Part 24 (1/2)
”It figures,” he said. ”We've used up all the faces.”
”In which case we'll be back where we started -- closed loop, and n.o.body but ourselves.”
”I guess so, right now. The others must have gotten off. Is that bad?”
”I can't buy it. Who set up all these other projectors?”
Veg shook his head. ”Got me there! If they'd gotten off, they'd have taken back their projectors -- so they must be still on. And there can't be six Vegs and six Tammes.” He sobered. ”Or can there?”
”Suppose your hexaflexagon had twelve faces?”
”Sure. There can be any number of faces if you start with a long enough strip of triangles and fold it right.”
”A twelve-face construction would merely add one new face to each of the six exterior angles,” she said.
Veg shrugged. ”I'll take your word. I'd have to make a live hexaflexagon to check it out myself.”
”Don't take my word. Make your construction.”
”Here? Now? Why not get to a better alternate to -- ”
”No.”
”I don't have anything to -- ”
She took apart the six-faced hexaflexagon, straightened out the long folded strip of plastic, pried at the edge with a small knife that appeared in her hand, and peeled it into two layers lengthwise. She produced a little vial of clear fluid, applied it to the edges, and glued the strips together endwise. The result was a double-length strip.
Veg sighed. He took it and folded it carefully. He made a flat spiral so that the double length became the size of the original but with two layers instead of one. Then he fas.h.i.+oned a normal hexaflexagon.
”Run through it and number your faces,” Tamme said.
”Okay.” This was a more complicated process, involving thirty flexes, but in due course he had it. Meanwhile, Tamme had been making a new diagram.
”Now start at face One and flex,” she said. ”I will call off your numbers in advance. Five.”
He flexed. ”Five it is.”
”Seven.”
He flexed again. ”Right.”
”One.”
”Right again. Hey, let me see that diagram!”
She showed it to him. It was an elaborated version of the prior one, with new triangles projecting from each of the six outer points. One angle of each of the outermost triangles carried the number of a new face, bringing the total to twelve.
They flexed through the rest of the construct. It matched the diagram.
”As I make it,” Tamme said, ”We could be on this one instead of the six-faced one. In that case our starting point would be Seven, followed by One, Five, Two, Eight, Five, Two, One, and now Three. If so, both our next two stops may be new worlds, Six and Nine.”