The ongoing saga of Heather and the Heretic

“Do you believe in the fourth dimension?”

His eyes seemed to light up as the very number passed through his lips. I swallowed nervously, mentally steeling myself. “Like, time? Well... yeah, I guess...”

He had a little laugh; he hid it well, but I had learned to recognize it. “No, not time,” he chuckled. “The fourth spatial dimension. Ana, kata, the extra directions, that sort of thing.” His hands began to shake, as they tended to when he was excited. “I've worked it out. I know how to explain it, to draw it, to map it now!”

I stayed very still for a second, thinking very carefully. “Including time, wouldn't that make five dimensions?”

The shaking stopped. “You're getting ahead of me there. I haven't worked out how to draw five yet. But look!” he exclaimed as he drew two pieces of paper out of his pockets. “Watch... and learn.”

He scurried over to the desk and picked up a pair of scissors. Very carefully he placed the papers together and cut a perfect circle out of them both. He held up one circle and grabbed a pen. “This circle,” he said, drawing a letter K on it, “is called Kghroman. This one,” he drew an M on the other, “is called Morgoth.” Finally he lay one of the papers flat on the desk and wrote a letter G, with an arrow pointing to the empty circle. “This space, which has the potential to contain either circle, is called Gomorra.”

“Why the funny names?”

For this I received a glare. “What does it matter?” He returned to the desk, placing one circle on either side of the paper. “Now. Without removing any of the above from the surface of the table, I want you to place both circles into Gomorra.”

I walked over to the table. He looked at me expectantly. Half-heartedly I began to shuffle the papers around on the desk. I could see where this was going; clearly the task was impossible. I sighed. “Look,” I said, “I don't know what you're expecting me to do here. It obviously can't be done.” I tried to push Kghroman through the paper. It began to lift. “What, have you worked out how to move papers through each other?”

With a smug look he walked over. “Of course it's impossible,” he said. “But now... watch this.” Very carefully he picked up Morgoth and placed it directly into Gomorra, then put Kghroman perfectly on top of it. He looked back up at me. “Do you see?”

I gave him a confused look. “That's cheating. You said they weren't allowed to leave the table.”

“No!” The agitation in his voice suggested I was being thick again. “That's the point. The puzzle can only be solved by adding an extra dimension. Moving from two to three is the answer. Now look.” He held the circles between his fingers and lifted the paper. “You see? The edges stick out slightly. They don't quite fit in three dimensions. But...” he put the paper down. “When you go back to two, they go back to occupying the same space. And this proves my point.

“Actually,” he went on, “the mere act of cutting out a circle proves my point. Not the circle, but the space that's left. Creating a hollow object externally – easy enough, you say, but seamlessly? To take the inside out without cutting the object open? With scissors and paper it's easy; you poke a hole and just cut. But in three dimensions? Imagine trying to forge a hollow metal sphere. Just think about it. It can't be done.

“So,” he said, “Do you see?”

I gave him my best blank stare. “Sorry? What exactly am I supposed to be seeing?”

He rolled his eyes. “All I'm saying is that by moving from 2-D to 3-D, you can move solid objects through each other and even put two in the same place. But now... let's take it up another level.” He picked up Kghroman and Morgoth and rolled them into balls between his palms. Then he put them down and clasped his hands together. “Right,” he said. “Now the space between my hands is Gomorra. It's not seamless, but you get the idea. Same puzzle as before. Place both balls into Gomorra.” I stared at the balls, then at his hands, then back at the balls. “What are you waiting for? This time you know the answer.”