Just curious about 4d

[Moontanman]

If a two dimensional object that is 2 x 2 inches has four square inches and a 2 inch cube has 8 cubic inches how many would a four dimensional object that is 2 inches on each side hold? Would it be proper to assume it would have 16 cubic inches or would it contain another measurement like 16 tesseracts?

 

 

[Janus]

I would say that it has 16 quatric inches.

 

And since a square would have a perimeter of 8 in( four 2 in edges) and a cube a surface area of 24 square inches( six 4 sq in. sides), I would assume that a tesseract would something like a "surface" volume of 64 cubic inches(eight 8 cubic in. cubes)

[Moontanman]

I would say that it has 16 quatric inches.

 

And since a square would have a perimeter of 8 in( four 2 in edges) and a cube a surface area of 24 square inches( six 4 sq in. sides), I would assume that a tesseract would something like a "surface" volume of 64 cubic inches(eight 8 cubic in. cubes)

 

 

Thank you!

[Keen]

It kind of depends on how you measure things, but to keep it simple, yeah a tesseract with 2 inches on each side would have a volume of 16 quadric inches.

 

How to define volume is quite a large topic in mathematics called measure theory and the most common volume encountered is what is called the Lebesgue measure, Lebesgue measure is what people commonly refer to as "volume" or "area" and can be generalised to any number of dimensions.

[DevilSolution]

Its strange how we use multiple dimensions in maths yet were bound to 3 dimensional reality + time. Does this mean the definitions are different? And how do you comprenhend higher mathematic dimentions (dismissing time). Like on paper it can be formulated but can it be comprehended in the mind?

 

Sorry for the tangent.

Edited September 7, 2015 by DevilSolution

[Keen]

In math, a dimension higher than 3 is usually used, because geometric analogies help to solve a lot of problems. If you want to solve for example equations with lots of variables, you view those equations as some geometrical entity in a higher dimensional space and sometimes, you can find interesting results concerning these equations The definitions stay the same for all dimensions. In basic geometry you can define a dimension as the number of coordinates in a coordinate system. In dimension two, you only need two coordinates to find a point (x,y), in dimension 3 you need three coordinates (x,y,z), in dimension 4, you need four coordinates (x,y,z,t) and so on...

It's of course not obvious to imagine something that has more than 3 dimensions, but what usually helps is to work with such spaces and explore their properties on paper. You find out, that lots of things which are true in dimension 3 are also true in higher dimension as well, so there are some analogies that can be found.

If you want to acquire some intuition concerning higher dimensional spaces, I'd suggest to wait for the release of a quite clever video game called Miegakura in which you have to solve puzzles in 4 dimensions.