Why we should teach 4D geometry at school

Tesseract (Source: http://en.wikipedia.org/wiki/File:8-cell-simple.gif)

This is a four dimensional cube (projected into 3D-space, then rendered into 2D) rotating in the fourth dimension. Isn’t this absolutely fascinating? I think teaching kids 4D-stuff could completely change how they work with data and information. I’ll explain in a minute. But first a very very quick intro.

A tesseract or hypercube is an object, that has a width, a height, a depth and then the fourth direction – the one we can’t see or imagine as we are trapped in three dimensional space.

I got interested in n-dimensional objects when I wrote a little 3D-renderer, that could transform (or render) 3D-objects into 2D shapes and then draw those shapes on a 2D-image. Pretty basic stuff. But it seemed obvious, that you should be able to construct 4D-objects virtually (after all, you just need an additional axis) and then transform them into 3D objects. It is possible and people did this of course!

The 4D-maze

John McIntosh created a 4D Maze Game that let’s you roam a 4D maze. The interesting part is: although people can’t really imagine the fourth direction, after some practice they are able to orient themselves spacially.

In this video a player navigates through the maze. Whenever these weird “inside-out-turns” happen, the user moves or turns in the fourth dimension. (Of course this is just one of two stereoscopic images from the game. So what you’d see as 3D in the game is 2D in this video, which means a lot of information is lost. Just like you would if you render a 3D cube to 1D)

So why is this important for school?

First of all because it’s supercool.

Secondly because it teaches children that there are things we may not be able to fully grasp or imagine visually, but still we’re able to do calculations on them. E.g. it’s very simple to calculate the contents of a tesseract or – a bit more complex – a hypersphere

Thirdly because it completely challenges our view of the world. We are so good at living with three dimensions that it’s hard to imagine there could be anything we can’t describe with three directions. But the simple realization that we can’t show or even imagine the fourth direction brings a mindblowing concept right inside our reality.

4D diagrams

But most importantly it introduces young minds to multi-dimensionality. Think of how we work with data today. We make spreadsheets. But spreadsheets only have 2 axes and we already have to fake the third axis by having multiple spreadsheets. Many business intelligence queries though produce multi-dimensional data and we don’t really have a way to visualize them yet. Maybe 4D-literate people could work with 4D diagrams. (Not 3D + color, but truly 4D)

As an example of what happens when you map multidimensional data to 2D graphics watch Garrett Lisi’s TED talk. It’s an excellent example for transforming multi-dimensional data to 2D and when you roughly understand how movement in multi-dimensional space translates to 2D or 3D, you will be able to make a bit of sense of the funny rotating dots.

Bring on the 4D curriculum

I think we should devise a course that makes people truly 4D-literate. In the sense of “they are able to grasp 4D visualisations of data”. Maybe it’s impossible. But maybe people can learn to break out of 3D space and see connections that are hidden to us today. After all, creating the number zero or painting funny symbols for abstract concepts must have seemed mind-boggling back then…

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