Area enclosed by a tesseract (4th Dimensional Hypercube)

[jmanela]

I was reading this book the other day The Time Machine by H.G. Wells, and he described moving through dimensions. I pondered the thought a little bit, and began wondering. If the area of a square is L x W, and the volume of a cube is L x W x H, then what is the area enclosed by a tesseract or a fourth dimensional hypercube. Is it the amount of space it takes up within multiple dimensions? Maybe… What do you think?

[DiscoKittyPrime]

Well, if you take the idea from The Time Machine, then you would take time to be the fourth dimension. In which case, the volume occupied by the tesseract would be L x W x H x T with “T” being the amount of time occupied.

[BornaE]

WXY*Z

W is the extra dimension in the Minkowski space.

[RyanCahoon]

@BornaE

WXY*Z

W is the extra dimension in the Minkowski space.

I would agree this holds for Euclidean 4-space, but is this measure of hyper-volume valid for Minkowski space, since only three of the dimensions are Euclidean?

@H. S. M. Coxeter

Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H. G. Wells in The Time Machine, has led such authors as J. W. Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski’s geometry of space-time is not Euclidean, and consequently has no connection with the present investigation.

[gblake]

@jmanela

If the area of a square is L x W, and the volume of a cube is L x W x H, then what is the area enclosed by a tesseract or a fourth dimensional hypercube.

It looked like you were going to ask for a formula for a hypercube’s 4-dimensional hypervolume (WxXxYxZ), but then you asked about area. Which did you want to learn?

This web site claims to have some answers for you. http://www.physicsinsights.org/hypercubes_1.html

[jmanela]

@gblake

It looked like you were going to ask for a formula for a hypercube’s 4-dimensional hypervolume (WxXxYxZ), but then you asked about area. Which did you want to learn?

This web site claims to have some answers for you. http://www.physicsinsights.org/hypercubes_1.html

thanks, that site answered my question. I was just confused about what the volume of the tesseract would be.

[Karthik]

@RyanCahoon

I would agree this holds for Euclidean 4-space, but is this measure of hyper-volume valid for Minkowski space, since only three of the dimensions are Euclidean?

It’s very common for people to confuse the properties a Euclidean Space and a Minkowski Space. Many properties carry over from a Euclidean Space, but some crucial ones do not. When studying Minkowski Space it’s critical that the student takes the time to understand the differences and why they exist. The biggest one of course being the differences in the definition of the space’s inner product.