Beauty in Mathematics

[picture of Niles]

Here I've collected some information from my public talk about beauty in mathematics. This talk was part of the Ohio State Newark series Faculty Talks Outside the Box.

Announcement

Johnson will present “Beauty in Mathematics: From the Concept of Zero to Seven-dimensional Shapes” as part of the Faculty Talks Outside the Box lecture series. The talk will take place on Friday, Jan. 27 from 11 a.m. to 12 p.m. in the Norman R. and Alethea E. Sleight Room in the John L. and Christine Warner Library and Student Center, 1219 University Drive, at Ohio State Newark.

If you have trouble with math, Johnson is here to help. Mathematics is about patterns, and he will explain how patterns in elementary arithmetic, such as addition, subtraction, multiplication and division, explain patterns in high-dimensional geometry. It is in these patterns that math meets art with aesthetically pleasing results. No calculators will be needed for the talk; just bring a spirit of curiosity.

During Faculty Talks Outside the Box, Ohio State Newark professors discuss recent research in their fields and how it relates to the community. The next talk will be held in February featuring Assistant Professor of Psychology Bradley Okdie, Ph.D. All talks are free and open to the public.

Slides from my talk

You can download the slides for my talk about Beauty in Mathematics here! The animations of a cube passing through a plane are from here. Henry Segerman has much, much more 3d math, both virtual and printed, on his website.

The Hopf fibration

The Hopf fibration shows how the three-sphere can be built by a collection of circles arranged like points on a two-sphere. In this animation we see specific points on the two-sphere synchronized with the circles (fibers) over them. The animation has been choreographed to demonstrate a number of special properties related to the symmetry and beauty of the three-sphere (see below); I hope you enjoy it.

If you want, you can watch Niles explain the Hopf fibration to the UGA Math Club. Warning: That talk is about an hour long, so get your popcorn ready!

I have another full page about the Hopf fibration here.

The 7-spheres, standard and exotic

[seven-sphere slice]

A slice of S^7.

In 1956 John Milnor startled the mathematical community by constructing smooth seven-dimensional manifolds that are homeomorphic but not diffeomorphic to the standard seven-sphere. His discovery opened a new branch of research in topology and won him the Fields Medal in 1962.

The animation below gives a quick overview of the construction and shows a visualization of exotic seven-manifolds. If you want, you can jump to the end of the slides at 3:10. You can also download a pdf copy of the introductory slides.

This animation was presented at the Second Abel Conference: A Mathematical Celebration of John Milnor, in January 2012.

If you want, you can watch Niles explain 7-spheres to the UK Math Club. That video is only a clip, so you should probably view it as a sequel to the Hopf fibration videos.

I have another full page about exotic 7-spheres here.

Further Resources

If you want to know more mathematics, here are some suggestions!

Videos

Books

Math Education

We didn't talk about it much today, but mathematics education is another subject I care deeply about. Take a look at Lockhart's Lament for a good read about what's at stake.

Jo Boaler of Stanford University is leading a revolution in how teachers and children think about mathematics, and I strongly encourage it!. You can learn about it at youcubed.org.

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