Here is a challenging problem for anyone with an interest in geometry. This project requires background research to solve it, but it is an excellent illustration of visual thinking in mathematics.
Figure 1 below shows a series of circles (iC₁, iC₂, iC₃, ..., iC₃₀), inscribed inside an arbelos. What is an arbelos? The arbelos is the white region in the figure, bounded by three semicircles. The diameters of the three semicircles are all on the same line segment, AC,…
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Average (6-10 days)
Good grasp of Euclidean geometry, a firm understanding of how to construct a mathematical proof, determination
Math can make you money! If you understand some basic math, you can make good decisions about how to keep, spend, and use your hard earned dollars. Try an experiment comparing the same balance in different types of bank accounts. How much better is a savings account than a checking account? What difference does the interest rate make? Which is better, an account that earns compound or simple interest? Can you compare the short and long term costs of borrowing money compared to saving the cash…
A magic square is an arrangement of numbers from 1 to n2 in an n x n matrix. In a magic square each number occurs exactly once such that the sum of the entries of any row, column, or main diagonal is the same. You can make several magic squares and investigate the different properties of the square. Can you make an algorithm for constructing a Magic Square? Can you show that the sum of the entries of any row, column, or main diagonal must be n(n2+1)/2? Are there any other hidden properties of a…
If you've played catch with both Aerobie flying rings and Frisbees, you know that the rings fly much further than the Frisbees with the same throwing effort. Why is that? Investigate the aerodynamics of flying rings and flying disks and find out!
A fractal is, "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole" (Mandelbrot, 1982). There are many different fractal patterns, each with unique properties and typically named after the mathematician who discovered it. A fractal increases in complexity as it is generated through repeated sets of numbers called iterations. There are many interesting projects exploring fractal geometry that go beyond…
How do you turn a 2-dimensional piece of paper into a 3-dimensional work of art? Origami, the classical art of Japanese paper folding, is loaded with mathematical themes and concepts. What are the common folds in origami, and how do they combine to create 3-dimensional structure? Can you classify different types of origami into classes based upon the types of folds they use? Can you show Kawasaki's Theorem, that if you add up the angle measurements of every other angle around a point, the sum…
For this project, you'll use a baseball as a pendulum weight, studying the motion of the ball with and without spin. Wrap a rubber band around the ball, and tie a string to the rubber band. Fasten the string so that the ball hangs down and can swing freely. Mark a regular grid on cardboard, and place it directly beneath the ball to measure the motion. You can also time the oscillations with a stopwatch. Lift the ball along one of the grid axes, and let it go. Observe the motion and record…
Here's a puzzle you may have heard before which you can build as a simple electric circuit. First, the puzzle: a farmer is traveling to market with his cat, a chicken and some corn. He has to cross a river, and the only way to cross is in a small boat which can hold the farmer and just one of the three items he has with him. The problem is, he has to be very careful about what he chooses to leave behind at any time. If the cat and chicken are left alone, the cat will eat the chicken. If…
How much difference does the spiraling motion of a well-thrown football make on the distance of the throw (compared to wobbling, or end-over-end motion of the ball)? Think of a way to reproducibly produce the desired ball motion and launch it with a constant force to find out. (For more information on the physics, see Gay, 2004.)
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