if i were to visit a world with four-dimensional geometry and look up at the night sky there, i might ask my travel guide, "where are the stars?"
"Can you not see them?"
I would use the rotator mechanism provided to me and see the occasional speck of light blink into and out of existence as I spun. "Only briefly, when I spin," I would say.
"I see," my travel guide will say, "the volume occupied by your eyes is too thin to intersect any stars unless you're aimed perfectly."
My travel guide knows I am an astronomer and am disappointed by this, and so sets to work configuring a planetarium program on a projector. This will allow me to see the surface of the hyperspherical night sky projected to a 'flat' three dimensional display.
It is still difficult for me to wrap my head around double rotation. My travel guide assures me it'll come in time. Easy for her to say--she's understood double rotation intrinsically since she was a child!
The four dimensional hyperspherical planet double-rotates of course. I stand little chance of understanding four-dimensional astronomy without understanding double rotation.
The four dimensional people are no more intelligent or technologically advanced than we are, in fact if anything they seem to be about a decade behind in computer technology (well, sort of.) Their transistors are many decades behind ours, but you can fit a lot more transistors in a given footprint than you can in three-dimensional space.
But I wonder what may become possible in 20 or 30 years, with humans and 4D-people collaborating. I tell my travel guide about my ideas.
"On Earth we have virtual reality headsets, but there's also some research into brain-computer interfaces. Imagine if we could bypass my eyes altogether and plug one of your cameras into my visual cortex!"
"I'm not sure that would work," she says.
"Brains can adapt to all sorts of visual stimuli. I'm fundamentally limited to viewing only a three-dimensional slice of your world projected into a two dimensional plane, so long as I am using just my eyes. But maybe if the data coming into my mind was three-dimensional instead of two-dimensional, I would be able to train my brain to interpret it."
My guide shifts something in her face. Reading an alien's facial expressions is hard enough even when you can see their entire face at once, let alone when you can't. But I think it's something like a smile.
"And what if it worked? How would you handle being flattened back into your own world?" she says.
This gives me pause. "Perhaps I could find other ways to fill up the extra dimension. Like, maybe I could plot the last ten seconds of visual input on the W axis. Or I could use it as a volumetric display for a 3D video-game, and see an entire game world all at once."
"Computer games render only the surfaces of objects, don't they? When I look at your world I see the interior in its entirety," says my travel guide. I feel a weird sensation in my stomach, and realize that she has again poked me on my inside.
"Maybe I could just stay in the fourth dimension forever," I say.
Later, it's morning (due to double rotation the length of day time is never constant) and I am sketching something in my sketchbook--plans for a device that might aid visitors like myself in interpreting 4-dimensional geometry, if only slightly. It's a kind of periscope to be worn on one eye or the other, shifting depth perception out to be in the axis perpindicular to my visual volume.
When my travel guide wakes up, I show her the drawing, but she can't understand it. That makes sense--I wouldn't be able to understand a flatlander's drawing after all. So I describe the device to her.
She creates a drawing, and places it upon the projector. It looks more or less like what I had in mind.
"We tried that once," my guide says. "If we go to the office we might be able to find the exact device we used. The test subject couldn't really make sense of it, but we haven't tried it on a human who has spent as much time in our world as you have."
Over the next few days I acclimate to the periscope (and the strange pressure in my head from the four-dimensional bulk of the helmet trying to pull my skull out of my volume.)
At first it was just confusing. I no longer had binocular depth perception. Though I didn't absolutely need it--binocular vision is not the only means of depth perception available to my brain. But the overlaid images kinda 'z-fighted' if you like, like when you cross your eyes on two different colors.
Eventually, with the help of my guide, I became able to sense depth again, but I was disappointed to find that it didn't really give me any real insights on four-dimensional geometry. It was still fundamentally just a pair of 2D projections of 3D slices of 4D space.
I adapted much more quickly to normal vision when I finally removed the helmet.
This world is so fantastical I just wish I could understand it better. I wish I had more time here. Only three weeks remain in my stay here before I have to be reprojected, and I still can't wrap my head around double-rotation!
