''Fourth Dimension''

Several news stories recently are about Einstein and a ‘‘Fourth dimension’’; ‘‘three of dimensions of space, and one of time’’. What are the first three? Direction, distance, and what?

up, sideways, forwards.

Or, height, length, and depth.

The x, y, and z axes.

A point is dimensionless.

A line has one dimension.

A plane has two dimensions,

A cube has three dimensions.

Time is the biggie!

You cant have more than one dimension without time being involved.

It takes time to go from one point to another.

A Tesseract has 4 dimensions.
Marilyn McCoo had the Fifth Dimension, but left, over artistic differences.

I think the intrigue about four dimensions is that they all contain mostly space but some time. You can’t really separate x, y, and z from each other and from time, except in the limit as velocities become tiny fractions of the universally important velocity c. This limit handles almost everything humans work with, so most of us can use this approximation, but I think that’s what Einstein was getting at.

But you need those special glasses to see them.

I discovered tesseracts when I was in 7th grade, from the book One, Two, Three . . . Infinity by George Gamow.

It occurred to me that, just as a 3-cube is bounded by 2-dimensional squares, which are bounded by 1-dimensional line segments, which are bounded by 0-dimensional points; likewise, a 4-cube must be bounded by 3-cubes! Try to wrap your mind around that. If you look at the picture in the above Wiki link and squint a bit, you can actually see them. Can you count them? How many 3-cubes are there? (Hint: Some of them are distorted and don’t look quite like cubes.)

Eight.So. A 5-cube must be bounded by 4-cubes, right? And a 6-cube must be bounded by 5-cubes, right? And each n-cube boundary is itself bounded by (n-1)-cubes, each of which bounded by (n-2)-cubes, all the way down to the 0-D vertices.

How many of each of these are there in cubes of each dimension?

(ETA: All of this was a year or so before I discovered Flatland.)

I worked out a formula for that! I made a chart, very similar to Pascal’s triangle, but with different numbers in it of course. The formula was a recursive formula, where each number was computed from the two numbers just before it.

What I can’t wrap my mind around, however, is the concept of a universe with more than one time dimension. What would that be like?

There’s a cool iPad app that does a nice job of helping you visualise a tesseract.

This is it.

Think of it like a plane of time, except you can only turn as far as the perpendicular (90°) to the direction of time to the other (since you can’t go backwards in time).

For a thought experiment, try this:

Adam and Bob are in the vicinity of a black hole. Because gravity warps spacetime, the closer Bob gets to the black hole, the more he rotates on the plane of time—more and more toward the perpendicular to Adam’s arrow of time. To Adam, this appears that Bob is slowing down. Until he’s reached the event horizon of the black hole, his image will show Bob completely motionless.

In that way, if Adam and Bob were really points moving along a plane, tracing a line behind them, Bob’s line would appear to get shorter and shorter, until it was just a point, relative to Adam’s.

Weird enough that there isn’t a simple description. With multiple time dimensions, the concept of ‘future’ and ‘past’ aren’t well formed, and the math describing physical phenomena don’t function in a deterministic way. It’s not clear that there is a way to make predictions or determine cause and effect in such a world, which would make reasoning impossible, so you couldn’t have intelligent life.

These two articles go into some detail, though your eyes will likely glaze over:

Interestingly, a scientist has put forward the idea that it’s only possible to form intelligent life in a 3+1 dimensional space. Basically more than one time dimension has the predictability problems above, in one or two spatial dimensions you can’t form complicated enough structures, and in four or more you can’t have stable orbits or atoms.

Generalization:

Take a line-segment and rotate it around one end. It sweeps out a circle. You can calculate the area of the circle by calculating the distance travelled by the midpoint of that line-segment, and multiplying it by the length of that line segment.

Take a semi-circle and rotate it around the flat part. It sweeps out a sphere. The volume is the distance travelled by the center of area of the semi-circle, multiplied by the area of the semi-circle.

Fourth dimension!

Take a hemi-sphere and “rotate it” around the flat part! It sweeps out a hypersphere. The hyper-volume is the distance travelled by the center of volume of the hemisphere, multiplied by the volume of the hemisphere.

What’s really fun is that you can actually calculate the “height above the base” of the center-of-hypervolume of this new hypersphere! Given that, you can calculate the h5volume of a h5sphere. The process is repeatable and generalizable, to any number of dimensions.

(You’ll end up using numerical integration, so your accuracy will be limited by the number of “slices” you’re willing to use.)

What’s really creepy is that for spheres of unit diameter, the volumes increase…right up to the ninth dimension…at which point the volumes start to decrease again! In essence, the h-volume is becoming so very concentrated in the middle of the object, that this contributes more and more to the volume. A h20sphere is almost all “center” and has only a diffuse and tenuous “outer” part.

(This is not my observation, but was mentioned by Martin Gardner in a Scientific American “Mathematical Games” article.)

You cant have change without time, but you dont have to actually draw a line from one point to another in order to talk about the properties of a line.

On a surface such as the ground on earth, you can find a position with two references. We use latitude and longitude. If you are flying in an airplane, you would have to include altitude. In most instances a stationary thing can be located with three references. Often referred by X, Y,Z. But if the thing is moving, then you have to include the time of it’s position as well.
It is at positions X, Y, Z, at this time.
It gets far more complex in certain areas of physics though, and under some of the extreme locations in the universe. When brought up along with Einstein, it usually refers to how time is not constant from one observer to another, if their circumstances are different enough.

Roger Penrose was able to show, mathematically, how, close to rotating black holes, space becomes “timelike” and time becomes “spacelike.” The dimensions twist around a little.

Ordinary special relativity changes the measurement of space and time dimensions, but they remain space and time, in the usual sense. I might not observe an event at the same place and time you do, but we’re still looking at events at a place and at a time. But under general relativity, the dimensions can get bent around, and start to change places.

We experience space differently than we experience time, but they are aspects of the same thing, space-time.

The first statement doesn’t make sense, and the second sentence doesn’t justify it. You can have an arbitrary number of dimensions without considering time. A cardboard box exists in three dimensions at any point in time without considering any other point in time or going anywhere.

It does, but it is a mathematical construct and we are unaware of any actual 4-space where such a thing physically exists.