Koan: Sphere

“What would an inverted sphere look like?” my Zen master asked me at our third meeting. Ah, the koan, I realized.

Immediately I set to work on it, and just as quickly came upon a slew of dead ends. But I kept at it.

Finally, I imagined a normal basketball. It has that tiny black rubber air hole, right? Where you put the needle from the pump in to inflate it. What if you were able to reach in, grab the insides of the basketball, and pull them out through the air hole? And then reinflate it. Voilà! Inverted sphere.

My Zen master slapped the side of my face with his cane. Back to the drawing board.

Then I thought a little more about what a sphere is. An idealized sphere, a Platonic sphere. It seemed to me there are four components to a sphere: Its center, its surface, its interior and its exterior. What makes it a sphere is that every line in every conceivable infinite direction outward from the center reaches a point on the surface in the same amount of distance. But how to invert it?

Could this have something to do with the curvature of space? Let’s see. If you drew a triangle on the side of a flat, stretched out sheet of rubber, all three angles would add to 180 degrees. But if you made that flat, stretched out sheet of rubber the side of a massive balloon and inflated it, well, then, the sides of the three angles would add to something greater than 180 degrees. Positive curvature, I’m told. Similarly, if you drew a triangle on the inside of a balloon, those angles would add to something less, due to the negative curvature. I wondered: could you warp spacetime sufficiently enough to invert a sphere?

That triangle on the side of the balloon … let’s grow it, eh? In my mind’s eye I watched it expand until – until what? If you imagine a triangle where each side is bowing outward, eventually you’ll have a circle – 360 degrees. You’ve doubled it, degree-wise. Does this mean it occupies one complete hemisphere of a sphere? Half a sphere? My mind seemed to stall, like a clutch thrown into a gear way too high for the current revving of its engine. But what do you expect – I haven’t slept in days, I need a drink, and all this rice farming is making me go –

To me, inversion meant something opposite, not something doubled, or tripled, or integer-multiplied. The inverse of X is not 2X, but – X. So if curvature was involved, it would have to be of a negative direction. The inside of the balloon. But how to extrapolate it out to three-dimensions?

As I pondered all this distraction, my mind wandered back to the definition of the O Perfect Sphere. Those rays, in particular, extending from the center outward towards infinity. To invert a sphere you’d have to have those outward-directed lines from the center converge upon a new center. Yet this center could not be in one particular region beyond the sphere – it had to be in all regions in all directions at once! And the old center would cease to exist, right? What would happen to the surface? The old surface would be annihilated, but instantaneously recreated as the sphere flip-flopped into inversion.

Hold it! If the sphere is a star, does not such an inversion lead to a black hole, where rays of light (the lines extending outward from the sphere’s center) cannot overcome gravity and are bent 180 degrees backward ( – X instead of X)? And might an inverted sphere not really be a sphere at all, but a hole in realtime ourspace? A bottomless infinite well into Somewhere Else? These wormholes those physicists are always yammering about?

My Zen master happened to be walking by as I was in such intense thought, weeding in the tomato garden. He asked me how I was coming along, and I explained my ruminations to him.

He kneed me in the groin, and I doubled over in agony.

“The sphere is not a star,” he said. “It is you.”

My master reached down and helped me up. And I was enlightened.