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`

BP953

Image of this Bongard Problem vs. empty image.

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	COMMENTS	`"Image of Bongard Problem with solution X vs. empty image" where X is the phrase in quotes.`
	CROSSREFS	`See BP959, BP902.` `Adjacent-numbered pages: BP948 BP949 BP950 BP951 BP952 * BP954 BP955 BP956 BP957 BP958`
	KEYWORD	`nice, precise, meta (see left/right), miniproblems, overriddensolution, right-full, right-null, perfect, infinitedetail, experimental, funny`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	WORLD	`zoom in left (bp953_image) \| zoom in right (blank_image)`
	AUTHOR	`Leo Crabbe`

BP954

Solution could appear in a Bongard Problem that has itself as a panel vs. not so.

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	COMMENTS	`Loosely speaking, examples on the left are "Bongard Problems that can be self-similar". However, Bongard Problems with images of themselves deeply nested in boxes or rotated/flipped are not here considered "self-similar"; the Bongard Problem must use itself, as-is (allowing downward scaling and allowing infinite detail, ignoring pixelation--see keyword infinitedetail), as a panel.` `Bongard Problems fitting left evidently come in three categories: 1) the Bongard Problem could only appear on its own left side, 2) the Bongard Problem could appear on its own right side, or 3) the Bongard Problem could appear on its own left or the right side. See BP987.` `Meta Bongard Problems appearing in BP793 that are presentationinvariant necessarily fit left here.` `All examples here are in the conventional format, i.e. white background, black vertical dividing line, and examples in boxes on either side. (A more general version of this Bongard Problem might allow many formats of Bongard Problems, sorting an image left if a self-similar version is possible having the same solution and format. This more general version would no longer be tagged presentationinvariant, since sorting would not only depend on solution, but also format.)` `It would hint at the solution (keyword help) to only include images of Bongard Problems that, as it stands, are already clearly categorized on one side by themselves. (That is, images of Bongard Problems that belong on one of the two sides of BP793.) It is tricky to come up with images that are categorized by themselves as it stands but that could NOT be recursively included within themselves. EX7967, EX7999, EX7995, and EX6574 are some examples.`
	CROSSREFS	`See BP987 which narrows down the left-hand side of this BP further based on whether or not the BP could contain itself as a panel on both sides.` `Adjacent-numbered pages: BP949 BP950 BP951 BP952 BP953 * BP955 BP956 BP957 BP958 BP959`
	KEYWORD	`hard, abstract, challenge, meta (see left/right), miniproblems, infinitedetail, presentationinvariant, visualimagination`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	AUTHOR	`Leo Crabbe`

BP959

This image of this Bongard Problem vs. empty image.

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	CROSSREFS	`See BP953, BP902.` `Adjacent-numbered pages: BP954 BP955 BP956 BP957 BP958 * BP960 BP961 BP962 BP963 BP964`
	KEYWORD	`meta (see left/right), miniproblems, left-finite, right-finite, left-full, right-full, right-null, perfect, infinitedetail, finished, experimental, funny`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	WORLD	`zoom in left \| zoom in right (blank_image)`
	AUTHOR	`Aaron David Fairbanks, Leo Crabbe`

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