Originally shared by Robert Jacobson
Don’t forget to join us tonight in the Mathematics Hangout On Air! You can ask us what this crazy shape is and what it has to do with the fourth dimension.
Required reading: “—And He Built a Crooked House—” by Robert A. Heinlein
Except there is no orthogonal 4th dimension.
4D cube projected on 3D hypersurface.
Tessaract, a 4 dimensional cube. Also known as hypercube. (hypercube is term referred to many multidimensional objects.)
Hypercube ? Interesting
This visualization of 3D onto 2D might help this make sense:
Imagine a wire frame cube. Imagine it’s shadow (which is a 3D object projected onto 2D.) The back of the cube will be smaller than the front in the shadow, even though they are the same size in 3D. You can orient it such that the smaller back side square is directly inside the larger front side square — with the corners of both connected by diagonal lines. The front and back squares are the same size, but perspective makes the back side look smaller (and inside) the front square. Rotate it, and it will look like the back square stretches and swells then “swallows” the front square, which is now small.
Another way to think about it is a 3D cube drawn in perspective. All of the lines in the 3D wire frame are actually the same length in 3D, even though perspective foreshortens them in the projection.
The same is true of this hypercube rendering (as well as the 3D cube rendered in 2D):
All faces are the same size and shape.
All sides are the same length.
All angles are right angles.
Closer things look bigger, but aren’t — trick of the perspective projection.
Projections of rotations look exactly like this animation: it looks like one side stretches out then “swallows” the rest. Not true — the back side merely moves in front of the rest.
This is not a hypercube, this is a projection of a hypercube. You cannot see 4 dimensional object, nor you can imagine it. When you draw a cube on a paper, your imagination helps you to see it as a three dimensional object. You cannot draw 4d object on a paper because you’v never seen it.
There’s no difference between a cube and a tesseract??
That’s like asking “what’s the difference between a square and a cube?” and being told, “none.”
Difference between a square and a cube is similar to the difference between a cube and a tesseract. The problem is we can’t “see” the 4th dimension directly, only in our imagination.
Renders like this can be misleading unless you understand what you’re looking at (like understanding the shadow of a wire frame cube rotating.)
Four mutually orthogonal directions is a physical impossibility.
A cube is 3D and a square is 2D. A tesseract is 3D (just look at the OP image if you are struggling with this).
Rick Chandler (and Terry Jiang ) I don’t mean to sound like an authority. I meant only that it sure took me a while to figure out how to visualize it. I looked at my first tesseract and it looked like a cube within a cube. But as I thought more and more about it, and I realized that all the sides are the same size, that all sides are on the outside, that all corners are 90deg, I was finally able to touch it in my imagination.
I really didn’t mean to elevate myself over anyone, but only to caution people that it’s not trivial to do — nor should people give up because at first it makes no sense. For me, it took a lot of thinking through the topology carefully.
There are many things in math that you can’t just hold in your hands and that are very difficult to imagine in your mind. But don’t write those things off for other people just because you can’t do it. Einstein could “see” things that we/most of us can’t. That doesn’t mean what he “saw” didn’t exist.
It doesn’t really matter if there is any such thing as the 4th dimension — we certainly can’t go there any more than a Flatlander could understand or go to our world. It may not exist anywhere in this Universe or beyond our Universe. However, it is a true mathematical construction, and so it does really exist in mathematics.
Think of it this way: the Mandelbrot Set cannot exist in the real world — real, solid things are limited by the finite reality of atoms and sub-atomic particles. So how do you imagine an infinitely long boundary within a finite space? Because fractals are infinitely “curly” and as you zoom in — forever — you never get to a straight line.
For some, this is a terribly hard thing to imagine — and I don’t blame them! A person’s experience is that things have a minimum smallness that stops at quarks. (Ok, honestly, no one’s true experience goes through that — we can “observe” quarks by using instruments, but no one can see them with their eyes.) So how can imagine something that can’t exist within our real life experience? It’s not impossible. And it takes a good bit of letting go of your common sense understanding of the world.
So much of physical science is the understanding of counter-intuitive things through theory, proven only after cleverly crafted experiments. Is light a particle or a wave? It is both, and that’s terribly hard to imagine. But the theory holds, both in the mathematics and also in experiment! We can’t really understand something that is both, but we have to trust that this is how it is, because this theory actually holds water. It’s provable.
The 4th dimension is real in mathematics, and tesseracts are a real representation of what a 4th dimensional “cube” would look like. _Of course_ it is outside of our experience! And it may be impossible for us to experience, even if such a dimension were actually physically real, because our brains are wired for 3D, and we have only 2 eyes for 3D stereo.
But we have a vastly creative imagination, and we can imagine things that are impossible to experience — light “wavicles,” infinite fractal boundaries, quarks, and the 4th dimension.
(At minimum, please don’t try to tell me what I can imagine or not imagine. Because I surely can imagine a tesseract all by myself. 🙂 )
Ah, this thread is bringing me back to my youth when I curled up with the old (when it was still good) Scientific American. It’s making me remember how much I really enjoyed topology — trying to imagine projections of non-Euclidian geometries into our world.
A circle drawn on the surface of a sphere does not have the same value of Pi as one drawn on a flat surface — the radius is longer. How does the projection of a spherical dimension work if you take it into the 3rd dimension? Is there such a space where a real circle that you and I could draw on paper would come out with a different ratio of circumference to diameter?
A circle drawn on a saddle shaped topology, similarly, has a different ratio too — the radius is shorter in that case. Can the 3rd dimension be saddle-shaped?
These questions are not necessarily theoretical. It just might be that our Universe is non-Euclidian! But that the circles we draw — like being on a very very very very large sphere — are as good as flat, and so we get Pi as our answer and we think our Universe is Euclidian.
But that’s not necessarily the case. Theory suggests that space might be curved by gravity.
Robert Jacobson you wanna back me up on any of this discussion? 🙂 (Still getting use to G+ and hope this will give you access to the entire thread, and discussion of the existence, or non-existence, of tesseracts.)
PS I’m atheist. 🙂 No god or magic necessary. 🙂
Paul James a tesseract is a 3D projection of a 4D object.
In fact, the movie above is a 2D projection of a 3D rotation of a 4D object. 🙂 We “see” 3D from the rotation because our brains can decipher 3D from movement (parallax, or course.)
(And I find it ironic that folks call that object 3D, since it obviously is viewed on a flat display. 🙂 Parallax is very convincing!)
The OP graphic is a representation of a 3D object. Four mutually orthogonal directions is physically impossible.
Here’s an interesting image to consider: a 3D cube projected into the 2nd dimension.
This is not a cube. It’s a projection of a cube into 2D.
Note that because of perspective, the back side of the cube looks smaller than the front side. But — because we have experience with the 3rd dimension — we know this is not true.
We know that the smaller square is the same size as the bigger one that encloses it. Similarly, we also know that the other 4 sides that look skewed, are also the same size and shape. We know that all lines in 3D are the same length. We also know that all angles are 90deg.
Our experience in 3D allows us to understand this projection, even though in 2D the back square is smaller in the projection, the sides are skewed, the lines are different lengths, and many of the angles are not 90deg.
It is so natural for us to understand this projection in 3D we take it for granted as a cube.
That understanding in 3D is very natural and we’re wired for it.
Understanding 3D projections of 4D objects is far far less natural.
Rick Chandler those links explain a lot about you. 🙂
If you’re limited by what you can experience directly, then most of science and mathematics is beyond your reach. Even history requires an imagination.
Hypercube I think.??
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