By Jonathan Heiles, HS Science

The First Pass course focuses on developing tools to to get a foundation in a range of counterintuitive ideas from advanced math and science. The semester started with practicing  this method through multiple approaches to thinking about multiple spatial dimensions. Students built models to supplement the understanding (and to get a break from the intense mental efforts involved in the work).

Imagine a cube built out of straws. Its shadow would look something like the picture above. The shadow of an object has one fewer dimensions than the object itself, so the shadow of a cube is a square. You get, however, two nested squares (one from the front face and one from the rear face). The rear square, being farther away, is smaller. Students worked out that the six sides of the cube are represented by the two squares and four trapezoids in the figure.

By analogy (and by rigorous mathematical transformation), the shadow of the fourth dimensional equivalent of a cube (a hypercube or a tesseract) is three-dimensional. It looks like one cube nested in another. The central cube is from the back of the figure, so it appears smaller. The class built on the interpretation of the cube’s shadow to identify the eight cubes expected to be in a hypercube.

Confused?  If you email questions to jheiles@poughkeepsieday.org, students will work out answers for you.