"Any" here means any image of a Bongard Problem in the relevant format, i.e. with white background, black vertical dividing line, and examples in boxes on either side.

All examples shown in this Problem clearly sort themselves on the left or right.

A self-referential but maybe simpler solution is "would sort all examples in this whole Bongard Problem on one of its sides vs. not so." Users adding examples please try to maintain this: for any example you add to the right of this Bongard Problem, make sure it does not sort all the other examples in this Bongard Problem on just one of its sides. - Aaron David Fairbanks, Aug 26 2020

Adjacent-numbered pages:
BP939 BP940 BP941 BP942 BP943  *  BP945 BP946 BP947 BP948 BP949

hard, challenge, presentationinvariant

boxes_bpimage_sorts_self [smaller | same | bigger]
zoom in left | zoom in right

Jago Collins

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 considered "self-similar"; the Bongard Problem must use itself, as-is (allowing downward scaling and ignoring pixelation), as a panel.

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.

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

hard, stub, abstract, challenge, meta (see left/right), miniproblems, infinitedetail, presentationinvariant, visualimagination

Leo Crabbe

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually becoming ___"; gradually becoming thickly outlined.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually becoming "gradually becoming ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and call it "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more belabored way of putting it:

Call something like "is star-shaped" a "rule". An object can fit a rule.

Call something like "all but one are ___" a "rule-parametrized rule". A collection of objects, with respect to a particular rule, can fit a rule-parametrized rule.

A drawing on the right shows many collections. Every collection fits the same rule-parametrized rule (with respect to various rules); furthermore the collection of collections fits that same rule-parametrized rule (with respect to some rule collections can fit).

Likewise a drawing on the left shows a collection of collections, with some noticeable recurring rule-parametrized rule. The collection of collections must fit that rule-parametrized rule with respect to the rule of fitting that rule-parametrized rule (with respect to various rule).

An unintended solution to this BP is "not all groups share some noticeable property vs. all do." It is hard to come up with examples foiling this alternative solution because (what was above called) the rule-parametrized rule usually has to do with not all objects in the collection fitting the rule. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Some examples would fit left under a certain interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property, and all things in a list being the same is a palindrome). But those rules are not necessarily the most obvious ways of interpreting these pictures, so they have been marked as ambiguous. Either of these placed on the left would prevent the intended solution being overridden (see the previous paragraph).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

The right side of this Problem is a subset of BP999^left.

Adjacent-numbered pages:
BP993 BP994 BP995 BP996 BP997  *  BP999 BP1000 BP1001 BP1002 BP1003

"Odd one out with respect to what property is the odd one out" would not fit left in this Problem: even though this example does seem doubly-meta, it is not doubly-meta in the right way. There is no odd one out with respect to the property of having an odd one out.

Similarly, consider "gradual transition with respect to what the gradual transition is between", etc. Instead of having the form "X 'X __' ", this is more like "X [the __ appearing in 'X __'] ". Examples like these two could make for a different Bongard Problem.

hard, unwordable, challenge, overriddensolution, infodense, contributepairs, funny, rules, miniworlds

zoom in right

Aaron David Fairbanks

Adjacent-numbered pages:
BP1006 BP1007 BP1008 BP1009 BP1010  *  BP1012 BP1013 BP1014 BP1015 BP1016

hard, precise, stretch, challenge, left-narrow, perfect, preciseworld

fill_polygon [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP1033 BP1034 BP1035 BP1036 BP1037  *  BP1039 BP1040 BP1041 BP1042 BP1043

hard, precise, allsorted, handed, leftright, math, challenge, meta (see left/right), miniproblems, assumesfamiliarity, structure, preciseworld, presentationinvariant

boxes_dots_bpimage_clear_set_of_numbers [smaller | same | bigger]

Aaron David Fairbanks

Adjacent-numbered pages:
BP1035 BP1036 BP1037 BP1038 BP1039  *  BP1041 BP1042 BP1043 BP1044 BP1045

hard, precise, allsorted, convoluted, notso, handed, leftright, math, challenge, meta (see left/right), miniproblems, assumesfamiliarity, structure, preciseworld, presentationinvariant

boxes_dots_bpimage_clear_set_of_numbers [smaller | same | bigger]

Aaron David Fairbanks

With mathematical jargon:

No distinct same-sized copies of self overlap on a subset with positive measure in the Hausdorff measure using the Hausdorff dimension.

For a covering of a fractal by finitely many scaled down copies of itself, the condition of that no two have an intersection with positive measure is equivalent to the condition that the Hausdorff dimension coincides with the similarity dimension.

(There is another similar condition in this context called the "open set condition" which implies this but is not equivalent. The open set condition is equivalent to the condition that the Hausdorff measure using the similarity dimension is nonzero.)

https://en.wikipedia.org/wiki/Hausdorff_dimension

https://en.wikipedia.org/wiki/Open_set_condition

Adjacent-numbered pages:
BP1115 BP1116 BP1117 BP1118 BP1119  *  BP1121 BP1122 BP1123 BP1124 BP1125

challenge, perfect, infinitedetail

[smaller | same | bigger]

Aaron David Fairbanks

Another wording: "can be repeatedly broken along 'fault lines' to yield individual pieces vs not."

Robert Dawson, A forbidden suborder characterization of binarily composable diagrams in double categories, Theory and Applications of Categories, Vol. 1, No. 7, p. 146-145, 1995.

All of the examples fitting left here would fit right in BP1199 except for (1) a single rectangle, (2) two rectangles stacked vertically, or (3) two rectangles side by side horizontally.

All of the examples fitting right in in BP1097 (re-styled) would fit right here (besides a single solid block, but that isn't shown there).

Adjacent-numbered pages:
BP1195 BP1196 BP1197 BP1198 BP1199  *  BP1201 BP1202 BP1203 BP1204 BP1205

hard, precise, challenge, proofsrequired, inductivedefinition, left-listable, right-listable

Aaron David Fairbanks

Adjacent-numbered pages:
BP1240 BP1241 BP1242 BP1243 BP1244  *  BP1246 BP1247 BP1248 BP1249 BP1250

hard, precise, convoluted, dual, rotate, boundingbox, hardsort, challenge, proofsrequired

Aaron David Fairbanks