	`Another solution is "evokes infinity vs. not so." Or "seen as having arbitrarily fine detail." All left examples show objects getting smaller and smaller and closer and closer together approaching some limit within the box.`
	CROSSREFS	`Adjacent-numbered pages: BP538 BP539 BP540 BP541 BP542 * BP544 BP545 BP546 BP547 BP548`
	KEYWORD	`nice, abstract, concept, infinitedetail`
	CONCEPT	`finite_infinite (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP852

Object shown below is the "limit" of the sequence above (end result after "infinite time") versus not so.

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	COMMENTS	`The conceptual limit of the sequence may not be the limit of the points in the image. For example in a sequence of halvings the limit value is never reached, so the bottom would never change color and thus its limit would not would not either.` `Sequences progress from left to right (and there is not usually a way to intuitively extend the sequence in the other direction).`
	CROSSREFS	`Adjacent-numbered pages: BP847 BP848 BP849 BP850 BP851 * BP853 BP854 BP855 BP856 BP857`
	KEYWORD	`notso, creativeexamples, perfect, infinitedetail, assumesfamiliarity, structure, contributepairs, rules`
	AUTHOR	`Aaron David Fairbanks`

BP961

Includes itself on the left vs. includes itself on the right.

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	COMMENTS	`Some examples are Bongard Problems with this solution.`
	CROSSREFS	`Adjacent-numbered pages: BP956 BP957 BP958 BP959 BP960 * BP962 BP963 BP964 BP965 BP966`
	KEYWORD	`nice, precise, dual, handed, leftright, perfect, infinitedetail, both, neither, preciseworld`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1058

Top layer is two vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP1053 BP1054 BP1055 BP1056 BP1057 * BP1059 BP1060 BP1061 BP1062 BP1063`
	KEYWORD	`arbitrary, perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search),` `specificity (info \| search),` `two (info \| search)`
	WORLD	`recursive_boxes [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP1059

No infinite nesting vs. some infinite nesting.

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	CROSSREFS	`Adjacent-numbered pages: BP1054 BP1055 BP1056 BP1057 BP1058 * BP1060 BP1061 BP1062 BP1063 BP1064`
	KEYWORD	`perfect, infinitedetail`
	CONCEPT	`finite_infinite (info \| search),` `fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	WORLD	`recursive_boxes [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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