We talked some more about hypercubes in class today, including drawing some pictures and building our own models out of gumdrops (take care of them!)

If you’re interested in reading more about the 4th dimension, check out the links below. I especially recommend the short story And He Built a Crooked House.

• The Adventures of Fred, Bob, and Emily – a detailed look at how the lives of a 2D (Fred), 3D (Bob), and 4D (Emily) creature interact with each other. See especially the “World” section, where the author, Garrett Jones, imagines how wheels, water, and war would work in these universes.
• And He Built a Crooked House – a short story by “Big Three” science fiction writer Robert A. Heinlein about an eccentric architect who designs a house in the shape of an unfolded hypercube. An earthquake hits, and the house folds back up on itself to concerning results (see also this student film version of the story)
• Some Notes on the Fourth Dimension – some animations and movies showing the geometry of the fourth dimension, including some of those featured in the Flatland dvd bonus features.

Your homework: You were given a sheet of graph paper at the end of class. This is what you should do with it:

• Fold it in half lengthwise (“hot dog” style).
• Unfold and put a dot on the left edge of your crease.
• Flip a coin (or use some other random procedure). If the coin lands heads, draw a diagonal that goes over and up one square. If the coin lands tails, draw a diagonal that goes over and down one square.
• Continue with this pattern, creating a zig-zag across your paper, until you reach the other side. Bring that in tomorrow.

Report your findings for our final box-counting project here.

After talking so much these past few weeks about fractional dimension—dimension values that fill in the gaps between 1, 2, and 3 dimensions, we turn our perspective in the other direction on the number line: towards the 4th dimension. It stands to reason that the trends we’ve observed could be extended past the 3rd dimension, but considering fractals, or even Euclidean shapes, is immensely challenging for us as 3-dimensional creatures.

What helps is to consider the perspective by analogy: we can better understand the fourth dimension by putting ourselves in the mindset of a 2-dimensional creature considering the third dimension. Fortunately, this is territory that has been well-covered.

In 1884, British teacher and theologian Edwin Abbott Abbott published Flatland: A Romance of Many Dimensions. It is told from the perspective of A. Square, a denizen of the titular 2-dimensional world, and starts off explaining aspects of their universe in great detail. The second part describes his first encounter with the 3-dimensional Spaceland, and the changes to his world-view as a result.

Abbott wrote the book partly as an exercise in geometry, but also partly as a satire on the regimented Victorian-era social hierarchy. As a result, there are some rather uncomfortable characterizations of women as lower-class citizens, among other shocking commentary.

You can read the full book here (I have paper copies if you’d prefer that), but for tomorrow please at least read this excerpt.