**Lessons Worth Sharing**

**Alex Rosenthal**Educator

**Cale Oglesby**Animator

**Chris Acker**Sound Designer

**David Housden**Composer

**George Zaidan**Educator

Here's a great video by Sixty Symbols -- it's all about dimensions.

This video really only skims the surface of Flatland. For the full experience, you can read the book for free.

The visualization of the hypersphere in the video was generated using the following equation:

x^2 +y^2 +z^2 +t^2 = 1

This describes a set of points at fixed distance from a central point (just as the equation for a circle is x^2 +y^2= 1 and for a sphere is x^2 +y^2 +z^2 = 1)

We subtracted t^2 from both sides to get

x^2 +y^2 +z^2 = 1 – t^2

We then treated t as a time variable and ran the model over the interval time t = -1 to time t =1. Try plugging in different values between -1 and 1 for t, and you’ll see that this gives a series of spheres that start as a point, grow into a unit sphere (of radius 1) and shrink back into a point, as shown in the video.

However, this is not the only way of visualizing 3D cross sections of a hypersphere. Another visualization may be seen in this animation. This was animated by using a different, more complicated variable for time and gives you some sense of the complexity of the hypersphere.

This article, written by Professor Thomas Banchoff, gives some historical background – both from a social and a mathematical perspective--into the circumstances surrounding Edwin Abbott when he wrote *Flatland*.

Here's a YouTube video that explores the tenth dimension.

How could creatures live in that world with similar constraints as in Flatland (i.e. they can’t pass through each other and have to be able to see or sense their world)?

In the video, we depicted Flatland as having very faint gridlines like those you would see on a sheet of graph paper (notice that our characters pass over the gridlines in their travels through Flatland).