The On-Line Encyclopedia of Bongard Problems

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BP1

Empty image vs. non-empty image.

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	COMMENTS	`The first Bongard Problem.` `All examples in this Bongard Problem are line drawings (one or more connected figures made up of curved and non-curved lines).`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 214.`
	CROSSREFS	`Adjacent-numbered pages: * BP2 BP3 BP4 BP5 BP6`
	EXAMPLE	`A circle fits on the right because it is not nothing.`
	KEYWORD	`easy, nice, precise, allsorted, unstable, world, left-narrow, left-finite, left-full, left-null, perfect, pixelperfect, finished, traditional, stableworld, deformstable, bongard`
	CONCEPT	`empty (info \| search),` `existence (info \| search),` `zero (info \| search)`
	WORLD	`zoom in left (blank_image) \| zoom in right (curves_drawing)`
	AUTHOR	`Mikhail M. Bongard`

BP4

Convex vs. concave.

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	COMMENTS	`All examples in this Problem are outlines of shapes or solid black shapes.`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 215.`
	CROSSREFS	`BP136 is the same solution (flipped) but with only polygonal outlines and also with extraneous dots distracting from the solution.` `Adjacent-numbered pages: BP1 BP2 BP3 * BP5 BP6 BP7 BP8 BP9`
	KEYWORD	`easy, nice, precise, unstable, right-narrow, finished, traditional, bongard`
	CONCEPT	`concave_convex_angle (info \| search)`
	WORLD	`outline_or_fill_shape [smaller \| same \| bigger]`
	AUTHOR	`Mikhail M. Bongard`

BP15

Closed shape outline vs. non-closed curve.

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	COMMENTS	`All examples in this Bongard Problem are non-self-intersecting curves.`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 218.`
	CROSSREFS	`Adjacent-numbered pages: BP10 BP11 BP12 BP13 BP14 * BP16 BP17 BP18 BP19 BP20`
	KEYWORD	`nice, unstable, world, finished, traditional, bongard`
	CONCEPT	`closed_open (info \| search),` `hole (info \| search),` `loop (info \| search)`
	WORLD	`curve_or_outline [smaller \| same \| bigger] zoom in left (shape_outline) \| zoom in right (curve)`
	AUTHOR	`Mikhail M. Bongard`

BP72

Ends of the curve are parallel vs. ends of the curve are perpendicular.

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	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 237.`
	CROSSREFS	`Adjacent-numbered pages: BP67 BP68 BP69 BP70 BP71 * BP73 BP74 BP75 BP76 BP77`
	KEYWORD	`nice, precise, unstable, ignoreimperfections, finished, traditional, bongard`
	CONCEPT	`slope_at_endpoint (info \| search),` `tracing_line_or_curve (info \| search),` `parallel (info \| search),` `perpendicular (info \| search)`
	WORLD	`curve [smaller \| same \| bigger]`
	AUTHOR	`Mikhail M. Bongard`

BP211

More black than white vs. more white than black.

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	CROSSREFS	`See BP196 for a version of this Problem restricted to uniform textures.` `Adjacent-numbered pages: BP206 BP207 BP208 BP209 BP210 * BP212 BP213 BP214 BP215 BP216`
	KEYWORD	`easy, nice, spectrum, dual, blackwhite, unstable, right-null, perfect, pixelperfect, traditional, continuous, viceversa, bordercontent`
	CONCEPT	`outlined_filled (info \| search),` `size (info \| search),` `texture (info \| search)`
	WORLD	`bmp [smaller \| same \| bigger] zoom in left \| zoom in right`
	AUTHOR	`Giuseppe Insana`

BP324

Left shapes can be placed on top of each other to make right shape vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP319 BP320 BP321 BP322 BP323 * BP325 BP326 BP327 BP328 BP329`
	KEYWORD	`precise, unstable, perfect, pixelperfect, structure, orderedtriplet, traditional`
	CONCEPT	`set_union (info \| search),` `same_shape (info \| search),` `same (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP325

Left shapes can combine by symmetric difference (XOR logical operator) to make right shape vs. left shapes can combine by intersection (AND logical operator) to make right shape.

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	CROSSREFS	`Adjacent-numbered pages: BP320 BP321 BP322 BP323 BP324 * BP326 BP327 BP328 BP329 BP330`
	KEYWORD	`precise, unstable, perfect, pixelperfect, structure, orderedtriplet, traditional`
	CONCEPT	`xor (info \| search),` `overlap (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP335

Tessellates the plane vs. does not tessellate the plane.

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	COMMENTS	`EX7152 is an example of a shape than can be stretched in such a way that it no longer tessellates the plane. This is a property that is only exhibited by shapes that tessellate with rotated copies of themselves. - Leo Crabbe, Mar 05 2021`
	CROSSREFS	`Adjacent-numbered pages: BP330 BP331 BP332 BP333 BP334 * BP336 BP337 BP338 BP339 BP340`
	KEYWORD	`nice, stretch, unstable, math, hardsort, creativeexamples, proofsrequired, perfect, pixelperfect, traditional`
	CONCEPT	`infinite_plane (info \| search),` `tessellation (info \| search),` `tiling (info \| search)`
	WORLD	`shape [smaller \| same \| bigger] zoom in left (fill_shape)`
	AUTHOR	`Aaron David Fairbanks`

BP344

Shape can tile itself vs. shape cannot tile itself.

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	COMMENTS	`Left examples are sometimes called "rep-tiles."` `The tiles all must be the same size. More specifically, all left examples can tile themselves only using scaled down and rotated versions of themselves with all tiles the same size. Right examples cannot tile themselves using scaled down rotated versions of themselves or even reflected versions of themselves with all tiles the same size.` `Without the puzzle piece-like shape EX4120 on the right side the current examples also allow the solution "shape can tile with itself so as to create a parallelogram vs. shape cannot tile with itself so as to create a parallelogram."`
	CROSSREFS	`See BP532 for a version with fractals.` `Adjacent-numbered pages: BP339 BP340 BP341 BP342 BP343 * BP345 BP346 BP347 BP348 BP349`
	EXAMPLE	`Go to https://oebp.org/files/yet.png for an illustration of how some left-sorted shapes tile themselves.`
	KEYWORD	`hard, nice, precise, notso, unstable, math, hardsort, creativeexamples, proofsrequired, perfect, traditional`
	CONCEPT	`recursion (info \| search),` `self-reference (info \| search),` `tiling (info \| search),` `imagined_shape (info \| search),` `imagined_entity (info \| search)`
	WORLD	`shape [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP348

Shape on the right is the convex hull of shape on the left vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP343 BP344 BP345 BP346 BP347 * BP349 BP350 BP351 BP352 BP353`
	KEYWORD	`nice, precise, unstable, perfect, orderedpair, traditional`
	CONCEPT	`convex_hull (info \| search)`
	WORLD	`2_fill_shapes [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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