MATHEMATICAL SURFACES AND ART
There’s more to the world of surfaces than spheres and paraboloids. Many beautiful and exotic surfaces have been created using various tools from multivariable calculus and beyond. Sometimes these surfaces are created for the sake of art, and other times they serve some distinct application. Examples include:
- Boy’s surface, an attempt to squeeze a mathematical space known as the projective plane into 3d space. You can even create a model from Legos.
- The Invisible Handshake is a mathematical sculpture located right outside Olin Rice. It was built by Helaman Ferguson. It is not just a fanciful shape, it has defining formulas etched into its surface (go take a look!). Its design is based on a “minimal surface” known as a Costa surface.
- Seifert surfaces are a general kind of surface constructed from knots or links.
Pick a particular surface and explore its mathematical properties and construction. Try making a model using Sculpey clay and/or other materials.
In 1569, G. Mercator (1512-1594) created a new map that had a significant impact on navigation. Until then, as sailors navigated the open ocean by following a fixed compass bearing, they could not map a straight line from point A to point B that would correspond to a path of constant compass bearing on the earth’s surface (except for a few special cases). Mercator’s accomplishment allowed navigators to chart paths of constant bearing, rhumb lines or loxodromes, between any two points on the map. You could explore this topic, and address issues such as these:
- Construct the Mercator projection,
- Determine the rhumb heading from one location to another
- Compute the rhumb distance between two locations on the earth.
There are many good resources, but you could start with this treatment of the third question. You can look at this link for the basics about the Mercator projection.
Mathematical crochet and knitting
Are you crafty? Do you like to work with yarn and needles? Thankfully, Cornell professor Daina Taimina discovered a way that you can explore ideas from geometry and multivariable calculus while practicing your craft. She invented techniques for crocheting surfaces of negative curvature, which are a bit like saddles, except cooler and harder to create. She gave a TED talk about it here.
In this project you could explore the mathematical definition of negative curvature and create some examples in crochet. See this website or this blog post for lots of good links and info.
Planimeters and mechanical integration
A planimeter is a measuring device that computes the areas of arbitrary shapes. In a way, it is a physical manifestation of Green’s theorem; tracing around the perimeter of the object allows the planimeter to perform a mechanical computation based on Green’s theorem and compute the area enclosed by this perimeter. There are several different kinds of planimeters. You can even make a “Prytz” planimeter with a spoon. A polar planimeter is available to borrow from your instructor. You can see how it works in this Mathematica demo.
In this project you could build a linear or a “Prytz” planimeter, and explain mathematically why it works. You could also explore “planimeter proofs”, which are ways of using a planimeter to establish interesting mathematical ideas, such as the isoperimetric inequality.
As you might know, there is a 3d printer lurking in the department. This project would be a journey from mathematical concept to a 3d printed object. That is, you would design something (either artistic or functional) using principles of multivariable calculus, create a model of that object using modeling software such as Sketchup, and print the object. You could also upload your object to Thingiverse when it’s done. Some ideas for things you could design include:
- A strangely shaped wine holder or other variation on this idea.
- Gears to create a mechanism for a certain type of motion (see Spirographs project below, for example).
- Constant diameter solids. There is a set of these in the MSCS Department office for you to see. See also this link.
Harmonographs, spirographs and strange mechanisms
Many interesting parameterized curves can be created by combining several different types of motion. Here are some examples to explore:
- The “road” for the square wheeled bike, called a catenary curve. This curve is obtained by rolling a square with its center at fixed height.
- Combinations of different wheels in a spirograph.
- A harmonograph uses pendulum motion to create a parameterized curve. A student built harmonograph that you can experiment with is hanging out in OLRI 205.
- Using fixed shapes on a rotating gear, you can create arbitrary horizontal and vertical motions. This means you can use gears to draw practically any picture you want. If you design these gears, they could be 3d printed.
What do you get when you intersect, at right angles, two cylinders of equal radius? What about three? Such solids are called Steinmetz Solids.
In this project you would figure out the volume of these, and you could also make them out of clay, or some other suitable material.
In class we began a study of infinite series. There is so much more to learn here! In this project you could learn the material of text sections of Chapter 9. You could read Bill Dunham’s Journey Through Genius Chapter 9 on “The Extraordinary Sums of Leonhard Euler.”
When two geometric patterns are superimposed, additional geometric patterns may become visible. These are common in the “Op Art” posters of the 1960s.
Read the paper Moire Fringes and the Conic Sections (M. R. Cullen, The College Mathematics Journal, Vol. 21, No. 5 (Nov., 1990), pp. 370-378), available via JSTOR. A Google search for “moire pattern math” will lead to lots more.
Explain how multivariable calculus explains these visual effects, and create your own!
Lowest bid auctions
Alice, Bob, and Charlie are chosen to participate in an auction for an economics class. Each submits a secret bid of 1, 2, or 3. The person with the lowest unique bid wins the auction and gets a prize of $1. For example: If Alice and Bob bid 1, and Charlie bids 2, then Charlie wins. Although Alice and Bob bid lower numbers, their bids were not unique so they do not win. Also, if they all bid the same number, no one wins. If each plays the same mixed strategy, what is the optimal strategy? (what is the symmetric mixed Nash equilibrium?)
To solve this problem, you’ll need to study a little bit of game theory. There’s a great youtube video series to get you started. The solution to the problem above can also be found here, but I very strongly recommend trying the problem yourself before reading the solution.
Dogs and Rabbits
A rabbit is running around (perhaps following a path given by a parameterized curve). A dog is chasing the rabbit, simply by running directly at the rabbit at any given moment. What sorts of paths will be traced out by the dog, given a specific rabbit path? This project would consist of writing down the equations that define the motion of the dog and using Mathematica to solve these equations and plot the paths of both rabbit and dog. You can explore defining different “escape” paths for the rabbit, and try to deduce the circumstances under which escape is possible. There are many fun possibilities to explore, and you might encounter many beautiful curves (Google image search “pursuit curves” for examples).
The prisoner problem
Consider the following puzzle. A prisoner is presented with two bowls, one with 50 white beads and one with 50 black beads. He is instructed to put some beads from the first bowl into the second, and some beads from the second bowl into the first. He is then blindfolded, and his captors stir the bowls and switch them around so that he is unaware of which bowl is which. He is asked to select a bowl, and then a single bead from that bowl. A white bead means he is set free, and a black bead means he will remain in prison. What kind of strategy should the prisoner adopt to obtain a better than 50/50 chance of freedom? What strategy should he adopt for the best chance at freedom? How can calculus be used to solve this problem? Can you generalize this problem with things like more urns, beads, colors, etc?
n dimensional spheres
The area of a disk is πr2. The volume of a sphere is 4πr3. What is the volume of a sphere in four dimensions? What about in five dimensions? What about n dimensions? Consider volumes of radius 1 spheres in 2, 3, 4, 5, 6, . . . dimensions. A very curious thing happens to these volumes as the dimensions get larger: the volumes get smaller!
This project could begin by exploring the definition of a sphere in 4 or more dimensions. How can we visualize such a thing? You could then explain how to compute the volume of an n dimensional sphere. (“Generalized spherical coordinates” may be helpful here.) Finally, is there any explanation for the strange behavior of the volumes as dimension increases?
Multivariable calculus was born to solve physics problems. Applying multivariable calc to an interesting physics problem would make a good project. Here are a few suggestions for interesting problems.
- Suppose that a very long drill is used to drill a straight into the ground until is comes out someplace else on the surface of the earth (not necessarily on the exact opposite side if the hole is not straight down). If I then drop an object into that hole, will it ever reach the other opening? If so, how long will it take?
- Suppose that we have two planets of equal mass, but one has a hollow core. Would the gravitational fields of these planets be the same? What would a person in the middle of the hollow planet experience?
- Before calculus, planetary motion was understood through Kepler’s laws. What are these laws, and how can they be explained using multivariable calculus?
- Explain Gauss’ Law and Coulomb’s Law and some of their consequences. (See p. 1045 in the text for example.)
- Find your own problem in mechanics, electrodynamics or quantum mechanics and show us how multivariable calculus saves the day.
Partial differential equations
A partial differential equation is an equation involving an unknown multivariable function and its partial derivatives. Pick a simple partial differential equation (examples include the heat equation, advection equation, wave equation or Laplace equation) and explain some of its properties and why it might be important to study. Find an interesting application of your chosen partial differential equation and explain it. You might want to use Mathematica to solve these equations for various boundary or initial conditions. Some very important (and cool) partial differential equations are:
- The heat equation, which describes diffusion of heat in a solid object or a pollutant in water.
- The wave equation, which describes the movement of sound and water waves.
- Laplace’s equation, which describes potentials in physical theories.
- Schrodinger equation, which describes the motion of quantum particles.
Soap bubbles and minimal surfaces
Investigate the mathematics of soap bubbles. There is a fascinating connection between soap bubbles and minimal surfaces (surfaces which have a minimal area relative to their boundaries). You can use wire and soap to make some of your own.
Choose a calculus topic and explore its history. Below are a few example topics.
- The history and development of the use of vectors.
- Lagrange multipliers and constrained optimization.
- The mathematical work of a prominent figure in multivariable calculus, such as Euler, Gauss, Green, Lagrange.
You’re the teacher now!
It is often said that teaching is one of the best ways of learning. For this project, you should choose a particular topic from multivariable calculus and design your ideal “lesson” to teach this subject. This lesson could consist of a video lecture, some examples and exercises, a helpful visual aid, and even a Mathematica activity.
Choose your own adventure
Create a topic of your own choosing. You can scour the internet for topics, or look in the project sections in the textbook. One idea would be to examine how multivariable calculus can be applied to a topic important to your major.