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BP1119 Tiled by finitely many smaller copies of itself (different sizes allowed) vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

These are sometimes called "irreptiles".

CROSSREFS

See BP532 for the version with only one size of tile allowed.

Adjacent-numbered pages:
BP1114 BP1115 BP1116 BP1117 BP1118  *  BP1120 BP1121 BP1122 BP1123 BP1124

KEYWORD

hardsort, proofsrequired, perfect, infinitedetail

CONCEPT fractal (info | search),
recursion (info | search),
self-reference (info | search),
tiling (info | search)

WORLD

[smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP1120 No same-sized copies of self overlap vs. distinct same-sized copies overlap.
(edit; present; nest [left/right]; search; history)
COMMENTS

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.)

REFERENCE

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

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

CROSSREFS

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

KEYWORD

challenge, perfect, infinitedetail

CONCEPT fractal (info | search),
recursion (info | search),
self-reference (info | search),
overlap (info | search)

WORLD

[smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP1121 Bongard Problems that were added to the OEBP to be used as examples in particular meta-BPs vs. other Bongard Problems,
BP570
BP868
BP915
BP1042
BP1043
BP1105
BP1141
BP1150
BP1163
BP1168
BP1227
(edit; present; nest [left/right]; search; history)
COMMENTS

Bongard Problems sorted left have the keyword "example" on the OEBP.


Plea to the reader: We need a good counterexample for this Problem. Could you please make a good example of a Problem this meta BP would sort on its right? Don't forget to tag it appropriately. - Leo Crabbe, Dec 13 2021

CROSSREFS

Adjacent-numbered pages:
BP1116 BP1117 BP1118 BP1119 BP1120  *  BP1122 BP1123 BP1124 BP1125 BP1126

KEYWORD

meta (see left/right), links, keyword, oebp

AUTHOR

Aaron David Fairbanks

BP1122 Content of any square is an image of the whole panel vs. not so.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Similar to BP818.

Adjacent-numbered pages:
BP1117 BP1118 BP1119 BP1120 BP1121  *  BP1123 BP1124 BP1125 BP1126 BP1127

KEYWORD

nice, minimal, size, boundingbox, infinitedetail, preciseworld, absoluteposition

CONCEPT fractal (info | search),
recursion (info | search),
self-reference (info | search)

AUTHOR

Leo Crabbe

BP1123 Can be cut into tiles forming a checkerboard pattern vs. not so.
?
(edit; present; nest [left/right]; search; history)
COMMENTS

All examples in this Problem are grids consisting of two objects.

CROSSREFS

Adjacent-numbered pages:
BP1118 BP1119 BP1120 BP1121 BP1122  *  BP1124 BP1125 BP1126 BP1127 BP1128

EXAMPLE

EX9124 shows a 9 square by 9 square grid. Take each tile to be 3 squares by 3 squares; there is a 3 tile by 3 tile checkerboard pattern. (One of these tiles is itself a checkerboard pattern; the other is all black squares.)

KEYWORD

hard, nice, precise, allsorted, hardsort, grid, miniworlds

AUTHOR

Jago Collins

BP1124 Bongard Problems such that examples are always by default sorted left, until some unforeseen way of fitting right is noticed (a person is never "sure" something should fit left, but can be "sure" something fits right) vs. vice versa.
BP347
BP829
BP1127
BP801
BP1155
BP1163
(edit; present; nest [left/right]; search; history)
COMMENTS

Left-sorted Bongard Problems have the keyword "left-unknowable" on the OEBP.

Right-sorted Bongard Problems have the keyword "right-unknowable".


Think of searching for needles in endless haystacks. You can be sure a haystack has a needle by finding it, but you can never be sure a haystack does not have a needle.


When a Bongard Problem is "left-unknowable", individual examples cannot be determined for certain to fit left, by any means. The author of the Bongard Problem just chooses some examples that seem to fit left. (See also the noproofs keyword.)


It is very extreme for this to apply to all examples without exception. Often a Bongard Problem is close to being purely left-unknowable, but a few examples spoil it by being obviously disqualified from the right side for some reason.


It is natural for a person to guess the solution to an unknowable Bongard Problem before actually understanding all the knowable examples, taking some of them on faith.

As a prank, take a left- or right- unknowable Bongard Problem and put an example that actually belongs on the unknowable side on the knowable side. The solver will have to take it on faith there is some reason it fits there they are not seeing.

(The property of having this kind of sorting mistake is unknowable for left- or right- unknowable Bongard Problems.)



One interpretation of topology (a subject of mathematics -- see https://en.wikipedia.org/wiki/Topology ) is that a topology describes the observability of various properties. (The topological "neighborhoods" of a point are the subsets one could determine the point to be within using a finite number of measurements.) The analogue of a property that is nowhere directly observable is a "subset with empty interior". Furthermore, the fact that the negation of the property is observable corresponds to the subset being "closed".

CROSSREFS

Left- or right- unknowable Bongard Problems are generally notso Bongard Problems: an example fits on one side just in case it cannot be observed to fit on the other.


Although the descriptions of left-couldbe and right-couldbe sound similar to "left-unknowable" and "right-unknowable", they are not the same. It is the difference between a clear absence of information and perpetual uncertainty about whether there is more information to be found. For any example sorted on a "could be" side, there is a clear (knowable) absence of information whose presence would justify the example being on the other side.

Sometimes an unknowable BP can be turned into a couldbe BP by explicitly restricting the amount of available information. For example, if there were a hypothetical Bongard Problem with infinitely detailed pictures, using a low resolution for all pictures could simplify the issue of detecting some properties that would be "unknowable". Many fractal-based BPs are this way (e.g. BP1122). See keyword infinitedetail.


Right-unknowable Bongard Problems are generally left-narrow (and left-unknowable Bongard Problems are generally right-narrow).


A Bongard Problem with examples on both sides cannot be tagged both proofsrequired and left- or right- unknowable.


Many Bongard Problems are about finding rules (keyword rules)--in each panel a rule is to be found, and there are no specified limits about what kind of rule it can be or how abstract it can be. (Just like a Bongard Problem.) "There is a rule vs. there isn't" (resp. vice versa) are right- (resp. left-) unknowable. (That is, disregarding cases that obviously do not define a rule because of some trivial disqualifying reason.)


Actually, I think there is something more to be said about this. It is possible to design examples that signal there is no rule to be found. See for example EX9138 in BP1127 and EX6829 in BP829. (Related: keyword help.) Each of these examples communicates a clear rule that "doesn't count". And there is so little information shown that a person can feel confident they've noticed all the relevant details. So, contrary to how they are currently tagged, these Bongard Problems aren't strictly "unknowable"; there are some exceptional knowable cases. But being too strict about the definition of "unknowable" makes it so there aren't any examples of unknowable Bongard Problems, so it's probably better to be a bit loose. - Aaron David Fairbanks, Apr 20 2022

Adjacent-numbered pages:
BP1119 BP1120 BP1121 BP1122 BP1123  *  BP1125 BP1126 BP1127 BP1128 BP1129

EXAMPLE

The perfect example is BP1163.


Interesting example of a Bongard Problem that is neither left-unknowable nor right unknowable in particular, but for which it is impossible to know whether any example fits on either side: BP1229 (translational symmetry vs. not) made with examples that can be expanded to any larger finite region the solver wants to look at. In this case, examples could only be sorted based on what they seem like (see seemslike), trusting they appear in a way that hints psychologically at what they actually are (see help).

KEYWORD

dual, meta (see left/right), links, keyword, side, viceversa

CONCEPT semidecidable (info | search)

AUTHOR

Aaron David Fairbanks

BP1125 BP pages on the OEBP (with a criterion for sorting examples that in some cases may be very difficult to work out) where users should be certain (i.e. know a proof) about how examples are sorted vs. users can include examples on a side as long as nobody has seen a reason it does not fit there.
BP335
BP344
BP532
BP850
BP1119
BP1137
BP1200
BP1245
(edit; present; nest [left/right]; search; history)
COMMENTS

Left-sorted Bongard Problems have the keyword "proofsrequired" on the OEBP.

Right-sorted Bongard Problems have the keyword "noproofs" on the OEBP.


For every "noproofs" Bongard Problem there could be made a stricter "proofsrequired" version. This stricter version will be hardsort.


Deciding to make a Bongard Problem noproofs adds subjectivity to the sorting of examples (keyword subjective).



One interpretation of topology (a subject of mathematics -- see https://en.wikipedia.org/wiki/Topology ) is that a topology describes the observability of various properties. (The topological "neighborhoods" of a point are the subsets one could determine the point to be within using a finite number of measurements.) The analogue of restricting to just the cases where a property is observably true (i.e. "proofsrequired") corresponds to taking the topological "interior" of that property.



TO DO: It may be better to split each of these keywords up into two: "left-proofsrequired", "right-proofsrequired", "left-noproofs", "right noproofs".


CROSSREFS

See keyword hardsort.


Bongard Problems that are left-unknowable or right-unknowable will have to be "noproofs".

Adjacent-numbered pages:
BP1120 BP1121 BP1122 BP1123 BP1124  *  BP1126 BP1127 BP1128 BP1129 BP1130

EXAMPLE

In "proofsrequired" BP335 (shape tessellates the plane vs. shape does not tessellate the plane), shapes are only put in the Bongard Problem if they are known to tessellate or not to tessellate the plane. A "noproofs" version of this Bongard Problem would instead allow a shape to be put on the right if it was just (subjectively) really hard to find a way of tessellating the plane with it.

KEYWORD

meta (see left/right), links, keyword, oebp, instruction

AUTHOR

Aaron David Fairbanks

BP1126 Meta Bongard Problems in which examples are pages on the OEBP vs. meta Bongard Problems in which examples are pictures of Bongard Problems.
BP501
BP503
BP504
BP506
BP507
BP508
BP509
BP510
BP511
BP512
BP513
BP514
BP515
BP516
BP517
BP518
BP519
BP520
BP521
BP522
BP526
BP534
BP535
BP537
BP539
BP541
BP542
BP544
BP546
BP547
BP549
BP550
BP552
BP553
BP554

. . .

BP200
BP793
BP795
BP796
BP802
BP803
BP827
BP828
BP829
BP830
BP831
BP832
BP833
BP834
BP835
BP836
BP868
BP871
BP872
BP873
BP874
BP875
BP876
BP877
BP878
BP879
BP880
BP881
BP894
BP948
BP952
BP953
BP954
BP955
BP957

. . .

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

Bongard Problems sorted left have the keyword "links" on the OEBP.

Bongard Problems sorted right have the keyword "miniproblems" on the OEBP.


The keyword "links" is automatically added to a Bongard Problem on the OEBP if a BP number is added as an example.


Meta Bongard problems that sort Bongard Problems purely based on their solutions (keyword presentationmatters) usually have two versions in the database: one that sorts images of Bongard Problems and one that sorts links to pages on the OEBP. If both versions exist, users should make them cross-reference one another.

CROSSREFS

All the examples of miniature Bongard Problems within any meta Bongard Problem tagged "miniproblems" would fit left on BP1080 (which is a showcase of the various formats for images of Bongard Problems).

Adjacent-numbered pages:
BP1121 BP1122 BP1123 BP1124 BP1125  *  BP1127 BP1128 BP1129 BP1130 BP1131

KEYWORD

meta (see left/right), links, keyword, world, left-self, metameta

WORLD

metabp [smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP1127 There is no rule for how the objects in a cluster interrelate vs. there is.
(edit; present; nest [left/right]; search; history)
COMMENTS

Other ways of phrasing this:

"Local" vs. "global" properties of collections: to check a collection satisfies a "local" property, it is only necessary to check each individual thing in it satisfies some property.

The rule all collections satisfy is just "every object is a ___" vs. the rule is something more.

CROSSREFS

Adjacent-numbered pages:
BP1122 BP1123 BP1124 BP1125 BP1126  *  BP1128 BP1129 BP1130 BP1131 BP1132

KEYWORD

abstract, creativeexamples, left-unknowable, rules, miniworlds

CONCEPT local_global (info | search)

WORLD

[smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP1128 Bongard Problem with inductive definition of solution vs. other Bongard Problems.
BP956
BP1129
BP1200
(edit; present; nest [left/right]; search; history)
COMMENTS

Bongard Problems sorted left have the keyword "inductivedefinition" on the OEBP.


An inductive definition is like, "Call certain basic objects 'blurps', and call combinations of blurps 'blurps' too."

CROSSREFS

Adjacent-numbered pages:
BP1123 BP1124 BP1125 BP1126 BP1127  *  BP1129 BP1130 BP1131 BP1132 BP1133

KEYWORD

meta (see left/right), links, keyword

WORLD

bp [smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

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