I came up with many of these problems based on your research projects, and pulled some from our more recent work in Stillwell. Do any 2, but pull them from different sections. The other usual rules and expectations apply. Revisit the problem solving template if needed. This is due 12-2.
Fractals
- What are the various camera moves that you see in a fractal video, and what are the mathematical transformations that define them?
- The group showed us what the Mandelbrot set and Sierpinski triangle really are. Find another fractal and share with us the true details of its definition. What else can you easily show or tell us about it?
Music
- Create a system of tones within an octave, starting with a single tone. What are the benefits and drawbacks of your construction compared to another?
- Show that fifths can never add up to an octave.
- With musical frequency, what sounds to the ear like adding is in reality multiplying. Besides musical frequency, what’s another system of perception whose base phenomenon follows a multiplicative rule but whose perception is additive in nature?
Topology
- What’s a shape you can draw on a torus that you can’t draw on a plane?
- What’s another weird shape that you can make using the “identification and gluing edges” strategy showed in the presentation? What’s different about this shape from a plane?
- How do you find an inscribed square in a square? an ellipse?
- If you have two diagonals that are equal, how do you know that you have a rectangle?
- Explain “The surface which represents all pairs of points on the curve is the Möbius strip”
Geometric constructions
- Before considering the problem of doubling the cube using a compass and straight-edge, it helps to know what all you can make and how. We saw for instance that you can make $\sqrt 2$ by drawing a simple diagonal of a unit square.
At the time we didn’t stress the details, but this really does connect back to the very simple abilities you start with (drawing circles with a given radius, drawing lines between two points, and extending those lines indefinitely). Early on, you show that you can make perpendiculars, and that you can copy line segments and radii elsewhere.
So given two line segments for $x$ and $y$, it is not too hard to make $x+y$. you just make $x$ a bunch longer, then use the compass to copy y to the extended version of $x$.
Minus works about the same. And times as long as your doing whole multiples. But what about division? How would you construct something like 1/5, given 1? It is possible, and discoverable, but not immediately obvious.
Can you do this?
What about arbitrary multiplication? Given lengths x and y, construct $x \cdot y$.
- What are the rules of origami construction and how do you use them to construct the cube root of 2?
- Supposing you can construct a parabola, how do you construct the cube root of 2? Who first did this?
Differential Equations
- What is the most general function that solves the differential equation
$y^{\prime \prime }=-k^2y$?
Take some functions you know and love (and can find the derivatives of) and reimagine them as solutions to differential equations. Does seeing them from this perspective give you a sense for their shape?
Infinitesimals
- Derive the formula for the area of a circle using the formula for its circumference and an exhaustion argument based on inscribed polygons of increasing numbers of sides.
- What is the history of our ability to approximate $\pi$?
- Derive a formula for the volume of a tetrahedron.
Factoring and Primes
- Show that if a prime $p$ has remainder 3 when divided by 4, i.e. $p \equiv 3 \text{ (mod 4)}$, then $p$ remains prime in the Gaussian integers, i.e. $p$ is not a sum of squares of regular integers.
- How did Euler prove Fermat’s claim that 27 is the only cube to exceed a square by 2?
- Show that there are no rational sided right triangles whose angle is a rational multiple of $\pi$.
