A Polyhedral Demonstration of the Fact That Nine Times Thirty Equals 270, Along with Its Interesting Dual

30 times 9 is 270

It would really be a pain to count the faces of this polyhedron, in order to verify that there are 270 of them. Fortunately, it isn’t necessary to do so. The polyhedron above is made of rhombus-shaped panels which correspond to the thirty faces of the rhombic triacontahedron. Each of these panels contains nine faces: one square, surrounded by eight triangles. Since (9)(30) = 270, it is therefore possible to see that this polyehdron has 270 faces, without actually going to the trouble to count them, one at a time.

The software I used to make this polyhedron may be found at http://www.software3d.com/Stella.php, and is called Stella 4d. With Stella 4d, a single mouse-click will let you see the dual of a polyhedron. Here’s the dual of the one above.

30 times 9 is 270 -- the dual

This polyhedron is unusual, in that it has faces with nine sides (enneagons, or nonagons), as well as fifteen sides (pentadecagons). However, these enneagons and pentadecagons aren’t regular — yet — but they will be in the next post.

Count the Dots!

I’ll be nice, and not name names, but there was a time, when I taught at another school, when I had in-school suspension for half a day, every school day, for an entire semester. It sucked.

There was a senior in this high-school version of, well, jail, who was working on Geometry homework. I was trying to help him — and to get him to think for himself. (I’m always trying crazy stuff like that.)

To do the problem he was working on, he had to know what three times eight equals. He was 18, or perhaps 19, and a senior. I was not willing to simply tell him the product of three and eight, because . . . that’s just ridiculous.

I told him to draw a row of eight dots on his paper. He did. I told him to draw another such row beside the first row. He did. I told him, finally, to add a third row. He did, and gave me an utterly blank look.

I said, “Now count the dots.” He did.


Even though he got the correct answer, he was still furious at me for, well, possibly the rest of his life, which may or may not still be going on. We haven’t kept in touch.