The Dilettante: Draw a Hypercube

I learned to draw this hypercube from Rudy Rucker’s Geometry, Relativity and the Fourth Dimension. I highly recommend this book, and wish I had read it as a high school senior or college freshman.

Geometry, Relativity and the Fourth Dimension by Rudy Rucker

Geometry, Relativity and the Fourth Dimension

The hypercube figure is on the cover of the book. According to Rucker, “This design for the hypercube is taken from a little 1913 book, A Primer of Higher Space, by Claude Bragdon, an architect who incorporated this and other 4-D designs into such structures as the Rochester Chamber of Commerce Building.”

At first it looks difficult to draw, but with a little practice, you can actually draw this freehand. Here is how I do it:

First, draw a cube:

Hypercube step 1

That’s pretty easy. I’m using grid paper here, but if you have a steady hand and a good eye, you don’t need it.

Now, draw a second cube with the same dimensions, down and to the right of the first cube. Make them overlap so:

Especially notice the position of the squares that make up the the “front/back” portion of the second cube. Make those squares overlap the the squares from the first cube in the same way. This can be a little tricky at first, but use a different color to draw the second cube, and it will be easier.

Finally, connect the corners of the first cube with the corresponding corners of the second cube:

I’ve used a third color to connect the corners. This can get confusing, but just focus on one pair of corners at a time. The outside corners are the easiest, so start there, and that will give you a way to proceed with the more overlapping parts.

Here’s one I drew in Inkscape with shaded sides:

Hypercube

One fun way to get lost in this figure is to try to count all the cubes that make it up.

The Dilettante on Living in a Möbius World

Currently reading One Two Three… Infinity by George Gamow.

Here is a little science demonstration inspired by a section on the topology of the cosmos.

Some things I discovered while working on this video:

You can’t make the kind of Möbius strip that I did by nesting two Möbius strips together: you can’t nest two Möbius strips and have one follow the contours of the other all the way around.

If you construct this, and pull it apart so that it hangs loosely as a single loop of paper, it’s very difficult to put it back the way it was. It’s kind of a puzzle.

Here are some other cool Möbius strip related videos I like:

A superconducting vehicle that runs on a magnet covered Möbius strip.

One of my favorites, Vi Hart, showing off a Möbius music box.

This awesome 3D printed round Möbius strip.