Continuing our discussion from yesterday, if I were a Flatland creature trying to understand a cube, it’s not good enough merely to look at it. I need to do things with it.
Before going on, it is important to clarify something: In this discussion I sometimes use color. The color itself is not important. Color just provides a way to label things, to make the pictures easier to talk about. The only thing that is really important is the geometry: Where things are. That’s where all of the insight will come from.
Our Flatland creature, after staring at her picture of the 2 × 2 × 2 cube, decides she would like to rotate it and see what happens. Being a rather clever Flatlandian, she knows that a cube can be rotated in at least three different ways: (1) about its Left/Right dimension; (2) about its Up/Down dimension, and (3) about its Front/Back dimension.
To make it easier to see what happens when the cube rotates, we’re going to add a little more color for labeling. Specifically, we will make the colors of the four “bottom” cubes more vivid:
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Now let’s see what happens when our Flatland friend rotates this cube 90o in each of the three ways:
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What does the result tell us, if anything, about what rotation in three dimensions might seem like to a Flatland creature? More tomorrow.