The On-Line Encyclopedia of Bongard Problems

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BP805

Bongard Problem sorts example below on the left versus Bongard Problem sorts example below on the right.

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	COMMENTS	`See BP372 for the version of this with BPs with black shapes as examples instead of boxes.`
	CROSSREFS	`Adjacent-numbered pages: BP800 BP801 BP802 BP803 BP804 * BP806 BP807 BP808 BP809 BP810`
	KEYWORD	`nice, dual, handed, leftright, creativeexamples, structure, contributepairs, rules`
	CONCEPT	`bongard_problem (info \| search),` `categorization (info \| search),` `choice (info \| search)`
	WORLD	`boxes_bdimage_six_per_side_extra_below_clear_sort [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP806

Image of repeating wallpaper with only 3-fold rotational symmetries versus image of repeating wallpaper with some other symmetries.

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	COMMENTS	`Left examples have wallpaper symmetry group p3 (a.k.a. 333).`
	CROSSREFS	`Adjacent-numbered pages: BP801 BP802 BP803 BP804 BP805 * BP807 BP808 BP809 BP810 BP811`
	KEYWORD	`nice, math, bordercontent`
	CONCEPT	`rotational_symmetry (info \| search),` `three (info \| search)`
	WORLD	`wallpaper [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP809

Figures can be transformed into one another by smooth stretching (before and after there are the same crossroad-points; there is a curve connecting points before if and only if there is a curve connecting those points after) vs. not so.

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	COMMENTS	`Left examples are topologically homeomorphic figures.` `For some examples one can imagine pulling the shape "out of" the 2d square in 3d in order to transform it, and then laying it flat back in the 2d square. See BP810 for the version where this is not allowed.`
	CROSSREFS	`Adjacent-numbered pages: BP804 BP805 BP806 BP807 BP808 * BP810 BP811 BP812 BP813 BP814`
	KEYWORD	`nice, math, unorderedpair, traditional`
	CONCEPT	`topological_transformation (info \| search)`
	WORLD	`two_figures_made_of_curves [smaller \| same \| bigger] zoom in left (two_homeomorphic_figures_made_of_curves)`
	AUTHOR	`Aaron David Fairbanks`

BP810

Figures can be transformed into one another by smooth stretching (intersection points stay constant; paths connecting those points remain), while remaining within the 2d box vs. movement out of the plane required.

(edit; present; nest [left/right]; search; history)

	CROSSREFS	`All examples here fit left in BP809, a version where the figures are allowed to pass through themselves while being deformed.` `Adjacent-numbered pages: BP805 BP806 BP807 BP808 BP809 * BP811 BP812 BP813 BP814 BP815`
	KEYWORD	`nice, math, unorderedpair, traditional`
	CONCEPT	`topological_transformation (info \| search)`
	WORLD	`two_homeomorphic_figures_made_of_curves [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP811

Archimedean tiling (regular polygons, all vertices look the same) versus two-uniform tiling (regular polygons, two different kinds of vertex).

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	CROSSREFS	`Adjacent-numbered pages: BP806 BP807 BP808 BP809 BP810 * BP812 BP813 BP814 BP815 BP816`
	KEYWORD	`nice, math`
	CONCEPT	`infinite_plane (info \| search),` `tiling (info \| search),` `symmetry (info \| search)`
	WORLD	`wallpaper_tiling [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP818

Dot's position within square is center of square's position within panel vs. not so.

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	COMMENTS	`All examples in this problem are dots inside square outlines.`
	CROSSREFS	`Similar to BP1122 (the "fractal" version).` `Adjacent-numbered pages: BP813 BP814 BP815 BP816 BP817 * BP819 BP820 BP821 BP822 BP823`
	KEYWORD	`nice, notso, boundingbox, absoluteposition`
	CONCEPT	`bounding_box (info \| search),` `self-reference (info \| search),` `center (info \| search),` `distance_from_center (info \| search)`
	WORLD	`point_inside_square [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP820

Shape can be combined with a copy of itself to form a convex shape vs. not so.

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	CROSSREFS	`For the generalization of this property, see BP991.` `Adjacent-numbered pages: BP815 BP816 BP817 BP818 BP819 * BP821 BP822 BP823 BP824 BP825`
	KEYWORD	`nice, precise, allsorted`
	CONCEPT	`tiling (info \| search)`
	WORLD	`fill_shape [smaller \| same \| bigger] zoom in left`
	AUTHOR	`Isaac Hathaway`

BP827

Image of noisy Bongard Problem vs. image of Bongard Problem with minimal noise.

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	COMMENTS	`"Noise" is a property that changes between examples which is independent to the solution property.`
	CROSSREFS	`See BP511 for the version with links to pages on the OEBP instead of images of Bongard Problems (miniproblems).` `See BP845 for noise in sequences of quantity increase.` `Adjacent-numbered pages: BP822 BP823 BP824 BP825 BP826 * BP828 BP829 BP830 BP831 BP832`
	KEYWORD	`nice, meta (see left/right), miniproblems, creativeexamples, presentationmatters, assumesfamiliarity, structure, contributepairs`
	WORLD	`boxes_bpimage_three_per_side [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP828

Image of Bongard Problem with one simple solution vs. image of Bongard Problem with two "independent" simple solutions.

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	COMMENTS	`Properties are considered independent if none logically implies the other (or its opposite).` `Another way of putting it is that on the right, the most descriptive good solution is of the form "[property 1] AND [property 2] vs. neither".`
	CROSSREFS	`Adjacent-numbered pages: BP823 BP824 BP825 BP826 BP827 * BP829 BP830 BP831 BP832 BP833`
	KEYWORD	`nice, meta (see left/right), miniproblems, creativeexamples, assumesfamiliarity, structure`
	CONCEPT	`exists_one (info \| search),` `two (info \| search)`
	WORLD	`boxes_bpimage_three_per_side [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP829

Image of a Bongard Problem with no simple solution versus image of a Bongard Problem with a simple solution.

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	COMMENTS	`Left examples have no solution, but they do not break the rules in ways so extreme that it is plainly impossible for them to have a solution, such as including the same image on both sides or including no images per side. (See such as including the same image on both sides or including no images per side.`
	CROSSREFS	`See BP522 for the version with links to pages on the OEBP instead of images of Bongard Problems (miniproblems).` `See BP968 (flipped) for a version of this Bongard Problem including examples of invalid Bongard Problems that don't even admit a convoluted solution (the same image appears on both sides).` `Also see BP1080, which is similar to BP968, but including various different formats of Bongard Problems, distinguishing them from arbitrary images that are not Bongard Problems.` `Adjacent-numbered pages: BP824 BP825 BP826 BP827 BP828 * BP830 BP831 BP832 BP833 BP834`
	KEYWORD	`nice, meta (see left/right), miniproblems, creativeexamples, left-unknowable, right-narrow, assumesfamiliarity, structure, help, presentationinvariant`
	CONCEPT	`existence (info \| search),` `simplicity (info \| search),` `zero (info \| search)`
	WORLD	`boxes_bpimage_three_per_side_nosoln_allowed [smaller \| same \| bigger] zoom in right (boxes_bpimage_three_per_side)`
	AUTHOR	`Aaron David Fairbanks`

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