In this project you could build a linear or a “Prytz” planimeter, and explain mathematically why it works. You can see how it works in this Mathematica demo. A polar planimeter is available to borrow from your instructor. You can even make a “Prytz” planimeter with a spoon. There are several different kinds of planimeters. 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. Planimeters and mechanical integrationĪ planimeter is a measuring device that computes the areas of arbitrary shapes. See this website or this blog post for lots of good links and info. In this project you could explore the mathematical definition of negative curvature and create some examples in crochet. She invented techniques for crocheting surfaces of negative curvature, which are a bit like saddles, except cooler and harder to create. Mathematical crochet and knittingĪre 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. You can look at this link for the basics about the Mercator projection. There are many good resources, but you could start with this treatment of the third question. Compute the rhumb distance between two locations on the earth.Determine the rhumb heading from one location to another.You could explore this topic, and address issues such as these: 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. Mercator (1512-1594) created a new map that had a significant impact on navigation. Try making a model using Sculpey clay and/or other materials. Pick a particular surface and explore its mathematical properties and construction. Seifert surfaces are a general kind of surface constructed from knots or links.Its design is based on a “minimal surface” known as a Costa surface. It is not just a fanciful shape, it has defining formulas etched into its surface (go take a look!). The Invisible Handshake is a mathematical sculpture located right outside Olin Rice.Boy’s surface, an attempt to squeeze a mathematical space known as the projective plane into 3d space.Sometimes these surfaces are created for the sake of art, and other times they serve some distinct application. Many beautiful and exotic surfaces have been created using various tools from multivariable calculus and beyond. There’s more to the world of surfaces than spheres and paraboloids.
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