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BP1118 Self-similar only scaled about one point vs. multiple centers of self-similarity.
(edit; present; nest [left/right]; search; history)
COMMENTS

There is only ever one such center of self-similarity or infinitely many.

CROSSREFS

Adjacent-numbered pages:
BP1113 BP1114 BP1115 BP1116 BP1117  *  BP1119 BP1120 BP1121 BP1122 BP1123

KEYWORD

nice, perfect, infinitedetail

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

WORLD

[smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

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

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

BP1147 Columns of the table could be respectively labeled "Number" and "Number of times number appears in this table" vs. not so.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP1142 BP1143 BP1144 BP1145 BP1146  *  BP1148 BP1149 BP1150 BP1151 BP1152

KEYWORD

nice, precise, notso, handed, leftright, left-narrow, grid, preciseworld, left-listable, right-listable

CONCEPT self-reference (info | search)

AUTHOR

Leo Crabbe

BP1148 Number of dots in the Nth box (from the left) is how many times the number (N - 1) appears in the whole diagram vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

Left-sorted examples are sometimes called autobiographical or self-descriptive numbers.

REFERENCE

https://oeis.org/A349595

https://en.wikipedia.org/wiki/Self-descriptive_number

CROSSREFS

See BP1147 for a similar idea.

BP1149 was inspired by this.

Adjacent-numbered pages:
BP1143 BP1144 BP1145 BP1146 BP1147  *  BP1149 BP1150 BP1151 BP1152 BP1153

KEYWORD

nice, precise, unwordable, notso, handed, leftright, left-narrow, sequence, preciseworld, left-listable, right-listable

CONCEPT self-reference (info | search)

AUTHOR

Leo Crabbe

BP1149 Number in the Nth box (from the left) is how many numbers appear N times vs. not so.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Inspired by BP1148.

Adjacent-numbered pages:
BP1144 BP1145 BP1146 BP1147 BP1148  *  BP1150 BP1151 BP1152 BP1153 BP1154

KEYWORD

nice, precise, unwordable, notso, handed, leftright, left-narrow, sequence, preciseworld, left-listable, right-listable

CONCEPT self-reference (info | search)

AUTHOR

Aaron David Fairbanks

BP1241 Any point contained in (arbitrarily) smaller version of self vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

Note if any point is contained in some smaller version of the whole, then any point is contained in arbitrarily smaller versions of the whole.


It isn't possible to unambiguously communicate in a picture whether or not a few specific points are included in the fractal. The pictures are interpreted as what is intuitively simplest. To make matters less ambiguous, all the fractals here contain all points arbitrarily close to points in them. (They are topologically closed. See also BP1239.)


The left hand side of this is a stronger condition than the left hand side of BP1116.

CROSSREFS

Adjacent-numbered pages:
BP1236 BP1237 BP1238 BP1239 BP1240  *  BP1242 BP1243 BP1244 BP1245 BP1246

KEYWORD

notso, perfect, infinitedetail

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

WORLD

connected_fractal [smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP1246 Any symmetry exhibited by some non-empty subset of the objects is also a symmetry of the whole thing vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

In other words, placing the image over itself (rotation and flipping allowed) so that any parts match up makes the whole image match up to itself vs. not so.

CROSSREFS

See BP965 for a variation on this idea where the "parts" are allowed to be arbitrary regions of the image instead of individual objects shown in the image.

Adjacent-numbered pages:
BP1241 BP1242 BP1243 BP1244 BP1245  *  BP1247 BP1248 BP1249 BP1250 BP1251

KEYWORD

precise, allsorted, unwordable, notso, traditional

CONCEPT local_global (info | search),
self-reference (info | search),
symmetry (info | search)

AUTHOR

Aaron David Fairbanks

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