Hypercubes

[parksie]

After reading the book 'Hyperspace' by Michio Kaku and playing around with POV-Ray, I was wondering what sorts of thoughts people here had on the subject of Hypercubes, and different ways of visualising them.

[Sam Finch]

You can't Visualise an actual hypercube as sutch, there are ways of thinking about them, one way is to have Time as a 4th Dimension, just imagine a cube apearing, staying still for a while and then disapearing, It doesn't actually matter how many Seconds you decide represent a metre, relitivity sees the ratio of Distance/time = the speed of light. (eg a year of time is equal in length to a light year) that can be a bit too hard to imagine so just pick a set speed to convert with. You can do this for any 4D shape you like

you don't have to use Time as a 4th Dimension, you can use anything you want as long as you can take a point and define a range, or set of ranges, sutch that a condition can be true inside that range an false outside that range. fo example you could use colour, you select a scale so that the dimensions of your shape never pass outside the visible spectrum. then look at 3 Dimensional x Sections of your shape at every value of the wavelength, if a given point in your shape exists at a particular wavelength then that point emits that wavelength, when you've finished your x sections then mix all the colours each pint reflects to get the colour at that point.

If you ever actually have to imagine a 4D shape the extra Dimension will usually be given, otherwise there would be no reason to consider the shape.

If you want to consider the possibility of 4D space then you will just have to resort to maths, there's no real way of visualising it.

[parksie]

Yeah, I sort of knew about that, but was wondering if people had any differnt methods other than the 'take a 3d slice' method.

[Sam Finch]

Well, it's not something you're gonna be able to think of all at once, I doubt anyone can even picture the entire of a 3D object (including the inside) Just imagine something like a car, try and think of the whole object at once, including all the stuff n the engine, the insides of fuel pipes etc, or a book, try and see every page at once, there's just too much to think about all at once, that's why we can't visualise 4D shapes, there's just to much of them.

[parksie]

Good point. I wonder what sort of methods a 4D person would use to try and visualise a 5D object...

[Sam Finch]

same ones we use to picture 5D objects. try playing 4 or 5 dimensional noughts and crosses(tic tac toe) that might help get a general picture. I've never managed to play properly in anything above 9D.

[Juan Carlos Rey]

Sticking on the geometric side, one can imagine an hipercube the same way one draws a cube on paper (two dimensions): there are, generally speaking, two ways:

1- Draw a square inside a bigger one, and join every corner of the inner square to the corresponding corner of the outer square.

2-Draw a couple sqares off center, and also join corners.

Taking this approach a little further, imagine a cube in perspective. Now draw another cube inside this first one, and join every corner of the inner cube to the corresponding corner of the outer cube. You will end with 8
cubes, all deformed (by effect of the perspective) but still recognizable, and a lot of faces, lines and corners. This can be named an "hypercube" in a two dimensions perspective. That hypercube could also be built in three dimensions, off wire, timber, etc. And then you´ll have an hypercube in a three dimensions perspective.

Sorry I couldn´t draw them here. (And sorry my bad English)

[Edited by Juan Carlos Rey on 08-06-2000 at 11:40 PM]

[parksie]

I've seen number 1 before, but your second suggestion is very intriguing.

[Guv]

Just as you can visualize a cube unfolded as 6 squares in the form of a cross, there is a 3D object which is a 4D Hypercube unfolded. This figure looks like a "4-Story Building" made up of 4 cubes, with four more cubes attached to the faces of the second or third story. There is a Sci-Fi story about a house built like this in California. An earthquake causes the house to fold up into a Hypercube, and the people inside become very confused. I do not know what a 5D Hypercube looks like unfolded.

Many years ago I decided to develop the ability to visualize 4D objects. The follies of youth! After about a year or so I gave up.

I have since concluded that a million or more years of evolution in a 3D world dominated by classical physics has "hard wired" the human brain. It just cannot visualize 4D objects and cannot visualize what goes on in the Quantum world.

However, there are a lot of interesting facts & formulae which can be understood. Ignoring the Quantum world, let us think about Unit Hypercubes, Hyperspheres, .a perhaps a few other subjects in more than 3D.

First, it is obvious that the Hypervolume of a Unit Hypercube is one unit no matter how many dimensions there are. It can easily be proven that the longest diagonal is the square root of the number of dimensions. Approximately 1.414 in 2D, 1.732 in 3D, 2.236 in 5D; Exactly 2 in 4D 3 in 9D, 10 in 100D, 20 in 400D.

Now imagine the largest Hypersphere which can fit inside a Unit Hypercube tangent to all the faces. It has a radius of ½ unit. The formulae for Hypersphere volumes is not easy for a non-mathematician to determine, but if you know Integral Calculus and have time on your hands, you can determine a formula for any number of dimensions. The first few formulae are: Pi*R*R, 4*Pi*R*R*R/3, Pi*Pi*R*R*R*R/2, 8*Pi*Pi*R*R*R*R*R/15, Pi*Pi*Pi*R*R*R*R*R*R/6 (See http://mathworld.wolfram.com/Hypersphere.html & sorry I do not know how to format the formulae better). I hope I did not make a type in any of the formulae. Notice that ½ is raised to the power of the number of dimensions. Also, another factor of Pi shows up every other dimension. There is a fractional term which gets smaller.

If you consider the formulae you are led to the strange (or non-intuitive) fact that as the number of dimensions increase, the fraction of Hypercube volume used by the embedded Hypersphere approaches zero. Consider that in 100D, there is a 10 unit diagonal: The middle unit of the diagonal is inside the embedded Hypersphere, with 9 units outside the Hypersphere. All this indicates that the Hypersphere volume itself approaches zero as the number of dimensions increases.

Next, think about the largest Hyperspheres which can be put into the corners of a Hypercube, but tangent to & outside the embedded Hypersphere. In 2D, you can put 4 circles in the corners of a square tangent to the embedded circle; In 3D, you can put 8 spheres in the corners of a cube tangent to the embedded sphere; In 4D, 16 hyperspheres; Et cetera. Now here comes the weird fact. At some point (well less than 10D, I think) each of the Hyperspheres in the corners are bigger than the embedded central Hypersphere.

Another weird concept from 4D can best be explained by analogy, using imaginary creatures confined to 2D. Imagine little 2D creatures whose universe is the surface of a very large sphere. If the sphere is large enough it seems like a plane to them, but they wonder if their universe is infinite. They start a project to paint bigger and bigger circles centered on a point. They expect to either keep the process up indefinitely or else find themselves between the last circle painted and some barrier. Think of them as starting with a little circle around what we can think of as the North Pole of the earth. Until they draw an equatorial circle, they need more paint for each circle. Then they start needing less and less paint for each circle, which is confusing to them, but obvious to us. As they approach the south Pole, there is a point at which they realize that they are inside the last circle they painted!!!! They now realize that their universe is finite, but unbounded. Their minds are somewhat boggled because they cannot visualize a 3D Hypersphere. I think (not absolutely certain) that there are curved 3D spaces which have an analogous property. If you started building bigger and bigger concentric spheres, you would eventually end up inside the last spheres you built.

[Sam Finch]

I've never really thought about volumes in higher dimensions, (probably because of all the calculus)

When you talk about the Idea of the hyperspheres in the corners of a hypercube, you are right about that, they are equal at 9D


the radius of the corner spheres = (the long diagonal of the hypercube - the radius of the centre sphere) / 2

for a hypercube of side 1 it's long diagonal = sqr(D)
where D is the number of Dimensions

so Rc = (Sqr(D) - 1) / 2 the spherese are of equal volume when Rc = 1

1 = (Sqr(D) - 1) / 2

2 = Sqr(D) - 1

Sqr(D) = 3

D = 9

I disagree with the Idea that the volume of the contained sphere tends to zero as the number of dimensions increases, clearly it does for a unit hypercube, but although I didn't work the equations through it looks like there is a value for the side of the hypercube above which the volume of the contained hyperspheres is strictly increasing with dimensionality.

what is happening is that tha proportion of the hyper-sphere's volume to the volume of the hypercube is decreasing, but we are keeping the volume of the hypersphere the same, so essentially we are comparing volumes of objects in 2 different dimensions (in a round about way) which is invalid, so saying the volume tends to zero is not true.

[Sam Finch]

Sorry, I lied a bit in my last post, there is no side length you can set by which the volume of the internal sphere always increces, whatever side length you set it to the volume will always tend to 0 as the number of dimensions increces. However, for any number of Dimensions you pick you can always pick a side length above which the volume of the internal sphere increces with dimensions up to the number of dimensions you pick. I also still stand by my point about not being able to compare volumes in different dimensions.

[Guv]

Sam, you are correct & incorrect. It is true that the volume of an embedded Hypersphere divided by the volume of the enclosing Hypercube approaches zero with increasing number of dimensions. You are right & I was wrong when I said this implied that the volume of any Hypersphere decreases with increasing dimensions.

You forced me to think a little more about Hypervolumes. The further thought made me realize the implications of the exponential formula for the volume of a Hypercube. For side less than one, the volume approaches zero with increasing dimensions! For side equals one, it is constant (This I always realized). For side greater than one, it grows without bound (I never thought about this). For a Hypercube of side greater than two, embedded Hypersphere has a radius greater than one. Both Hypervolumes grow without limit with increasing dimensions. The ratio still approaches zero.

It is true that a Hypersphere of radius ½ has a volume which approaches zero with increasing dimensions. This is also true for other (perhaps not all) radii less than one.

I do not know how to calculate the radius of the Hyperspheres in the corners of a Hypercube, tangent to & outside the embedded Hypersphere. I disagree with the formula you gave. Your formula assumes that the center of the Hyperspheres is half way between the corner of the Hypercube & the surface of the central hypersphere. Such a Hypersphere would extend outside the Hypercube, rather than being tangent to the faces.

[parksie]

Guv - I read about that story. Didn't they end up over the Empire state building? It's called a tesseract, BTW.

I like all this maths that's floating around - it finally gives me something concrete to play with. Thanx for the replies.

[Sam Finch]

Guv, Good point, I missed that, I think it's a combination of the bit of the diagonal being missed out being so mall in 2 and 3 dimensions and the fact it was 3 'o' clock in the morning when I wrotet it.

This thread looks interesting, but I only read the first post.

Does anyone know how to visualize a "hyper-cone"?

[Sam Finch]

yeah, it's a Sphere with its Radius Increacing/Decrecing at a constant rate. There's an important concept in special relitivity called the light cone. basicly imagine you turn a lamp on somewhere in space, with nothing shadowing it in any direction (it doesn't have a stand or anything, it's just a point light source), the light cone is the set of points in 4Space that are lit by the lamp. I can't remember exactly what it does, I remember that if you are inside it you can't leave it, I think they have something to do with black holes in general relitivity. But I can't really remember.

anyway, the light cone is a 4D hyper cone.

[Guv]

I have no ideas about a hypercone. Never thought about it, and not sure how to start. Here is a try.

Equations of a cone. Note: First 2 are equations of a circle in 2D (If R is constant).
X= R*cos(A)
Y = R*Sin(A)
Z=R*Constant
In the above, R & A are variables.

Equations of hypercone? First 3 are equations of a sphere in 3D (If R is constant).
X1=R*cos(A)*cos(B)
X2=R*cos(A)*sin(B)
X3=R*Sin(A)
X4=R*Constant
In the above, R, A, & B are variables. Thinking of A as latitude and B as longitude helps visualize the sphere.

I am pretty sure that the above are correct, but it does not help me much in visualizing it.

Instead of all points on a circle connected by lines to a point, maybe all points on a sphere connected by lines to a point without any line intersecting the sphere? I do not like this.

Instead of a succession of ever larger circles connected to each other, a succession of ever larger spheres connected to each other. This seems correct, but I cannot visualize it. Imagine just two of the circles from an ordinary cone. Every pair of corresponding points must be connected. Now consider two spheres with every pair of corresponding points similarly connected by lines which do not intersect any of the other points on the two spheres. I cannot do it.

Instead of a 2D object (triangle) twisted in 3D space so that two sides coincide, a 3D object (??) twisted in 4D space so that two Planes (??) coincide? This does not seem right, even if I could decide what 3D object to twist. All I get from this is one cone inside another from twisting a solid triangle.

The hypercylinder is not much different from the hypercone. For a cylinder, the circles are the same size instead of being different sizes like the cone. For a hypercylinder, the spheres are all the same size.

[Sam Finch]

The volume of a HyperCone is easy if we have the volume of a hypersphere

say the volume of an N dimensional hypersphere of radius r is S(N,r) then the volume of a hypercone in N dimensions of base radius R and height h

C(N,r,h) = (0.5)^N-1 * h * S(N,r)

this can be proved with fairly simple calculus

in order to construct a hyper cone you have to rotate a cone about a plane thruogh its central axis (imagine half a cone placed on a mirror to look like a whole cone, the surface of the mirror is the plane you have to rotate it about.

how to rotate a 3d object about a plane is something that I can only describe in maths (and maths with too many symbols to post here) It's not something that I or any normal person could visualise.

you could do a 3D perspective of it in the same way we could do a 2D perspective of a 3D cone, of course you would have to do some shading on it in the same way as you would have to do some shading of a 2D image of a 3D cone. It would probably look something like a 3D cone with funny colours (the trouble is you would need colours on all points inside the cone, and this is probably impossible to visualise)

Within the cone is the only spacetime coordinates that are reachable from the spacetime coordinate at the vertex, says SR.

I'm wondering if it can be visualized with the time axis orthogonal to the three spacial dimensions (instead of just 1 or 2 spacial dimensions).