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