The Geometry of Mop-Dog and Boy



Mop-Dog was leader of the pack. Didn’t matter that there were only two members of the pack—Mop-Dog’s head and Mop-Dog’s tail—because a good time was always had by all.

On this particular day, Mop-Dog’s pack was running around in two circles at once. One tight circle was around himself because he was chasing the other member of his pack as fast as he could run. It was a tight circle, indeed, because Mop-Dog was a tiny being with very short legs.

As Mop-Dog circled himself, he also ran around his best friend, Boy—but in the opposite direction. It made Boy’s eyes cross to watch Mop-Dog for very long.

A group of Boy’s friends came running around the cor ner in a line playing ‘Follow the Leader.’ The first friend ran up to Boy, bisecting the circle made by Mop-Dog, and jumped over Boy! Boy had to scrunch down a little for comfort.

It was a good thing, too, because each of the other six friends followed along, leaping over Boy as they went. Boy’s friends then did a strange thing: they started circling both Boy and Mop-Dog, but in the opposing direction of the outside circle!

There were now three circles going at once: the smallest circle moved in a middle-sized orbit around Boy, with the largest circle of Boy’s friends on the outside. How many different directions were the three circles going? You can count them for yourself.

Then the friends started making a figure 8, using Boy as the center of the 8. How many circles were going at once now? And in how many different directions? Hah! You’ve probably figured out that there can never be more than two circular directions on a flat plane. (Of course, if the friends were able to run underneath Boy, and then fly over him in a circle , that would add another dimensional plane and yet another possibility of direction.)

Anyway, we’ve got four circles going now, in two directions. Be sure you can see this in your mind, because they all did something now completely unexpected.

The first friend started making a large circle around the whole pattern, and of course everybody else followed. I mean, everybody followed, so that Boy got up after his last friend jumped over him, and also got in the circle line. Even Mop-Dog, running in after Boy, helped complete the larger circle, so that all the parts became a whole circle.

They ran ‘til they got tired, then Boy peeled off to make a zig-zag line to the corner store for an ice cream cone, with everybody following like a long dragon. Boy and his friends all crowded into the store, while Mop-Dog waited outside, patiently chasing his tail.