Exploring 4D Space: Mysteries of Hypercubes

[onethatyawns]

I've read the descriptions and watched the videos over and over that describe these shapes, and my skepticism has simply not gone away. Can anybody explain what I might be missing?

Here is my reasoning:
Physics supercedes, or at the very least, is parallel to math. Math is not something that produces meaningful results on its own. Math is merely abstracted physics; it's never really led the pack in terms of revolutionizing scientific knowledge. Any great math discovery seems to be paired with a great physics advancement. Therefore, the math was simply the language of the physics discovery itself. It was not a separate thing.

And with that said, the hypercube, as I understand it, is not a shape that exists. There may be some equations that point to more than 3 spatial parameters, such as the Kaluza-Klein, but I intuitively feel that this is misunderstood. This is an abstraction gone astray. It cannot mean 4 literal spatial dimensions.

 

[jedishrfu]

Math and Physics don't supercede one another. Math is the language of Physics. It allows physicists to analyze and predict things from their experiments. Its more accurate to say physics theory drives experiment and experiment drives physics theory and all this is done thru math and measurement.

With respect to Math, it can be used as the language of choice for many other fields as well. Its a tool that provides us with a logical and self-consistent means to study problems. However, Math is even more than that and has been used to extend itself into other even more abstract areas that someday may find some practical use in our everyday lives.

 

[Mark44]

I've read the descriptions and watched the videos over and over that describe these shapes, and my skepticism has simply not gone away. Can anybody explain what I might be missing?

Here is my reasoning:
Physics supercedes, or at the very least, is parallel to math. Math is not something that produces meaningful results on its own. Math is merely abstracted physics; it's never really led the pack in terms of revolutionizing scientific knowledge.

I disagree. There are many concepts in mathematics that did not arise out of physics, with just a couple being the Pythagorean theorem and the Quadratic formula, not to mention all of trigonometry.

Any great math discovery seems to be paired with a great physics advancement. Therefore, the math was simply the language of the physics discovery itself. It was not a separate thing.

There are a number of concepts in mathematics that came about with no connection to any applications, let alone physics. One of these is the algebra developed by George Boole, that bears his name, that eventually became the basis for th logic used in computer circuits. Another example is number theory, that was at first considered "pure mathematics." Much of it has more recently found to be applicable in cryptography.

And with that said, the hypercube, as I understand it, is not a shape that exists. There may be some equations that point to more than 3 spatial parameters, such as the Kaluza-Klein, but I intuitively feel that this is misunderstood. This is an abstraction gone astray. It cannot mean 4 literal spatial dimensions.

We cannot perceive more than three spatial dimensions. However, mathematics has no such limits, and can deal with spaces with much higher dimensions. One of the applications of high-dimension vector spaces is error-correction algorithms in digital media such as CDs. In this case, the mathematics came first, and the application of it came much later.

Your argument that physics supersedes (I think you mean "precedes") does not take into account the examples I have mentioned. There are many more.

 

[nsaspook]

A classic example of 4D space math is in most computer games for 3D rotations using quaternions. Simple XYZ rotations will 'gimbal lock' when two planes overlap and is sequence order dependent. Using '4D' space is one solution this problem. Three.js is a very easy to use JavaScript API that using quaternions internally.

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/geometric/unit4dSphere/
http://www.gamasutra.com/view/feature/131686/rotating_objects_using_quaternions.php
http://threejs.org/

 

[onethatyawns]

A classic example of 4D space math is in most computer games for 3D rotations using quaternions. Simple XYZ rotations will 'gimbal lock' when two planes overlap and is sequence order dependent. Using '4D' space is one solution this problem. Three.js is a very easy to use JavaScript API that using quaternions internally.

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/geometric/unit4dSphere/
http://www.gamasutra.com/view/feature/131686/rotating_objects_using_quaternions.php
http://threejs.org/

Going through that first link, I'm not reading that there are 4 spatial dimensions, as the hypercube concept attempts to explain. It appears to just be an algorithm for moving within 3 spatial dimensions.

 

[Mark44]

Going through that first link, I'm not reading that there are 4 spatial dimensions, as the hypercube concept attempts to explain. It appears to just be an algorithm for moving within 3 spatial dimensions.

The information that nsaspook linked to uses a four-dimensional space (which we cannot perceive, being creatures of strictly three spatial dimensions) to perform various transformations of objects in three dimensions.

 

[onethatyawns]

I disagree. There are many concepts in mathematics that did not arise out of physics, with just a couple being the Pythagorean theorem and the Quadratic formula, not to mention all of trigonometry.
There are a number of concepts in mathematics that came about with no connection to any applications, let alone physics. One of these is the algebra developed by George Boole, that bears his name, that eventually became the basis for th logic used in computer circuits. Another example is number theory, that was at first considered "pure mathematics." Much of it has more recently found to be applicable in cryptography.

We cannot perceive more than three spatial dimensions. However, mathematics has no such limits, and can deal with spaces with much higher dimensions. One of the applications of high-dimension vector spaces is error-correction algorithms in digital media such as CDs. In this case, the mathematics came first, and the application of it came much later.

Your argument that physics supersedes (I think you mean "precedes") does not take into account the examples I have mentioned. There are many more.

Thank you for the solid reply.

 

[homeomorphic]

I disagree. There are many concepts in mathematics that did not arise out of physics, with just a couple being the Pythagorean theorem and the Quadratic formula, not to mention all of trigonometry.

I don't think those are very good examples because geometry can be considered as a kind of physics. It's just not traditionally called that, but it is physics because it makes physical predictions. Take a right triangle. Measure the sides other than the hypotenuse. Hypothesis: the length of the hypotenuse will agree with the value obtained from the Pythagorean theorem. Experiment: measure the hypotenuse. Result: agrees with hypothesis, within reasonable error bounds. And just as with Newtonian mechanics, if you extend the theory beyond its reach, it won't work. For example, if you tried this near a black hole.

On some level, I'm not sure there's really anything outside of physics, math or not. That's what's so great about physics. It covers everything. That doesn't mean that mathematicians are always thinking about physics, though, at least in the sense of the formal discipline of physics, when they come up with things.

This is an abstraction gone astray. It cannot mean 4 literal spatial dimensions.

It is helpful to think of things as being higher dimensional spaces. This isn't so clear to someone who hasn't developed their higher dimensional intuition, but it allows us to think of complicated things that have many degrees of freedom, using visual intuition and analogies with lower dimensional space. It's not unphysical. One of the big examples of a higher-dimensional space is a configuration space in classical mechanics.

http://en.wikipedia.org/wiki/Configuration_space

This doesn't mean that these spaces actually exist physically, only that they are a useful conceptual tool for reasoning about physics.

 

[onethatyawns]

I don't think those are very good examples because geometry can be considered as a kind of physics. It's just not traditionally called that, but it is physics because it makes physical predictions. Take a right triangle. Measure the sides other than the hypotenuse. Hypothesis: the length of the hypotenuse will agree with the value obtained from the Pythagorean theorem. Experiment: measure the hypotenuse. Result: agrees with hypothesis, within reasonable error bounds. And just as with Newtonian mechanics, if you extend the theory beyond its reach, it won't work. For example, if you tried this near a black hole.

On some level, I'm not sure there's really anything outside of physics, math or not. That's what's so great about physics. It covers everything. That doesn't mean that mathematicians are always thinking about physics, though, at least in the sense of the formal discipline of physics, when they come up with things.

This is exactly what I was arguing. You can't just make up a shape that could never exist in real life, in the three dimensional world. You can create an equation with more dimensions, but that's just an algorithm like you explained in the other part of your post. I don't know if higher dimensional intuition is really valuable. Maybe it's better to take higher dimensional algorithms and still imagine them as 3D shapes? Perhaps we are arguing semantics, but I would consider a 3D projection of a 4D object to merely be a 3D object. That doesn't count as actually visualizing 4D space, which is why I refer to that as seeing a 3D algorithm.

Thanks for the link though. I'll check it out.

 

[homeomorphic]

This is exactly what I was arguing. You can't just make up a shape that could never exist in real life, in the three dimensional world.

Why not? If it helps you to think about things, then you should absolutely go ahead and make up shapes that could never exist in real life.

You can create an equation with more dimensions, but that's just an algorithm like you explained in the other part of your post. I don't know if higher dimensional intuition is really valuable.

Then, you haven't done enough math or physics. One reason why it's valuable is because visualization is always helpful as a memory aid, if nothing else. I'm pretty good at linear algebra. Why? Because I can imagine higher dimensional vector spaces and linear transformations. I understand it better, remember it better, and see more beauty in it because I can visualize it. The fact that I'm only visualizing it by analogy doesn't make much difference. If you prefer, I think about 4 dimensional spaces visually, rather than actually visualizing them. That's a subtle, but important distinction.

Maybe it's better to take higher dimensional algorithms and still imagine them as 3D shapes? Perhaps we are arguing semantics, but I would consider a 3D projection of a 4D object to merely be a 3D object. That doesn't count as actually visualizing 4D space, which is why I refer to that as seeing a 3D algorithm.

Yeah, but it's a little more than that because you have to remember that what you are picturing isn't the real deal. It's just an analogy for your brain to grab onto that helps you think about it. Otherwise, you may risk getting confused and coming to false 3-dimensional conclusions about your 4-dimensional object. That's part of why I think mathematicians emphasize rigor and writing down formal proofs to make sure you are getting it right. But as far as the inspiration goes, it may well be visual. Not always, but it is a powerful way of thinking.

 

[onethatyawns]

Why not? If it helps you to think about things, then you should absolutely go ahead and make up shapes that could never exist in real life.

Name a shape that you can imagine but can't exist in real life.

 

[homeomorphic]

Name a shape that you can imagine but can't exist in real life.

That's what I've been doing when I said configuration spaces. Of course, you may say that you are only imagining a shadow of the actual shape to get a handle on it. So, maybe you're getting hung up on the semantics of what we mean by a shape. Maybe what we really mean is a mathematical object whose lower dimensional-analogues are things we can visualize.

 

[onethatyawns]

I think if we are to argue that a 3D projection of a hypercube (or some other higher dimensional shape) is a real object, then I would make a counter point. Hypercubes account for their 4th dimension by moving. Therefore, motion is a dimension. This means that a point moving for a period of time is a line segment, a line segment moving for a period of time is a polygon (assuming the line doesn't move precisely in the direction that its original point moved in), a polygon moving in a direction is a 3D Euclidian object (again, assuming it doesn't move in the direction of its previous two movements), a 3D object moving in space is a 4D object similar to a hypercube, a 4D object moving in space (essentially a 3D object with two orthogonal motions, a direction and a spin) is a 5D object.

That's actually how I perceive derivatives/antiderivatives, so that perfectly matches my thoughts. In fact, it makes it clear that electrons are 5 dimensional objects, with a height, width, depth, linear velocity, and angular velocity. I can't speak with authority, but maybe that correlates to the Kaluza-Klein unification that gave space 5 dimensions.

Going beyond 5 dimensions is a bit harder, but maybe we can do it. A 6D object would be like a 5D object that oscillates in one dimension orthogonal to its linear velocity (like up and down). A 7D object would be like a 6D object except it is now oscillating in two directions orthogonal to its linear velocity (now it's moving in circles towards its linear vector while also spinning about its own axis). I'm not sure what to do next with this model of dimensions, but I will think...

**PS I feel quite certain now that this is a superior way to make space and motion intuitive. Therefore, 6D objects DO exist in our world. We can see them everyday in the form of photons, which have a wavelength, a spin, and a velocity. It's argued they are massless and therefore width/height/depth-less, but that's besides the point. A 5D object exists in our world. It's an electron. Also, adding more dimensions to our typical conception of 3 spatial dimensions DOES bring more accurate predictions, as per string theory, because there can be abstracted layers of motion within motion, and each of those motions are a dimension.

Try modeling an electron in a carbon atom, on Earth which is spinning around itself, on Earth which is elliptically moving around the sun, in the solar system which is moving around the center of the milky way, in the milky way which is moving away from the center of the universe. Each layer of abstraction must be an actual dimension according to the seemingly logical marriage of dimension and motion.

 

[onethatyawns]

That's what I've been doing when I said configuration spaces. Of course, you may say that you are only imagining a shadow of the actual shape to get a handle on it. So, maybe you're getting hung up on the semantics of what we mean by a shape. Maybe what we really mean is a mathematical object whose lower dimensional-analogues are things we can visualize.

Yes, I think this is semantics, like I said either. My quabble, which is seemingly insignificant to you but is the entire point I am making, is that "lower dimensional-analogues" are all that exist. That is all it could be. To say that there exists a world which has 4 orthogonal spatial dimensions is pure fantasy. However, look at the post that I just made. Motion may be the solution to this contradiction if we take motion to be a literal dimension, like we do with the hypercube "projection".

 

[homeomorphic]

I think if we are to argue that a 3D projection of a hypercube (or some other higher dimensional shape) is a real object, then I would make a counter point. Hypercubes account for their 4th dimension by moving. Therefore, motion is a dimension.

There are 4-d objects in space-time is what you could say.

In fact, it makes it clear that electrons are 5 dimensional objects, with a height, width, depth, linear velocity, and angular velocity. I can't speak with authority, but maybe that relatives to the Kaluza-Klein unification that gave space 5 dimensions.

Well, electrons are either 4-d if you're thinking of them as things that exist in space time, or if you're thinking more along the lines of configuration space, that's going to be an infinite-dimensional Hilbert space.

Going beyond 5 dimensions is a bit harder, but maybe we can do it. A 6D object would be like a 5D object that oscillates in one dimension orthogonal to its linear velocity (like up and down). A 7D object would be like a 6D object except it is now oscillating in two directions orthogonal to its linear velocity (now it's moving in circles towards its linear vector while also spinning about its own axis). I'm not sure what to do next with this model of dimensions, but I will think...

That's something like what configuration spaces do. One of the examples is just n point particles. Each one has 3 directions it can wiggle in, so if you have n particles, they can wiggle in 3n different directions, so you get a 3n-dimensional configuration space and 6n-dimensional phase space if you want to describe velocities as well as positions. If you have particles that aren't just points, you would get more degrees of freedom.

 

[homeomorphic]

In some sense, these configuration spaces do exist in real life, by the way, provided you realize that the mathematical configuration space is only a model for the real life thing.

 

[onethatyawns]

Well, electrons are either 4-d if you're thinking of them as things that exist in space time, or if you're thinking more along the lines of configuration space, that's going to be an infinite-dimensional Hilbert space.

Why isn't an electron 5 dimensional? Yes, it moves linearly in 4D space, having x, y, z in displacement over t time, but does it not also have an angular velocity accounted for as its charge? That means 5 pieces of information or 5 dimensions.

That's something like what configuration spaces do. One of the examples is just n point particles. Each one has 3 directions it can wiggle in, so if you have n particles, they can wiggle in 3n different directions, so you get a 3n-dimensional configuration space and 6n-dimensional phase space if you want to describe velocities as well as positions. If you have particles that aren't just points, you would get more degrees of freedom.

Awesome. Do you have any suggestions on reading material?

 

[homeomorphic]

Well, they are still sort of imaginary. It's like the set of positions that your object COULD find itself in. So, we actually only observe one point in the configuration space at a time. And over time, we just see things moving along paths through the configuration space. So, it's kind of Platonic scenario, like how real circles are only reflections of some idealized circle that we can imagine. I don't know if I'm a Platonist or not. What it really means is that there's distinction between the actual things and the mathematical model of them.

 

[onethatyawns]

In some sense, these configuration spaces do exist in real life, by the way, provided you realize that the mathematical configuration space is only a model for the real life thing.

I don't agree; that's my fundamental disagreement. Something in math has to exist in the world (or it could exist if we created it). See my point about going from the electron to the universe. To model that motion, you would need a dang ton of dimensions. If you want to say "but they're still moving in the same spatial dimensions if you average it out", then I would point you to the very first abstraction of spatial size to movement, space-time.

 

[onethatyawns]

Well, they are still sort of imaginary. It's like the set of positions that your object COULD find itself in. So, we actually only observe one point in the configuration space at a time. And over time, we just see things moving along paths through the configuration space. So, it's kind of Platonic scenario, like how real circles are only reflections of some idealized circle that we can imagine. I don't know if I'm a Platonist or not. What it really means is that there's distinction between the actual things and the mathematical model of them.

If that object itself is motion (which correlates to my point-->line-->2D-->3D model) (which, thanks to modern physics, we now know is true: most of materials are empty space with occasional moving chemicals, which are full of empty space but occasionally moving particles), then it's configuration space is not just an imaginary model; it's the real thing.

 

[homeomorphic]

If that object itself is motion (which, thanks to modern physics, we now know is true: most of materials are empty space with occasional moving particles), then it's configuration space is not just an imaginary model; it's the real thing.

Maybe things are getting too philosophical here.

 

[onethatyawns]

Maybe things are getting too philosophical here.

The only philosophical point is that all math must exist in the real world. I can't prove that, but I haven't been disproven either. Just an intuition.

I think I'm using solid definitions of what a dimension is. I've been told that time is a dimension because a point can exist at the exact same coordinates two different times, and it will be in separate places in time-space. Everything else is just an antiderivative of that concept.

 

[homeomorphic]

Why isn't an electron 5 dimensional? Yes, it moves linearly in 4D space, having x, y, z in displacement over t time, but does it not also have an angular velocity accounted for as its charge? That means 5 pieces of information or 5 dimensions.

If you're going to think quantum mechanically, it's going to be a wave function. That's very infinite-dimensional. It's like a vibrating string. Infinitely many degrees of freedom because you have to specify what each point on the string is doing and there are infinitely many points. The Schrodinger equation does something similar to the wave equation.

Awesome. Do you have any suggestions on reading material?

The Road to Reality by Roger Penrose or maybe here:

http://math.ucr.edu/home/baez/classical/#lagrangian

 

[homeomorphic]

The only philosophical point is that all math must exist in the real world. I can't prove that, but I haven't been disproven either. Just an intuition.

Here's a "proof" (well, not quite): write down a mathematical statement without making it appear in the physical world. Better yet, don't even write it down. Just try to think a mathematical thought that transcends the confines of your physical brain. But then I'm a philosophical materialist, so...anyway, we'd better not get any more philosophical or else they will get mad at us, if they aren't already (hopefully not, since this is both short and intended to cut off further discussion in this direction).

I think I'm using solid definitions of what a dimension is. I've been told that time is a dimension because a point can exist at the exact same coordinates two different times, and it will be in separate places in time-space. Everything else is just an antiderivative of that concept.

Well, time is not really a dimension because space and time are somewhat interchangable in relativity. It's just that space-time is 4-dimensional. Not sure what you mean by anti-derivative.

 

[onethatyawns]

Well, time is not really a dimension because space and time are somewhat interchangeable in relativity. It's just that space-time is 4-dimensional. Not sure what you mean by anti-derivative.

Antiderivative, meaning to iterate more orders. I realized after saying that, there are orders of magnitude and there are orders of direction (dimension, like x, y, z), which are not necessarily the same thing. However, perhaps by following my previous explanation of dimensions, orders of magnitude can be modeled with symmetric orthogonal orders of direction too.

 

[Philip]

Ah, yes, higher dimensional objects. These entities are strange, aren't they? So unimaginable, they defy intuition and common 3D sense. I know the feeling. Even though it's impossible to represent a true 4D object in it's full 4D glory, we can still cancel out a dimension and view it in 3D. Either cross sections or shadow projections can do the trick. Depending on the type of shape you're working with, one works better than the other.

But, before one goes exploring 4D shapes in 3D slices, it's good to get your bearings with what you're going to be seeing. The problem is, we have to break our 3D sense and start thinking in 4 dimensions. Since we have a natural grasp on basic 3D shapes, viewing them in 2D slices will make more sense to us, with how it comes out. We understand quite well how the surface of the 3D object is a way of predetermining all 2D slices.

The trick is to imagine the whole 3D shape, while exploring in 2D. While imagining the 3D whole, try to place where the 2D plane is slicing. It's important to understand how we are unable to see the rest of the object, even though we know it's still there. So, when we move on to exploring 4D objects in 3D slices, our minds will be better prepared for this method of aided visualization.

Watching the 3D slices morph by moving a 4D shape will make more sense this time, instead of only thinking in 3D. We have to consider how we are unable to see the 4D whole, even though it is still there. And how the shape of this 4D object predetermines what all the 3D slices will be, depending on angle and depth.

Exploring 3D Torus in 2D: https://www.desmos.com/calculator/otpsrtlnww
-- adjust 'm' to slide, 'n' to rotate in 3D

Exploring 3D Cylinder in 2D: https://www.desmos.com/calculator/otpeykbx8g
-- adjust 'm' to slide, 'n' to rotate in 3D

Exploring 3D Cone in 2D: https://www.desmos.com/calculator/w3xptfnyhb
-- adjust 'a' to slide, 'b' to rotate in 3D

Exploring 3D Tetrahedron in 2D: https://www.desmos.com/calculator/ymijhsdgxc
-- adjust 'a' to slide, 'b' and 'c' to rotate in 3D

Exploring 3D Triangle Prism in 2D: https://www.desmos.com/calculator/1mmbj9339n
-- adjust 'a' to slide, 'b' and 'c' to rotate in 3D

Exploring 3D Square Pyramid in 2D: https://www.desmos.com/calculator/xupqv2qyq7
-- adjust 'a' to slide, 'b' and 'c' to rotate in 3D

Exploring 3D Cube in 2D: https://www.desmos.com/calculator/fgqzkxuntu
-- adjust 'a' to slide, 'b' and 'c' to rotate in 3D

And, since we can write an equation for a discrete three dimensional object, we can also write them for 4D shapes and beyond. These are objects of finite extent, having four true dimensions of space. Instead of thinking " how is that possible, it's only imaginary" , it's more to do with allowing it to be possible, and writing an equation that uses four dimensions:

http://www.reddit.com/r/hypershape/comments/2p1q3l/the_math_of_shapes_the_shape_of_math/

Here's a gallery of passing several types of 4D hyperdonut shapes through 3D. These are the four fundamental toroidal rings in 4D:

http://imgur.com/a/asnRZ

For some other types of shapes, see here:

Cyltrianglinder, cartesian product of triangle and circle : http://imgur.com/a/DvMEj

Tesseract, 4D hypercube : http://imgur.com/a/9rdLp

But, of course, there's no guarantee in wrapping one's mind around the concept. Usually not on the first couple times you try. It does actually take time, to the point where you have memorized the visuals. The main point is to compare what's happening in a lower dimension when an object gets scanned. We tend to see some of the same shape morphing patterns when going from 2D -> 3D -> 4D, especially with the toroidal shapes. We have to make the connection that the 3D things morphing is just a scan of an object that extends further than 3D. Even though we can't see all of it, the shape still has a precise physical structure, of finite extent.