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A Dozen Unsolved Problems in Geometry Erich Friedman Stetson University 9/17/03

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1. The Hadwiger problem 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)? L(2)=5, as shown below.

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D. Hickerson showed L(3) = 47. The last partition to be found was the division into 54 smaller cubes, as shown to the right. Partitions into 49 and 51 cubes are also challenging. 1. The Hadwiger problem

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L(4)≤853 and L(5)≤1890. The best bound known (due to Erdös) is L(d)<(e-1)(2d) d. Smart $: L(d) is probably o(d d ). 1. The Hadwiger problem

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2. The Polygonal Illumination Problem 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?

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2. The Polygonal Illumination Problem It is conjectured that for every S and P, the answer is yes. No counterexample is known, but no one has a proof. Even this easier problem is open: Does every polygon S have any point P where a light source would illuminate the interior?

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2. The Polygonal Illumination Problem For non-polygonal regions, the conjecture is false, as shown by the example below. The top and bottom are elliptical arcs with foci shown, connected with some circular arcs.

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2. The Polygonal Illumination Problem There are continuously differentiable regions where an arbitrarily large number of light sources are necessary. To get a region requiring an infinite number of light sources, you need one non-differentiable point (J. Rauch). Smart $: The conjecture is true.

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3. The Penny Packing Problem How can n non-overlapping d-dimensional spheres be arranged to minimize the volume of their convex hull? (The convex hull is the set of all points on a line segment between points in two different spheres.)

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3. The Penny Packing Problem In 2 dimensions, the answers are clusters, or “hexagonal as possible”.

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3. The Penny Packing Problem In 3 dimensions, the answers for n≤56 are sausages, with the centers in a straight line. For d=3 and n≥57, the answers are clusters. For d=4, the answers are sausages for n up to somewhere between 50,000 and 100,000!

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3. The Penny Packing Problem The Sausage Conjecture: (F. Tóth) In dimensions 5 and higher, the optimal configuration is always a sausage. U. Betke, M. Henk, and J. Wills proved the sausage conjecture for d≥42 in Smart $: The conjecture is true.

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4. The Chromatic Number of the Plane What is the smallest number of colors with which we can color the plane so that no two points of the same color are distance 1 apart? 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.

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4. The Chromatic Number of the Plane The chromatic number of this unit distance graph (which is called the Moser spindle) is 4, so ≥4.

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4. The Chromatic Number of the Plane The plane can be colored with 7 colors to avoid unit pairs having the same color, so ≤7.

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4. The Chromatic Number of the Plane If the sets of points of a given color have to be measurable, ≥5. If the sets have to be closed, ≥6. Smart $: =7.

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5. Kissing Numbers In d dimensions, the kissing number K(d) is the maximum number of disjoint unit spheres that can touch a given sphere. K(2)= 6 K(3)=12.

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5. Kissing Numbers J. Conway and N. Sloane proved K(5)=40, K(6)=72, and K(7)=126 in A. Odlyzko and N. Sloane proved K(8)=240, and K(24)=196,560 in All other dimensions are still unsolved. Smart $: K(9)=306.

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6. Perfect Cuboids A perfect cuboid is a rectangular box whose sides, face diagonals, and space diagonals are all integers.

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6. Perfect Cuboids It is not known whether a perfect cuboid exists. Several near misses are known: a=240 b=117 c=44 d ab =267 d ac =244 d bc =125 a=672 b=153 c=104 d ac =680 d bc =185 d abc =697 a = b=√ c = 7800 d ab =23711 d ac =20280 d bc =16511 d abc =24961

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6. Perfect Cuboids If there is a perfect cuboid, it has been shown that the smallest side must be at least 2 32 = 4,294,967,296. Smart $: There is no perfect cuboid.

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7. Cutting Rectangles into Congruent Non-Rectangular Parts For which values of n is it possible to cut a rectangle into n equal non-rectangular parts? Using triangles, we can do this for all even n.

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7. Cutting Rectangles into Congruent Non-Rectangular Parts This is harder to do for odd n. Here are solutions for n=11 and n=15.

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7. Cutting Rectangles into Congruent Non-Rectangular Parts Trivially, there is no solution for n=1. Solutions are known for all other n except n=3, 5, 7, and 9, which remain open. What is true in higher dimensions? Smart $: There are no solutions for these n.

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8. Overlapping Congruent Shapes Let A and B be congruent overlapping rectangles with perimeters A P and B P. What are the best possible bounds for length(A B P ) R = ? length(A P B)

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8. Overlapping Congruent Shapes It is fairly easy to prove 1/4 ≤ R ≤ 4. It is conjectured that 1/3 ≤ R ≤ 3. Same ratio defined for triangles? It is conjectured that the best bounds for a triangle with smallest angle are sin( /2) ≤ R ≤ csc( /2).

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8. Overlapping Congruent Shapes In d dimensions, is the best upper bound on the ratio of (d-1)-dimensional surface area equal to 2d-1? Of course, for circles, R = 1. Smart $: 1/3 ≤ R ≤ 3.

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9. Distances Between Points If we have n points in the plane, they determine …+(n-1) distances. Can we arrange n points in general position so that one distance occurs once, one distance occurs twice, … and one distance occurs n-1 times? (General position means no 3 points on a line and no 4 points on a circle.)

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9. Distances Between Points This is easy to accomplish for small n. An example for n=4 is shown below.

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9. Distances Between Points Solutions are only known for n≤8. A solution (by I. Pilásti) for n=8 is shown to the right.

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9. Distances Between Points Is there a solution for n=9? Is there a solution for all integers n? Erdös offered $500 for a proof of “yes” and $50 for a proof of “no”. Very little has been done on the same problem in higher dimensions. Smart $: There is a solution for n=9, but not for large n.

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10. The Kabon Triangle Problem How many disjoint triangles can be created with n lines in the plane? The sequence K(n) starts 0, 0, 1, 2, 5, 7, 11, 15, 21,.… The optimal arrangements for n≤9 are shown on the next slide.

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10. The Kabon Triangle Problem How many disjoint triangles can be created with n lines in the plane?

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10. The Kabon Triangle Problem What is K(10)? How fast does K(n) grow? S. Tamura proved that K(n) ≤ n(n-2)/3. Smart $: This bound can be improved.

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11. Aperiodic Tiles A tiling of the plane is called periodic if it can be translated onto itself with two non-parallel translations.

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11. Aperiodic Tiles A set of tiles is called aperiodic if they tile the plane, but not in a periodic way. Even though a square can tile the plane in a non- periodic way, it is not aperiodic.

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11. Aperiodic Tiles In 1966, Berger produced the first set of 20,426 aperiodic tiles, which he soon lowered to 104 tiles. In 1968, D. Knuth discovered 92 tiles. Shortly thereafter, R. Robinson reduced this to 35 tiles, R. Penrose found a set of 34 tiles, and R. Ammann lowered to 16 tiles.

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11. Aperiodic Tiles In 1971, R. Robinson found this set of 6 aperiodic tiles based on notched squares.

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11. Aperiodic Tiles In 1974, R. Penrose found this set of 2 colored aperiodic tiles, now called Penrose Tiles.

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11. Aperiodic Tiles The coloring can be dispensed with if we notch these pieces.

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11. Aperiodic Tiles This is part of a tiling using Penrose Tiles.

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11. Aperiodic Tiles Is there a single tile which is aperiodic? There is a set of 3 convex (meaning no notches) aperiodic tiles. Are there 2? 1? In 3 dimensions, R. Ammann has found 2 aperiodic polyhedra, and L. Danzer has found 4 aperiodic tetrahedra. Smart $: No single aperiodic tile exists.

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12. Heesch’s Problem The Heesch number of a planar shape is the number of times it can be completely surrounded by copies of itself. For example, the shape to the right has Heesch number 1. What’s the largest finite Heesch number?

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12. Heesch’s Problem A hexagon with two external notches and 3 internal notches has Heesch number 4!

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The highest known Heesch number is Heesch’s Problem Smart $: There are higher ones.

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References V. Klee, Some Unsolved Problems in Plane Geometry, Math Mag. 52 (1979) H. Croft, K. Falconer, and R. Guy, Unsolved Problems in Geometry, Springer Verlag, New York, Eric Weisstein’s World of Mathematics, The Geometry Junkyard,

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