Math Notes


Many of my published mathematics papers are available electronically.

Click here for papers appearing in Fibonacci Quarterly


I always carry a cup of Starbucks past Gary’s workstation on the way to mine through the maze of desk carrels at Games-du-Jour. The high-ceilinged room is divided into a warren of open cubicles by six-foot dark-gray rug-covered partitions, supposed to deaden the noise, and so much sameness that I spent three days learning the way to the water-cooler.

Gary is a super computer geek, writes code for computer games. Gary always looks spiffy, wears button-down collar shirts, slacks, and dress shoes, says he has to dress well to come to Games-du-Jour because his job is so much fun that he forgets he’s working. He always wears a tie.

I’m the one who looks nerdy—beard, long hair, thick glasses, comfortable rumpled clothing and sandals, but I’m the pseudo-nerd who writes how to assemble electronic gear and how to use the software. Boring as Hell but pays my bills. Sometimes I let my mind play, surfing the web; other than that, this job is definitely ho-hum.

Today, however, Gary’s desk is such a mess that I almost drop my coffee. Crumpled pieces of white paper litter his desk and floor space, crumpled white paper everywhere. “Looks like the city dump in here,” I say.

“It’s research, research for a computer game based upon flexagons.” Gary holds up a folded paper hexagon, pinches it on a corner, and opens it out from the center into a hexagon again.

“Looks like a cootie-catcher from the sixth grade,” I say.

“Not the same at all. It’s a hexa-hexaflexagon, six faces and a six-sided shape. Four graduate students at Princeton University—one of them Richard Feynman—discovered them. It’s all written up [8] in the American Mathematical Monthly.”

“Richard Feynman, I’m impressed. But what’s the big deal?”

“Here, make one.” Gary handed me a piece of adding machine tape with nineteen equilateral triangles penciled in. “Crease across the lines, back and forth, to warm it up.”

The triangles are numbered 1, 2, 3, 1, 2, 3, … with a blank one on the end. On the reverse side, the first triangle under the 1 is blank; the others, numbered 4, 4, 5, 5, 6, 6, 4, 4, … I have to set down the coffee. I use a corner of Gary’s desk.

“Now roll it up, folding the same underside numbers to face one another: 4 on 4, 5 on 5, 6 on 6. Fold back again three triangles in, then back three more triangles. It makes a hexagon shape.”

“What’s the piece hanging out?” I ask.

“You glue the blank sides onto one another. Here, use my glue stick.”

True, it does look like a hexagon, six 1s on one face, six 2s on the reverse.

“Cool. But what’s it good for?”

“Six different faces come out. Flex it and open from the center, and repeat. If it won’t flex, move to the next corner.”

I try it, not difficult. “What’s the puzzle, then?”

“You can put pictures onto a face and watch the center move to an edge. See, this spot moves out.”

“I guess a picture of a rabbit would really mess up, put its tail next to its ears. ”

Gary smiled. “That makes it a good puzzle. Pull a rabbit out of a hat, so to speak.”

A stupid puzzle: I say, “How are you turning it into a computer game?”

“That’s the trouble. To write the program, I have to know what happens to each triangle when you flex it. Something’s wrong because the 5-face comes out only once in all the possible flexings, but the 1-face and 2-face come out again and again.”

“You need to ration the glue,” I say.

“It isn’t the glue. If I try to keep my finger on a triangle to follow it, the whole thing rips apart.”

“Come on, man. You can’t keep crumpling those up, over and over.”

“I have to find it, the 5-face.” Gary frowns and begins folding another flexagon. “As you can see, I have developed ways of folding without drawing triangles or numbering.”

I shake my head. Weird. I go back to work.

I come early the next day. Gary looks rumpled and his tie wears a grease spot. His floor wears two inches of crumpled brown paper toweling.

“Been here all night,” Gary says. “Bigger flexagons, from a roll of paper towels.”

“Lots of luck, buddy.” I center my coffee on my desk and turn on my computer.

“Help! Peter! I’m stuck!”

“Coming.” Gary, in some kind of trouble, is connected to a paper-towel roll, a brown flexagon folded around his tie.

“I can’t get the tie out without tearing the flexagon,” he says.

I hold the tie, he flexes, his orange tie comes out.

Gary shakes his head. “I almost had it when I stuck my hand inside.”

I shake my head. “It’s too flat to fold it with a finger inside.”

Gary doesn’t hear me, and he doesn’t stop folding. When I leave for the day, Gary’s still folding, two rolls of brown paper towels filling the floor, discarded flexagons two inches thick.

The next morning, I come in carrying my Starbucks like always. Gary isn’t in his work-carrel. A cardboard and plastic tape flexagon with three-foot triangles fills his floor space like a dirty brown rug. The end of his royal blue tie, the one with yellow circles, pentagrams and infinity symbols, sticks out between two triangles. Curious, I pull it out and out and out until I have Gary’s entire tie but no Gary.

I will always wonder, did Gary find the fifth face?