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# A Dozen Unsolved Problems in Geometry

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```A Dozen Unsolved Problems
in Geometry
Erich Friedman
Stetson University
9/17/03
1. The Hadwiger problem
вЂў In d-dimensions, define L(d) to be the
largest integer n for which a cube can not
be cut into n cubes. What is L(d)?
вЂў L(2)=5, as shown below.
1. The Hadwiger problem
вЂў D. Hickerson
showed L(3) = 47.
вЂў The last partition to be
found was the division
into 54 smaller cubes,
as shown to the right.
вЂў Partitions into 49 and 51
cubes are also challenging.
1. The Hadwiger problem
вЂў L(4)в‰¤853 and L(5)в‰¤1890.
вЂў The best bound known (due to ErdГ¶s) is
L(d)<(e-1)(2d)d.
вЂў Smart \$: L(d) is probably o(dd).
2. The Polygonal
Illumination Problem
вЂў Given a polygon S constructed with
mirrors as sides, and given a point P in
the interior of S,
we can ask
if the inside of S
will be completely
illuminated by a
light source at P?
2. The Polygonal
Illumination Problem
вЂў It is conjectured that for every S and P, the
answer is yes.
вЂў No counterexample is known, but no one
has a proof.
вЂў Even this easier problem is open: Does
every polygon S have any point P where a
light source would illuminate the interior?
2. The Polygonal
Illumination Problem
вЂў For non-polygonal regions, the conjecture
is false, as shown by the example below.
вЂў The top and bottom
are elliptical arcs
with foci shown,
connected with
some circular arcs.
2. The Polygonal
Illumination Problem
вЂў There are continuously differentiable
regions where an arbitrarily large number
of light sources are necessary.
вЂў To get a region requiring an infinite
number of light sources, you need one
non-differentiable point (J. Rauch).
вЂў Smart \$: The conjecture is true.
3. The Penny Packing Problem
вЂў How can n non-overlapping d-dimensional
spheres be arranged to minimize the
volume of their convex hull?
вЂў (The convex hull is the set of all points on
a line segment between points in two
different spheres.)
3. The Penny Packing Problem
вЂў In 2 dimensions, the answers are clusters,
or вЂњhexagonal as possibleвЂќ.
3. The Penny Packing Problem
вЂў In 3 dimensions, the answers for nв‰¤56 are
sausages, with the centers in a straight line.
вЂў For d=3 and nв‰Ґ57, the answers are clusters.
вЂў For d=4, the answers are sausages for n up to
somewhere between 50,000 and 100,000!
3. The Penny Packing Problem
вЂў The Sausage Conjecture: (F. TГіth)
In dimensions 5 and higher, the optimal
configuration is always a sausage.
вЂў U. Betke, M. Henk, and J. Wills proved
the sausage conjecture for dв‰Ґ42 in 1998.
вЂў Smart \$: The conjecture is true.
4. The Chromatic Number
of the Plane
вЂў What is the smallest number of colors c
with which we can color the plane so
that no two points of the same color are
distance 1 apart?
вЂў This is just the chromatic number of the
graph whose vertices are in the plane
and two vertices are connected if they
are unit distance from each other.
4. The Chromatic Number
of the Plane
вЂў The chromatic
number of this
unit distance
graph (which
is called the
Moser spindle)
is 4, so cв‰Ґ4.
4. The Chromatic Number
of the Plane
вЂў The plane can
be colored
with 7 colors
to avoid unit
pairs having
the same
color, so cв‰¤7.
4. The Chromatic Number
of the Plane
вЂў If the sets of points of a given color have
to be measurable, cв‰Ґ5.
вЂў If the sets have to be closed, cв‰Ґ6.
вЂў Smart \$: c=7.
5. Kissing Numbers
вЂў In d dimensions, the kissing number K(d)
is the maximum number of disjoint unit
spheres that can touch a given sphere.
вЂў K(2)= 6
вЂў K(3)=12.
5. Kissing Numbers
вЂў J. Conway and N. Sloane proved
K(5)=40, K(6)=72, and K(7)=126 in 1992.
вЂў A. Odlyzko and N. Sloane proved
K(8)=240, and K(24)=196,560 in 1979.
вЂў All other dimensions are still unsolved.
вЂў Smart \$: K(9)=306.
6. Perfect Cuboids
вЂў A perfect cuboid is a rectangular box
whose sides, face diagonals, and space
diagonals are all integers.
6. Perfect Cuboids
вЂў It is not known whether a perfect cuboid exists.
вЂў Several near misses are known:
a=240 b=117 c=44 dab=267 dac=244 dbc=125
a=672 b=153 c=104 dac=680 dbc=185 dabc=697
a = 18720 b=в€љ211773121 c = 7800
dab=23711 dac=20280 dbc=16511 dabc=24961
6. Perfect Cuboids
вЂў If there is a perfect cuboid, it has been
shown that the smallest side must be at
least 232 = 4,294,967,296.
вЂў Smart \$: There is no perfect cuboid.
7. Cutting Rectangles into
Congruent Non-Rectangular Parts
вЂў For which values of n is it possible to cut a
rectangle into n equal non-rectangular parts?
вЂў Using triangles, we can do this for all even n.
7. Cutting Rectangles into
Congruent Non-Rectangular Parts
вЂў This is harder to do for odd n.
вЂў Here are solutions for n=11 and n=15.
7. Cutting Rectangles into
Congruent Non-Rectangular Parts
вЂў Trivially, there is no solution for n=1.
вЂў Solutions are known for all other n except
n=3, 5, 7, and 9, which remain open.
вЂў What is true in higher dimensions?
вЂў Smart \$: There are no solutions for these n.
8. Overlapping Congruent Shapes
вЂў Let A and B be congruent overlapping
rectangles with perimeters AP and BP .
вЂў What are the best possible bounds for
length(Aпѓ‡BP )
RпЃї = ------------------ ?
length(AP пѓ‡B)
8. Overlapping Congruent Shapes
вЂў It is fairly easy to prove 1/4 в‰¤ RпЃї в‰¤ 4.
вЂў It is conjectured that 1/3 в‰¤ RпЃї в‰¤ 3.
вЂў Same ratio defined for triangles?
вЂў It is conjectured that the best bounds for a
triangle with smallest angle пЃ± are
sin(пЃ±/2) в‰¤ RпЃ„ в‰¤ csc(пЃ±/2).
8. Overlapping Congruent Shapes
вЂў In d dimensions, is the best upper bound
on the ratio of (d-1)-dimensional surface
area equal to 2d-1?
вЂў Of course, for circles, RпЃЏ = 1.
вЂў Smart \$: 1/3 в‰¤ RпЃї в‰¤ 3.
9. Distances Between Points
вЂў If we have n points in the plane, they
determine 1+2+3+вЂ¦+(n-1) distances.
вЂў Can we arrange n points in general
position so that one distance occurs once,
one distance occurs twice, вЂ¦ and one
distance occurs n-1 times?
вЂў (General position means no 3 points on a
line and no 4 points on a circle.)
9. Distances Between Points
вЂў This is easy to accomplish for small n.
вЂў An example for n=4 is shown below.
9. Distances Between Points
вЂў Solutions are
only known
for nв‰¤8.
вЂў A solution (by
I. PilГЎsti) for
n=8 is shown
to the right.
9. Distances Between Points
вЂў Is there a solution for n=9?
вЂў Is there a solution for all integers n?
вЂў ErdГ¶s offered \$500 for a proof of вЂњyesвЂќ and
\$50 for a proof of вЂњnoвЂќ.
вЂў Very little has been done on the same
problem in higher dimensions.
вЂў Smart \$: There is a solution for n=9, but not
for large n.
10. The Kabon Triangle Problem
вЂў How many disjoint triangles can be
created with n lines in the plane?
вЂў The sequence K(n) starts
0, 0, 1, 2, 5, 7, 11, 15, 21, .вЂ¦
вЂў The optimal arrangements for nв‰¤9 are
shown on the next slide.
10. The Kabon Triangle Problem
вЂў How many disjoint triangles can be
created with n lines in the plane?
10. The Kabon Triangle Problem
вЂў What is K(10)?
вЂў How fast does K(n) grow?
вЂў S. Tamura proved that K(n) в‰¤ n(n-2)/3.
вЂў Smart \$: This bound can be improved.
11. Aperiodic Tiles
вЂў A tiling of the plane is called periodic if
it can be translated onto itself with two
non-parallel translations.
11. Aperiodic Tiles
вЂў A set of tiles is called aperiodic if they
tile the plane, but not in a periodic way.
вЂў Even though a
square can tile the
plane in a nonperiodic way, it is
not aperiodic.
11. Aperiodic Tiles
вЂў In 1966, Berger produced the first set of
20,426 aperiodic tiles, which he soon
lowered to 104 tiles.
вЂў In 1968, D. Knuth discovered 92 tiles.
вЂў Shortly thereafter, R. Robinson reduced
this to 35 tiles, R. Penrose found a set of 34
tiles, and R. Ammann lowered to 16 tiles.
11. Aperiodic Tiles
вЂў In 1971, R. Robinson found this set of 6
aperiodic tiles based on notched squares.
11. Aperiodic Tiles
вЂў In 1974, R. Penrose found this set of 2 colored
aperiodic tiles, now called Penrose Tiles.
11. Aperiodic Tiles
вЂў The coloring can be dispensed with if we
notch these pieces.
11. Aperiodic Tiles
вЂў This is part of a tiling using Penrose Tiles.
11. Aperiodic Tiles
вЂў Is there a single tile which is aperiodic?
вЂў There is a set of 3 convex (meaning no
notches) aperiodic tiles. Are there 2? 1?
вЂў In 3 dimensions, R. Ammann has found 2
aperiodic polyhedra, and L. Danzer has
found 4 aperiodic tetrahedra.
вЂў Smart \$: No single aperiodic tile exists.
12. HeeschвЂ™s Problem
вЂў The Heesch number of a planar shape is
the number of times it can be completely
surrounded by copies of itself.
вЂў For example, the
shape to the right has
Heesch number 1.
вЂў WhatвЂ™s the largest
finite Heesch number?
12. HeeschвЂ™s Problem
вЂў A hexagon
with two
external
notches and
3 internal
notches has
Heesch
number 4!
12. HeeschвЂ™s
вЂў The
highest
known
Heesch
number
is 5.
вЂў Smart \$:
There are
higher ones.
Problem
References
вЂў V. Klee, Some Unsolved Problems in Plane
Geometry, Math Mag. 52 (1979) 131-145.
вЂў H. Croft, K. Falconer, and R. Guy, Unsolved
Problems in Geometry, Springer Verlag, New
York, 1991.
вЂў Eric WeissteinвЂ™s World of Mathematics,
http://mathworld.wolfram.com/.
вЂў The Geometry Junkyard,
http://www.ics.uci.edu/~eppstein/junkyard/.
```
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