Posts Tagged ‘euler’

Parabola or catenary?

Parabola or catenary?

In this lecture we will retrace the steps of Archimedes, Newton, Euler, and other great mathematicians and learn about important mathematical functions, their properties, history, and applications. We will look at several interesting competition topics that often show up on AMC 10/12, related to exponential, logarithmic, trigonometric, hyperbolic, and other functions. We will also see how these functions turn up in solutions of some fundamental problems in math, physics, and engineering. We will have fun using them to draw important curves: cicloids, cardioids, catenaries, circles, ellipses, hyperbolas, parabolas, and will discover which one of them is brachistochrone and which one is tautochrone.

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Alicia Prieto Langarica continued her tradition of actively engaging the students as they explored deep mathematical concepts. Ms. Langarica began with a discussion of the regular polyhedron and allowed the students to prove for themselves why there can be no more than 5. She then talked about these 5 Platonic Solids and gave some of the cultural context of these important objects.

From this basis, Ms. Langarica was able to describe the wide variety of non-regular polyhedron. As diverse as these objects are, they all have the common properties described by Euler. Ms. Langarica showed how the relationship between the number of vertices, faces and sides would be constant for all of these figures.

To involve the students more directly, the students built their own polyhedron and demonstrated their own diversity and talent. This break activity prepared them for listening to the more challenging portion of the lecture on Minimal Surfaces. Ms. Langarica described the very practical value of finding minimal surfaces to conserve cost or weight in construction projects. She then showed several beautiful examples of minimal surfaces.

By finding the surface normal of any point on a curved shape, Ms. Langarica showed how a minimal surface could be tested or created. To drive this point home, Ms. Langarica took the math circle outside with their polyhedron creations. By dipping these objects into soap bubbles she was able to beautifully demonstrate how minimal surfaces would form spontaneously as a result of the physical properties of the air and soap film.

Ms. Lanagrica has provided her slides from the lecture and answers to the problems. Members of the Metroplex Math Circle e-mail group can download these files from the group site. To join the e-mail group simply click below.

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