Ken’s Blog

Changing up the theme for a day — for the sake of those who are not interested in 4D — I had a wonderful thought today.

Imagine that a hotel owner got a hold of the names of various rock and roll groups, and decided to create a theme hotel, where each room honors one of the names on the list.

But here’s the kicker: The hotel owner doesn’t know anything about rock and roll, and in fact doesn’t realize that these are the names of music groups.

I imagine such a hotel would be a very interesting. One room would be filled with little stones that roll around, and another would contain giant sculptures of beetles. Yet another would feature crickets. Down the hall would be a room where all the furniture was made of lead, and shaped like little zeppelins.

I had some trouble with “Radiohead”, until I realized that a room could be filled with portaits of top executives from the radio industry. These would be the radio heads, of course.

I’m sure I’m missing lots more. Does anybody else have any suggestions?

One piece of advice: I’d stay away from the Heavy Metals. After all, who’d want to stay in a hotel room build around the theme of Anthrax or Poison?

I suppose I could just tell you what sort of things in four dimensions correspond to yesterday’s post. But that wouldn’t be much fun, would it?

Instead, let’s think about this a bit. Here we had a virtual trackball that looked like a circle. It had some straight lines on it, but when we looked at it from the side, it turns out those lines were actually curves. That’s because those straight lines were, in fact, painted onto the front of a sphere.

So what looked at first like a flat disk was actually an image of the outside surface of a sphere.

What if we bring this whole discussion up to one higher dimension? At first we see something that looks like a sphere floating in space. But in fact it is something more.

The sphere we are looking at is the “front” side of a four dimensional sphere. Anything that is drawn inside our sphere is actually being drawn onto the outside of this four dimensional sphere.

If any of this seems mysterious, just go back to the three dimensional case: Something that looks like a flat disk is actually the front face of a sphere.

We’re just taking the same argument to one higher dimension: Something that looks like a sphere is actually the front “face” of a four dimensional sphere.

That’s a pretty weird idea, so I’m going to give it a day to sink in before going on.

In order to understand what’s going on with a 4D virtual trackball, let’s take a more careful look at a 3D virtual trackball.

What you see is something like this:

You can move your finger up and down (ie: in the direction of the red arrow), or from side to side (ie: in the direction of the green arrow), or around the circle (ie: in the direction of the blue arrow).

In each case, you are causing a sphere to rotate. This might seem mysterious to a denizen of Flatland, who does not have any direct perception of a third dimension. After all, the first two gestures look like straight line movement — very different from the circular movement of the third gesture.

Fortunately we are, in fact, 3D creatures. Which means we can employ a great 3D power-up to see what’s going on: We can rotate our view of the object:

Now we can see, viewing from an oblique angle, that all three of the movements are actually circular arcs. The only reason the red and green gestures look straight is that the user cannot directly perceive the third dimension, since the z direction is perpendicular to our view. The red and green paths actually curve toward us, around the front of the sphere.

Which means that for a 4D trackball, some things that look like straight line gestures are actually going to be curved — in particular they will curve around the “front” of the 4D trackball.

Tomorrow we’ll see just how many such curves are needed for this.

It’s very cool that we can rotate things in three dimensions just by playing with a circular disk. After all, a disk is flat!

What’s also cool is that we can rotate things in four dimensions just by playing with a sphere. It’s not even that difficult to do — but it can be a little difficult to understand.

Just as you can rotate 3D objects by putting a finger down on a disk on a computer touch-screen, then moving the finger around inside the disk, there’s an equivalent action for a spherical region floating in space in front of you.

Namely, you can put your hand inside the sphere, pinch your fingers together to “grab” the sphere, then move your hand around inside the sphere (then unpinch your fingers to “ungrab”).

One way to think about this operation is that the volume inside the sphere is kind of like the area inside a circle, except in one higher dimension. As you move your pinched fingers around inside the sphere, you’re essentially rotating a 4D trackball.

But how do your movements inside that floating sphere translate to rotations in four dimensions? We’ll get to that tomorrow.

Continuing yesterday’s thought … if you were looking at a 4D object in front of you — holographically, the way Tera presented it to Trey — how would you rotate it, to see it from different points of view?

First let’s look at how people rotate 3D things on computers. One common way uses something called a “virtual trackball”. It’s basically just an image of a trackball, seen from above. When you drag your finger over the image, it does what a real trackball would do.

As you can see in the above image, a virtual trackball “rotates” around the vertical axis when you drag left and right (the red arrow), about the horizontal axis when you drag up and down (the green arrow), and about the axis that comes toward you when you spin around its edge (the blue arrow).

So here we have an example of a completely two dimensional thing that lets you rotate stuff in 3D. Perhaps we can do something similar for rotating stuff in 4D…

Sorry to switch the topic, but this evening I spent a delightful forty minutes with a friend, lost in conversation while walking through the rain in Paris.

If you’re going to walk through the rain anywhere, I highly recommend Paris. 🙂

Going back to the question of four dimensions… If you were a two dimensional being in Flatland, and you really wanted to understand 3D, you would want the most high quality visualization you could get of that mysterious higher dimensional world.

In that spirit, it seems to me that it would make sense to create the most vivid possible representation of our four dimensional object — even if it is impossible for such an object to exist in our world.

However we choose to simulate this four dimensional object, our simulation should appear before us with the same sense of “presence” as a real object. And if we reach out our hand, we should be able to touch it.

Sounds like a great opportunity to build a really good virtual reality experience.

Jumping out of our story for a little conversation…

I find this technique for visualizing higher dimensions — referred to as “Ariel’s design” in the story — to be very useful for my own understanding of 4D shapes. I am now much better able to, say, think about the parts of yesterday’s shape.

By the way, that shape is actually called a 16-cell, because it has sixteen three dimensional “sides”. It’s also sometimes called a hexadecachoron, which is just a fancy Greek way of saying “sixteen sided space”.

Using this approach, I can now understand the more complicated 24-cell much better than I could ten days ago (when I first wrote about it here).

Still, Trey’s question haunts me. To put the question simply: It is clear that a two dimensional square contains area. And it is clear that a three dimensional cube contains volume. But what sort of stuff does a four dimensional hypercube contain? What does it feel like to be ‘inside’ a hypercube? Is this something our poor human brains could ever get a feel for?

“Trey, I have something for you.”

Trey hears a voice, but he doesn’t see anybody. “Ah, Tera, how lovely to, um, hear you.”

“Thanks. I hope you still feel as positive when you see my picture of the four dimensional diamond shape.”

“Why, is there something wrong with it?”

“No, to me it looks like a perfectly lovely and symmetric shape. But your mileage may vary.”

Just then a shape materializes in the air front of him. But it’s not like any shape he’s ever seen before. “Are you sure that’s the four dimensional version of an octahedron?” Trey asks doubtfully.

“Well, yes, if we follow Ariel’s rules for how to draw these things. As you can see, I’m using little octagons for the corners.”

“I see a big octagon with some crosses inside it,” Trey says.

“Actually, nothing is inside. The big red corner is ‘near’ and the little dark blue corner is ‘far’. The edges that come ‘near’ to meet the red corner are orange, and the edges that go ‘far’ to meet the blue corner are teal.

“And what about all the stuff that’s gray?”

“Oh, those are the parts that lie inside your three dimensional space — neither near nor far.”

“So the fact that the red corner and the blue corner look like they’re in the same place…”

“…is an illusion,” she said, finishing his thought. “They look like they’re in the same place because our extra dimension is only seen edge-on in your world. Actually those two corners are as distant from each other as the distance from the bottom to the top of the whole shape.”

“Oh, I see now,” Trey says, “there are eight corners all together — the six in my world, plus near and far.”

“You’ve got it! And all eight corners are on the outside. How many edges do you see?”

Trey is starting to feel more confident. “Let me see … twelve grey edges, plus six orange edges, plus six teal edges … twenty four. Is that right?”

“Yes, that’s perfect! Any other questions?”

“Just one. A square or diamond has area inside. A cube or octahedron has volume inside. So what’s inside one of these?”

“Oh dear,” Tera says, “that’s a harder question.”

When Tera shows him what she has come up with, Trey is taken aback. “That’s amazing! It just appeared in the air, like a hologram.”

“No biggie — we just call that a picture. Those flat things that you call pictures are so lame.

“Well,” Trey says, “we also have sculpture.”

“True. I didn’t say you were hopeless. Anyway, what do you see?”

“I see two cubes, a fat red one and a skinny blue one, both in the same place. And I totally get that the red one is the one that’s ‘nearer’ in your fourth dimension. But shouldn’t the blue cube be smaller than the red cube, since it’s further away?”

“Ah,” Tera says, “that’s the clever thing about your friend Ariel’s design idea. She doesn’t make things bigger or smaller as they change in that extra dimension. She just makes the lines fatter or skinner — and of course she also uses color. If you make shapes bigger or smaller, which is what most people try to do, it can get a lot harder to see where everything is.”

“Oh, I see,” Trey says, “And the eight little grey cubes are the edges that go from ‘near’ to ‘far’, right? People in my world can only see them on-edge, which is why they look like little cubes.”

“Exactly!” Tera says. “I think you’re getting the hang of this.”

“OK then, what about something diamond shaped?”

“You mean something like your octahedron, but in four dimensions?”

“Yes please.”

“Hmm, I think I can have one ready for you by tomorrow.”

Trey thinks it’s a bit unfair that he can’t see his new acquaintance. “Where are you?” he asks.

“I’m just a little to the side of your three dimensional world. I could drop in — that is, part of me could — but that would sort of mess with the conservation of mass in your local vicinity, and the results might be unpleasant for all of us.”

“Well then,” Trey says, not wanting any trouble, “we shouldn’t go there.”

“Or more to the point,” Tera agrees, “I shouldn’t go there. I think you can hear me just fine from here — the air vibrations seem to do the right thing. And I can project 3D images into your world, so I don’t think we will have any problems communicating. Now tell me about this Ariel, and her wonderful design ideas.”