Hilbert Space Invaders

As an adjunct at a two-year college, I have been given a class called “Foundations of Quantitative Reasoning” to teach on several occasions. Essentially it is a throw-away breadth of knowledge class that the college offers students who don’t need any specific math classes for their degree but still need a course in “quantitative reasoning”. The supplied textbook is a hodge-podge, Frankenstein’s monster of a book that starts the first chapter with the basics of formal logic and set theory and, in what could be called the paragon of awful transitions, switches to units and percentages in the subsequent chapters. The idea was to create an overview of “real-world math”, but the result gets on my nerves as much as the phrase that describes its goal.

Anyway. Enough complaining. I’ve made the course my own and had fun with it.

One of the things I did was take the “Mathematics and Art” chapter of the book and use it to help introduce students to the idea of doing math as a recreational activity. I integrated that chapter with the previous chapter about fractals, talked about the dreaded Cult of the Golden Ratio, threw in a few Vi Hart videos and ta-daaa: arts and crafts day in a college math course that is secretly some of the most interesting math my students did in that course. (And they didn’t complain about it either….mostly.)

While researching different ideas for this lesson, I came across the Wikipedia entry for Hilbert Curves.

This idea must have stuck in my mind because some months after that math-art lesson I found myself doodling during a meeting.


After I made the third iteration of the curve, I noticed that if I continued my grid a little more, I could enclose the shape in a pleasing manner. After I colored it in with my pencil, it reminded me of an enemy ship in Galaga. Or a space invader. Hilbert Space Invader.

I went back and extended the second iteration as well.

Summers are a slow time at the tutoring center where I work full-time and so I found myself making a grid on construction paper.


Then I realized I had drawn too many boxes, so I erased a few.



It turns out I still had too many, but I didn’t realize that until later. I was so concerned about recreating what I had doodled on my meeting agenda, I wasn’t really thinking about the curves were made in the first place. I feel like my intelligence fluctuates and today was a low day, because then I forgot how to make the curve completely and had to scribble for a while.


Ignore the prime factorizations, I was tutoring a very kind, older woman yesterday who was taking an algebra class to, in her words, “keep her mind from atrophying”. Which I found very impressive. I’d tell you how old she was, but I didn’t want to be rude yesterday, so I didn’t ask her. So finally I remembered how I had done the curve and started sketching. I tried to connect the middle of each square for the lines. I almost used a ruler to keep the edges straight, but my personality has too much lazy and not enough perfectionist to take the time to do that. Here are some stages of the work.



It was at this point that I realized I still had too many squares. Oops. Like I said, low day. I cut out the grid.


Then I drew a second level curve on the leftover piece.


Then I cut out the shapes to make the invaders.


I wasn’t sure what to do with the first iteration of the curve since it actually looks like this.


And there didn’t seem to be a good way to cut it out, so I left it as the square. I’m open to suggestions. Here they are on my whiteboard.


I’ll probably spend way too much time today making more and then recreating a space invaders scene on the whiteboard. I’m thinking about trying to make space invaders clone, but with my Hilbert Space Invaders instead. Also, I wasn’t sure at first if Hilbert Space Invaders was really the proper term for them since technically the shapes come from Hilbert Curves, but I couldn’t come up with a funny play on words that involved Hilbert Curves. However! I figure since a Hilbert space is a generalized n-dimensional space (if I understand the wiki article correctly) then my curves are invading a 2-d place that could be considered a Hilbert space. I think Hilbert Space Invaders turns out to be a good name.

So anyway, that’s what I’ve been wasting my time on.




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