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# Geometry

1. Pool problems: if you have a rectangular table without friction and send a pool ball at an angle , will it return to the same spot? Investigate using a diagram in Sketchpad (or Cabri). If it does not return to the same spot, will it pass over all points on the table? Does the answer depend on the dimensions of the table? Make a sketch in which you can change the dimensions of the table and the direction of the ball, and explore the path through 10 or 20 bounces. What happens on a circular pool table? Make a dynamic geometry sketch.
2. Flatland and sphereland. If you lived in flatland (the plane) could you build a bicycle which exists in the plane and works? Could you do the same on the sphere? Explore other ``machines'' in a flat space. References: [Dew], [Hin]. There are good descriptions of the problem in [Gard1], [Gard2].
3. There are many aspects of spherical geometry that could be investigated.

Explore congruences of triangles on a sphere. Other useful tools that are also available are a plastic sphere, with hemispherical ``overhead transparencies'', great circle ruler, compass etc. One can also make very effective models with plastic spheres from a craft shop and cut-off plastic containers for rulers.

Explore quadrilaterals and their symmetries on a sphere. Is there a family which shares most of the properties of a parallelogram? What symmetry do they have? Which two properties (e.g. opposite angles equal) are sufficient to prove all the other properties?

4. What equalities of lengths and angles are sufficient to prove two sets of four points (quadrilaterals or quadrangles...) are congruent? (Leads directly to unsolved research problems in Computer Aided Design.) For further references contact
whiteley@mathstat.yorku.ca.)
5. Build models showing that parallelograms with the same base and height have the same areas. (Is there a 3-dimensional analogue?) This can lead to a purely visual proof of the Pythagorean theorem, using a physical model based on dissections. The formula for the area of a circle can also be presented in this way, by building an exhibit on the Pythagorean theorem but with ``The area of the semicircle on the hypotenuse is equal to the sum of the areas of the semicircles on the other two sides.'' Reference: [Jac].
6. Study the regular solids (platonic and Archimidean), their properties, geometries, and occurrences in nature (e.g. virus shapes, fullerene molecules, crystals). Build models. References: [Gard2], Volume 2 of [Gard3], [Jac].
7. Consider tiling the plane using shapes of the same size. What's possible and what isn't? In particular it can be shown that any 4-sided shape can tile the plane. What about 5 sides? Make sketches in a geometry program (Sketchpad, Cabri, or using Kali (available free from the Geometry Center, or Reptiles: demo version available at the Math Forum at Swarthmore - these can be found at web sites.) References: [G&S], [Stei]. Check the Martin Gardner books.
8. Draw, and list any interesting properties of various curves: evolutes, involutes, roulettes, pedal curves, conchoids, cissoids, strophoids, caustics, spirals, ovals, ... References: [C&R] (which has lots of other ideas too), [Lock].
9. Make a family of polyhedra, e.g., the Archimidean solids, or Deltahedra (whose faces are all equilateral triangles), or equilateral zonohedra, or, for the very ambitious, the 59 Isocahedra. References: [Bal] (which is full of many ideas), [CDF&P], [Wen], [S&W], [S&F].

What polyhedral shapes make fair `dice'? What are the physical properties? What are the geometric properties? What is the root of the word ``polyhedra'' (and why does this fit with the use as dice?) Can you list all possible shapes? What numbers of faces can appear? What other (non-polyhedral) shapes are actually used in games?

What polyhedral shapes appear in crystals? List them all. Why do these appear? Why don't other shapes appear? What is the connection between the big outside shape and the inside ``connections of molecules''? Reference: [Sen]

10. What is Morley's triangle? Draw a picture of the 18 Morley triangles associated with a given triangle ABC. Find the 18 more for each of the triangles BHC, CHA, AHB, where H is the orthocentre of ABC. Discover the relation with the 9-point circle and deltoid (envelope of the Simson or Wallace line).
11. Investigate compass and straight-edge constructions - showing what's possible and discussing what's not. For example, given a line segment of length one can you use the straight edge and compass to ``construct'' all the radicals? Investigate constructions using origami (paper folding). Can you construct all figures that are constructed with ruler and compass? Can you construct more figures? References can be found in articles in Math Monthly, Math Magazine.
12. The cycloid curve is the curve traced by a point on the edge of a rolling wheel. Study its tautochrone and brachistochrone properties and its history. Build models. Suppose all cars had square wheels. How would you design the road so that you always had a smooth ride? What about other wheel shapes? Reference: [Wag].
13. Find as many triangles as you can with integer sides and a simple linear relation between the angles. What about the special case when the triangle is right-angled?
14. What is a hexaflexagon? Make as many different ones as you can. What is going on? Reference: [Gard4], Volume 1 of [Gard3].
15. A kaleidoscope is basically two mirrors at an angle of or to each other. When an object is placed between the mirrors, it is reflected 6 or 8 times (depending on the angle). Construct one. Investigate its history and the mathematics of symmetry. Make models of kaleidoscopes in a dynamic geometry program (Cabri or Geometers Sketchpad). Demonstrate why only certain angles work. References: [Bal], [Hod].
16. You make a tangram puzzle by diving a 2- or 3-dimension object into many geometrical pieces, so that the original object can be reconstructed in more than one way. Burr puzzles are interlocking assemblies of notched sticks. For example, there are Burr puzzles that look like spheres or barrels when they are completed. See [Cof] for information on how to construct your own.
17. Build rigid and non-rigid geometric structures. Explore them. Where are rigid structures used? Find unusual applications. This could include an illustration of the fact that the midpoints of the sides of a quadrilateral form a parallelogram (even when the quadrilateral is not planar). Are there similar things in three dimensions? Are there plane frameworks (rigid bars and flexible joints) that are rigid but contain no triangles? Are all triangulated spheres rigid (either made of sticks and joints or of hinged plastic pieces ``Polydron''). What is the formula for the number of bars in a triangulated sphere, in terms of the number of vertices? How does this formula relate to other rigid frameworks in 3-space?

Consider a plane ``grid'' composed of squares (say 4 squares by four squares) made of bars and joints. Which diagonals of squares will make this rigid? What is the minimum number? Can you give a recipe for deciding which diagonals will work? [There is a COMAP module related to this problem.] If the grid is composed of a trapezoid and its image after a half turn, alternating, does the same recipe work? [This is a research problem which has NOT been thoroughly worked out! whiteley@mathstat.yorku.ca]

18. The Art Gallery problem: What is the least number of guards required to watch over all paintings in an art gallery? The guards are positioned at specific locations and collectively must have a direct line of sight to every point on the walls. References: [Tuc], [Wag].
19. The Parabolic Reflector Microphone is used at sporting events when you want to be able to hear one person in a noisy area. Investigate this, explaining the mathematics behind what is happening.

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