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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