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

BP812 Aesthetically pleasing vs. not so.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP807 BP808 BP809 BP810 BP811  *  BP813 BP814 BP815 BP816 BP817

KEYWORD

easy, fuzzy, abstract, notso, stretch, anticomputer, subjective, invalid, experimental, funny, dithering

AUTHOR

Aaron David Fairbanks

BP813 Representations of natural mathematical objects vs. representations of arbitrary objects.
(edit; present; nest [left/right]; search; history)
COMMENTS

This is a very fuzzy definition. Some left examples arguably should be placed on the right, since the particular way they are represented is arbitrary--the Platonic solids EX6730 and primes EX6734 especially, as these show arbitrary placement and arrangement of objects. Furthermore if arbitrary representations are allowed one cannot be sure for example the right hand drawing of random numbers EX6740 does not represent "numbers" in general. Still this Bongard Problem has been solved by people.

CROSSREFS

Adjacent-numbered pages:
BP808 BP809 BP810 BP811 BP812  *  BP814 BP815 BP816 BP817 BP818

KEYWORD

fuzzy, abstract, stretch, math, solved, collective, experimental, dithering

AUTHOR

Aaron David Fairbanks

BP816 Cross section of a cylinder vs. not cross section of a cylinder
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP811 BP812 BP813 BP814 BP815  *  BP817 BP818 BP819 BP820 BP821

KEYWORD

precise, notso, stretch, unstable, perfect

CONCEPT cross_section (info | search)

WORLD

fill_shapes [smaller | same | bigger]

AUTHOR

Leo Crabbe

BP818 Dot's position within square is center of square's position within panel vs. not so.
(edit; present; nest [left/right]; search; history)
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.
(edit; present; nest [left/right]; search; history)
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

BP822 Two drawn polyhedra are duals vs. not so.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP817 BP818 BP819 BP820 BP821  *  BP823 BP824 BP825 BP826 BP827

KEYWORD

math, 3d, unorderedpair

AUTHOR

Aaron David Fairbanks

BP823 Conic section (plot of solution to conic equation) vs. not so.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP818 BP819 BP820 BP821 BP822  *  BP824 BP825 BP826 BP827 BP828

KEYWORD

notso, math, left-couldbe

AUTHOR

Aaron David Fairbanks

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