How Many Dimensions Are There?
Date: 03/29/2001 at 19:24:09 From: Adam Subject: Four dimensions Doc, I don't get it. I was playing with some simple shapes and looking at some of Escher's work, and I started thinking about four dimensions. I asked some people and got some brief references, and I understand what everyone's saying, but there's still something I'm uneasy about. Let's say we have a picture of a four-dimensional cube. Now, we see things in three dimensions, so the picture is a representation of a four-dimensional object in a three-dimensional environment, right? So the picture is just a representation. Now on to the questions: 1. We're just representing four-dimensional objects, so how do we know there are even four dimensions? Is it an assumption? How many dimensions are there? 2. What is the significance of drawing four-dimensional cubes? 3. If I were in a two-dimensional environment/world/realm/universe/ whatever, how would I be able to represent three dimensions? Could you recommend any books to me that you think I might find interest in reading? Thanks for your time, Adam
Date: 03/29/2001 at 23:20:07 From: Doctor Peterson Subject: Re: Four dimensions Hi, Adam. First, I wonder if you have tried searching our archives for the phrase "four dimensions" or "fourth dimension"? We get a lot of questions in this area, and you can find some good insights there. Let's look at your questions: >1. We're just representing four-dimensional objects, how do we know > there are even four dimensions? Is it an assumption? How many > dimensions are there? Mathematically, it doesn't matter whether there "are" four dimensions. We can still talk about the concept. In fact, we can talk about infinite-dimensional spaces! Actually, any time we work with something that involves four different variables, we are working in a four-dimensional space; for example, a problem involving three space dimensions plus time involves four dimensions. And if we deal with both position and velocity, we are working with six dimensions. I think you're really asking about extra dimensions of SPACE. That takes you out of math and into physics; there are some answers about that in our archives, but I can't say much about it. >2. What is the significance of drawing four-dimensional cubes? It just stretches your mind. I don't know that it has any special use. >3. If I were in a two-dimensional enviornment/world/realm/universe/ > whatever, how would I be able to represent three dimensions? Well, you are probably already used to representing three dimensions in a two-dimensional world; that's what you're doing when you draw on paper. This is a two-dimensional representation of a cube: +----+ /| /| +----+ | | +--|-+ |/ |/ +----+ The usual representations of a four-dimensional "cube" are done very similarly. (In fact, if you've seen them in books, you were really looking at 2-dimensional versions.) If we actually lived in two dimensions, I don't think we could draw at all, unless everything were somehow transparent so we could see through it. Think about it - if you just drew a square, how much of it could you see from inside the piece of paper? Here are a few answers I found in our archives: The Fourth Dimension http://mathforum.org/dr.math/problems/dixon24.html Beyond the Third Dimension http://mathforum.org/dr.math/problems/rob5.16.96.html Fourth Dimension http://mathforum.org/dr.math/problems/zeto5.13.97.html Tesseracts and Hypercubes http://mathforum.org/dr.math/problems/smith5.22.97.html Visualizing a Klein Bottle http://mathforum.org/dr.math/problems/sean.8.19.99.html The third is especially relevant to your questions. You may also want to read _Flatland_ by Edwin A. Abbott, about worlds of different dimensions: http://www.math.brown.edu/~banchoff/gc/Flatland/ - Doctors Peterson and TWE, The Math Forum http://mathforum.org/dr.math/
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