*I shall just sit down for a moment and pop on my boots and then I'll be on my way.*

Speaking of ways, pet, by the way, there is such a thing as a tesseract.

Mrs. Murry went very white and with one hand reached backward and clutched

her chair for support. Her voice trembled. "What did you say?"

- A Wrinkle In Time, Madeleine L'Engle

Speaking of ways, pet, by the way, there is such a thing as a tesseract.

Mrs. Murry went very white and with one hand reached backward and clutched

her chair for support. Her voice trembled. "What did you say?"

- A Wrinkle In Time, Madeleine L'Engle

We're likely not as troubled by tesseracts as Mrs Murry, but it is a nice little surprise when you come across them when they're unlooked for. You'll encounter them (or at least their 1-skeletons, the vertices and edges) while drawing factor lattices.

A factor lattice for a number

*n*has as its nodes all factors of

*n*. Two nodes

*a*and

*b*have an arrow (directed edge) between them if

*a*divides

*b*. Usually we don't draw all the arrows, just the ones where

*b/a*is a prime factor of

*n*- all the other arrows are found by composition, or are the trivial arrows that are the loops on each node (

*a*divides

*a*). (In this way, factor lattices are nice examples of very simple and special categories). Some notes on drawing factor lattices are here.

The factor lattice for 1 is just a single node.

A number like 6, that is the product of two primes gives a factor lattice that looks like a square.

A number like 30 that is the product of three distinct primes produces a factor lattice that looks like a cube.

And 210 is, you guessed it, the smallest number that produces a factor-lattice that looks like a (squished frame of a) tesseract (the 4-dimensional hypercube).

You can go further (of course) but it gets harder to see what's going on in the diagrams. 2310 is the smallest number that gives us a 5-cube.

Great Stuff! I am trying to teach my middle schoolers all sorts of ways to think about numbers and get a better visual and structural understanding. We are making these, albeit simpler, with gumdrops and toothpicks today!

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