The On-Line Encyclopedia of Bongard Problems

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A curve with one self-crossing vs. a curve without a self-crossing.

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	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 223.`
	CROSSREFS	`BP1232 is almost exactly the same, but allowing for multiple curves and multiple self-crossings.` `Adjacent-numbered pages: BP25 BP26 BP27 BP28 BP29 * BP31 BP32 BP33 BP34 BP35`
	KEYWORD	`nice, finished, traditional, bongard`
	CONCEPT	`x-crossing (info \| search),` `existence (info \| search)`
	WORLD	`crosscurve [smaller \| same \| bigger] zoom in right (curve_or_outline)`
	AUTHOR	`Mikhail M. Bongard`

Outlines made by triangles and circles intersect vs. outlines made by triangles and circles do not intersect.

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	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 246.`
	CROSSREFS	`Adjacent-numbered pages: BP94 BP95 BP96 BP97 BP98 * BP100 BP101 BP102 BP103 BP104`
	KEYWORD	`finished, traditional, bongard`
	CONCEPT	`collinear (info \| search),` `x-crossing (info \| search),` `cluster (info \| search)`
	AUTHOR	`Mikhail M. Bongard`

Four X-like points vs. two X-like points.

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	CROSSREFS	`Adjacent-numbered pages: BP109 BP110 BP111 BP112 BP113 * BP115 BP116 BP117 BP118 BP119`
	KEYWORD	`traditional`
	CONCEPT	`x-crossing (info \| search),` `number (info \| search)`
	AUTHOR	`Douglas R. Hofstadter`

The tightest-curved section, out of all sections of curve that make a complete turn (360 degrees), contains an x-crossing point vs. not so.

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	COMMENTS	`All examples are connected smooth curves allowed to self-intersect that must curve in only one direction (starting at one end, either clockwise or counter-clockwise), i.e. there is no inflection point.`
	CROSSREFS	`Adjacent-numbered pages: BP208 BP209 BP210 BP211 BP212 * BP214 BP215 BP216 BP217 BP218`
	KEYWORD	`less, convoluted, perfect, traditional`
	CONCEPT	`x-crossing (info \| search)`
	AUTHOR	`Giuseppe Insana`

Straight line joining the curve ends does not cut the curve vs. straight line joining the curve ends intersects the curve.

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	CROSSREFS	`Adjacent-numbered pages: BP235 BP236 BP237 BP238 BP239 * BP241 BP242 BP243 BP244 BP245`
	KEYWORD	`traditional`
	CONCEPT	`x-crossing (info \| search),` `line_or_curve_endpoint (info \| search),` `imagined_line_or_curve (info \| search),` `imagined_entity (info \| search)`
	AUTHOR	`Matthew J. Howells`

Straight lines that join filled and outlined circles intersect vs. straight lines that join filled and outlined circles do not intersect.

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	CROSSREFS	`Adjacent-numbered pages: BP256 BP257 BP258 BP259 BP260 * BP262 BP263 BP264 BP265 BP266`
	KEYWORD	`noisy, traditional`
	CONCEPT	`lines_coincide (info \| search),` `x-crossing (info \| search),` `outlined_filled (info \| search),` `feature_cluster (info \| search),` `cluster (info \| search),` `imagined_line_or_curve (info \| search),` `imagined_entity (info \| search),` `texture (info \| search)`
	AUTHOR	`Michael Ihde`

At least one touch or cross point vs. no touch or cross point.

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	CROSSREFS	`Adjacent-numbered pages: BP261 BP262 BP263 BP264 BP265 * BP267 BP268 BP269 BP270 BP271`
	CONCEPT	`x-crossing (info \| search),` `or (info \| search),` `existence (info \| search),` `t-point (info \| search)`
	WORLD	`arial_font_centered_symbol [smaller \| same \| bigger]`
	AUTHOR	`Merse E. Gáspár`

Curves with at least one crossing vs. curves do not cross.

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	COMMENTS	`Based on BP30, "curve with exactly one self-crossing vs. curve without a self-crossing". This is just a slightly more general version.`
	CROSSREFS	`Adjacent-numbered pages: BP1227 BP1228 BP1229 BP1230 BP1231 * BP1233 BP1234 BP1235 BP1236 BP1237`
	EXAMPLE	`EX6881`
	KEYWORD	`stub, notso, traditional`
	CONCEPT	`x-crossing (info \| search),` `existence (info \| search)`
	WORLD	`crosscurves [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

Different curves cross each other vs. curves only cross themselves (or do not cross at all).

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	CROSSREFS	`Adjacent-numbered pages: BP1228 BP1229 BP1230 BP1231 BP1232 * BP1234 BP1235 BP1236 BP1237 BP1238`
	KEYWORD	`notso, traditional`
	CONCEPT	`distinguishing_crossing_curves (info \| search),` `x-crossing (info \| search)`
	WORLD	`smooth_crosscurves [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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