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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

BP854 Nothing vs. nothing.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP849 BP850 BP851 BP852 BP853  *  BP855 BP856 BP857 BP858 BP859

KEYWORD

left-finite, right-finite, left-full, right-full, finished, experimental, funny

WORLD

nothing [same | bigger]
zoom in left (nothing) | zoom in right (nothing)

AUTHOR

Aaron David Fairbanks

BP897 Wide angles connected to narrow angles vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

Another solution is that right examples can be folded down flat onto one isosceles triangle while left examples cannot.

All examples in this Problem feature four isosceles triangles connected by corners and/or edges.

CROSSREFS

This was conceived as a false solution for BP898.

Adjacent-numbered pages:
BP892 BP893 BP894 BP895 BP896  *  BP898 BP899 BP900 BP901 BP902

KEYWORD

precise, allsorted, notso, traditional, preciseworld

CONCEPT triangle (info | search)

WORLD

[smaller | same | bigger]

AUTHOR

Molly C Klenzak, Aaron David Fairbanks

BP898 Can fold into tetragonal disphenoid ("isosceles tetrahedron") vs. cannot.
(edit; present; nest [left/right]; search; history)
COMMENTS

Which two sides are the long sides and which side is the short side, or equivalently which angles are the wider angles and which angle is the narrower angle, is the only relevant information to consider for each triangle. Triangles are all assumed isosceles and congruent to one another.


All examples in this Problem feature four of these triangles connected by corners and/or edges.

CROSSREFS

BP897 was conceived as a false solution for this.

Adjacent-numbered pages:
BP893 BP894 BP895 BP896 BP897  *  BP899 BP900 BP901 BP902 BP903

KEYWORD

hard, precise, allsorted, notso, math, preciseworld

CONCEPT triangle (info | search)

WORLD

[smaller | same | bigger]

AUTHOR

Molly C Klenzak

BP899 Regions in drawing (ignore background) can be coloured using three or fewer colours such that no adjacent regions are coloured the same colour vs. four colours are required.
(edit; present; nest [left/right]; search; history)
COMMENTS

This Problem is related to the four colour theorem.

CROSSREFS

Adjacent-numbered pages:
BP894 BP895 BP896 BP897 BP898  *  BP900 BP901 BP902 BP903 BP904

KEYWORD

hard, nice, math

CONCEPT separated_regions (info | search)

WORLD

[smaller | same | bigger]

AUTHOR

Jago Collins

BP905 Graph can be redrawn such that no edges intersect vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

A graph is a collection of vertices and edges. Vertices are the dots and edges are the lines that connect the dots. On the left, all edges can be redrawn (curved lines are allowed and moving vertices is allowed) such that no edges cross each other and each vertex is still connected to the same other vertices. These graphs are called planar.

CROSSREFS

Adjacent-numbered pages:
BP900 BP901 BP902 BP903 BP904  *  BP906 BP907 BP908 BP909 BP910

KEYWORD

nice, precise, allsorted, notso, math, left-null, preciseworld

CONCEPT graph (info | search),
topological_transformation (info | search)

WORLD

connected_graph [smaller | same | bigger]

AUTHOR

Molly C Klenzak

BP924 Polygons where all sides are different lengths vs. Polygons where not all sides are different lengths.
(edit; present; nest [left/right]; search; history)
COMMENTS

All examples in this Problem are outlines of convex polygons.

This is a generalisation of scalene triangles to any polygon.

CROSSREFS

The left side implies the right side of BP329 (regular vs. irregular polygons), but the converse is not true.

The left side of BP329 implies the right side, but the converse is not true.

Adjacent-numbered pages:
BP919 BP920 BP921 BP922 BP923  *  BP925 BP926 BP927 BP928 BP929

EXAMPLE

Any scalene triangle will fit on the left, because no two sides are equal.

However, any regular polygon will not fit on the left, because all of its sides are equal.

A random convex polygon will "almost surely" fit on the left.

KEYWORD

nice, stretch, right-narrow, traditional

CONCEPT all (info | search)

WORLD

polygon_outline [smaller | same | bigger]

AUTHOR

Jago Collins

BP932 Every vertex is connected to every other vs. vertices are connected in a cycle (no other connections).
?
?
(edit; present; nest [left/right]; search; history)
COMMENTS

Complete graphs with zero, one, two, or three vertices would be ambiguously categorized (fit in overlap of both sides).


Left examples are called "fully connected graphs." Right examples are called "cycle graphs."

CROSSREFS

Adjacent-numbered pages:
BP927 BP928 BP929 BP930 BP931  *  BP933 BP934 BP935 BP936 BP937

KEYWORD

precise, left-narrow, right-narrow, both, preciseworld

CONCEPT graph (info | search),
distinguishing_crossing_curves (info | search),
all (info | search),
loop (info | search)

WORLD

connected_graph [smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP1016 Rigid vs. not rigid.
(edit; present; nest [left/right]; search; history)
REFERENCE

Henneberg, L. (1911), Die graphische Statik der starren Systeme, Leipzig

Jackson, Bill. (2007). Notes on the Rigidity of Graphs.

Laman, Gerard. (1970), "On graphs and the rigidity of plane skeletal structures", J. Engineering Mathematics, 4 (4): 331–340.

Pollaczek‐Geiringer, Hilda (1927), "Über die Gliederung ebener Fachwerke", Zeitschrift für Angewandte Mathematik und Mechanik, 7 (1): 58–72.

CROSSREFS

Adjacent-numbered pages:
BP1011 BP1012 BP1013 BP1014 BP1015  *  BP1017 BP1018 BP1019 BP1020 BP1021

KEYWORD

nice, physics, help

CONCEPT rigidity (info | search),
graph (info | search),
imagined_motion (info | search)

WORLD

planar_connected_graph [smaller | same | bigger]
zoom in left (rigid_planar_connected_graph)

AUTHOR

Aaron David Fairbanks

BP1099 Considering only the ways they are connected, anything that can be said about a given node can be said about every other node vs. not so.
(edit; present; nest [left/right]; search; history)
REFERENCE

https://en.wikipedia.org/wiki/Vertex-transitive_graph

CROSSREFS

Adjacent-numbered pages:
BP1094 BP1095 BP1096 BP1097 BP1098  *  BP1100 BP1101 BP1102 BP1103 BP1104

KEYWORD

precise, allsorted, notso, math, preciseworld

CONCEPT graph (info | search),
distinguishing_crossing_curves (info | search)

WORLD

graph [smaller | same | bigger]
zoom in left

AUTHOR

Leo Crabbe

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