Date: 05/13/97 at 19:19:15 From: alexszeto1 Subject: hi Hi Dr.Math, I'm a freshman in high school and I'm doing a research paper on the fourth dimension. Will you please help me try to understand what the fourth dimension is? If you could, can you also include any information you know about the fourth dimension? Thank you for your time, Jody
Date: 06/25/97 at 13:34:41 From: Doctor Barney Subject: Re: hi The fourth dimension is not so much a thing as it is an idea. This is like the fact that the number 3 isn't really a thing in and of itself. You can write a symbol to represent 3 on a piece of paper or a chalk board, you can use the idea 3 to describe how many apples you have, and you can even buy a birthday candle in the shape of a 3, but you haven't really bought "three," have you? Mathematics is a philosophy which we use primarily to describe physical phenomena. The most obvious example is the way Euclidean geometry (three-dimensional geometry) describes physical space. That is because the physical space as we understand it has three independent "degrees of freedom," or three directions in which we and the other objects in space are able to exist, expand, or move. For example: length, width, height; latitude, longitude, altitude; Elm street, fourth building down, second floor; x, y, z. Now, the idea of the fourth dimension (or the idea of the first four dimensions all together) is an idea we use to describe any quality, state, object, event, or concept which requires four independent degrees of freedom (ways in which it is able to be different) in order to describe it completely. For example: length, width, height, weight; latitude, longitude, altitude, temperature; Elm street, fourth building down, second floor, 9 O'clock on Thursday; x,y,z,w. It's really that simple. The only reason people get confused about it is because they cannot visualize it. If I tell you the length, width and height of an object, you can get an idea of what it looks like, perhaps a cube or a long slender rod or anything in between. But if I also tell you what temperature something is or how much it weighs, what does that look like? The problem with visualization becomes even more acute when we try to graph the data. You can make a drawing of a three-dimensional object on a flat piece of paper, and you can even make a model to represent three-space, like a relief map that shows the elevation of the ground at every point over a given area. But when you try to draw a picture of four-dimensional space it is impossible. For your paper, you might want to explore some potential applications in more detail. Here are some ideas to get you started. 1) Time is considered to be the fourth dimension in many applications, with the first three being the classical representation of physical space. For example, to study how an airplane travels through the sky, you would need a series of data telling where it was at any given time. The individual data points might look like (x,y,z,t) where t is the time that the plane was at point (x,y,z). In your paper you might discuss what types of functions could describe how the plane travels through this four-dimensional space. To try to visualize the function, picture a plane flying through the sky leaving a very long contrail. Now picture signs along the contrail depicting constant intervals of time. The signs will be farther apart when the airplane is going faster. Just as the slope of a line or other function plotted on a two-dimensional graph tells you one parameter of the function changes relative to another, so the distance between the signs on our imaginary contrail would tell you how the airplane's position changes relative to time. 2) You could create or describe a hypothetical representation of the temperature in the atmosphere as a function of location. (You could also use pressure, or humidity, or any other property of the air.) Create several layers, using a transparent media such as vu-graph foils. Color the different areas different colors on a spectrum from hot to cold to represent the temperature at that location. Be sure the temperature is a continuous function (does not jump from the hottest color to the coldest) as you go from one layer to another directly over the same point. In the February '97 National Geographic there is a fascinating article on the oceanic science of the Arctic. On page 49 is a really cool figure showing one layer from a multi-dimensional model scientists have created to describe the temperature as a function of position in the Arctic Sea. 3) You could also create a four-dimensional database that has nothing to do with physical space. Collect four different pieces of data on 20 or so of your classmates, such as age in months, shoe size, number of siblings, and amount of money on their person when they answered the questions. Then experiment with different tables, charts or graphs to present the data and explore any potential correlations. For the purpose of this study, these people would exist in the four- dimensional space which you define. I hope this helps. On a final note, you are welcome to use my suggestions and rewrite some of my ideas in your own words, but please don't copy large sections of my answer directly into your research paper. That would be plagiarism, which is a form of cheating. -Doctor Barney, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 06/25/97 at 14:34:25 From: Doctor Sydney Subject: Re: hi Dear Jody, Hello! I want to add just a little to Dr. Barney's response to you. I hope you find what I have to say interesting even though it's too late for your research paper! Dr. Barney did a nice job explaining how lots of people understand the fourth dimension. However, there is one major way that people think about the fourth dimension that was not addressed in his response to you. That is what I will talk about with you. Let's start with what we know. Let's consider first a plane. We know that a plane is 2-dimensional, right? What does that really mean? Well, one way to think of it is that we can put a pair of axes on the plane (an x-axis and y-axis, for example), and then we can assign to each point on the plane a pair of numbers, right? You've probably worked with graphing things on a plane, so you are probably familiar with what I am talking about. If I told you to graph the triangle with vertices (1,0), (0,0), and (0,1) you would know what to do, right? The space is 2-dimensional because we use PAIRS of numbers to define points in space. Each number in the pair tells us how far we should move from the origin in a certain direction, right? So, each number in the pair tells us where to go in space. Think now about the third dimension. In three dimensions, we assign TRIPLETS of numbers to each point in space, right? Again, each of the coordinates in the triplets assigned to points in 3-D space tells us how far to move in a certain direction from the origin, right? For instance, if we think of a 3-D space with the standard x, y, and z axes and origin (0,0,0), then the point (1,2,3) would tell us to move 1 unit down the x-axis from the origin, 2 units down the y-axis from the origin, and 3 units down the z-axis from the origin, right? So, all of the coordinates are space coordinates; they tell us how far to move in a specific direction. So, if we wanted to think of the fourth dimension as a space that has 4 SPACE parameters (just in the way that the plane has 2 space paramaters and the world we live in has 3 space parameters), then we have to use our imagination a little, right? If what I have said above confused you don't worry. Just start following from here. This is confusing stuff, believe me! One way to understand what the fourth dimension "looks like" is to carefully examine what the 3rd dimension looks like to "creatures" living in a 2-dimensional world. If we can understand this, then we can understand some of what the fourth dimension looks like to us creatures living in the 3-D world by using appropriate analogies. Here is what I mean. Suppose you live in a 2-dimensional world. For simplicity, let's say that you live in a plane. Let's call this world "Flatland." You don't see anything that is outside of this plane. As far as you know, that is the only space there is. Let's say that you are a solid yellow square. I am a hollow green circle. Now, what will you see when you look at me? Remember that you can see only things that are in the plane. It might be fun and helpful for you to cut out of construction paper a yellow square and a green circle. Then, to figure out what you see when you look at me, put the square and circle on a flat surface (like a table), and think about what you would see at eyelevel with the table. You might be surprised! Below, I will talk about what you see because it will help us to understand the fourth dimension; however, it will be more fun for you if you don't read what the answer is but figure it out yourself. Okay, so now you have done the experiment or perhaps thought about what you would see, so I can talk about what the answer is. As you probably found, a person who lives in this plane sees a green line when he or she looks at me. When I look at you, I see a yellow line. When I first see you, if we are both sitting still, I can't tell if you are a square, a circle, a line, or some other shape, right? All of these things look the same to a person in Flatland. How do you think people in Flatland determine what shapes their friends are? Or, do they not care? (: These are fun things to think about. Okay, so now we understand a little bit about what life is like in Flatland. It might be fun for you to think about what life is like in Flatland. What do houses look like? If I have a twin sister who is a solid green circle and is the same size as me, can you tell us apart (Recall that I was a hollow green circle)? Drawing pictures will help you. Let's think about some objects in Flatland. Think about a safe in Flatland. What would it look like? We would want it to be a container such that when locked, inhabitants of Flatland cannot get to the objects inside of it, right? What kind of shape would work for this? In our 3-D real world, we often use a hollow cube, right? Well, in Flatland, they use, among other things, hollow squares. Does that make sense? Think about it - an object that is in a hollow square cannot be taken by an inhabitant of Flatland; it is completely secure. Good. Now, we introduce the third dimension into our 2-D world. Think about someone who is an inhabitant of this world that we live in, our 3-D world. Suppose they walk by Flatland. They see Flatland as a flat land (duh!) with lots of flat creatures and objects in it, right? Now suppose that they come across the safe used by me. I have put a necklace in my safe. Is my necklace safe from someone in the 3-D world? No! A 3-D person can easily reach into my safe and take my necklace, right? He or she does not need to break down the barrier of the safe since he or she can just reach in from the top. Does that make sense? So, one way to understand the third dimension from the perspective of the second dimension is that enclosed spaces like safes can broken into by 3-D creatures without touching any of the barriers that 2-D creatures see. Let's think about how we could think about the fourth dimension from the perspective of the third dimension in an analogous way. Think about the safes we use in the 3-D world. They are, as we said before, often hollow cubes, right? Once we put something in them and lock the door, we can feel sure that no one will be able to get to them without breaking down one of the sides of the safe, right? Well, in the same way that a 3-D creature could break into a 2-D safe without breaking down any of the sides, a 4-D creature can break into our 3-D safe without breaking down any of the walls. Now, that is pretty wild, eh? It just doesn't seem to make sense. However, it is one way to think about the fourth dimension. I could go on for a long time about similar ways to understand the fourth dimension, but instead I will refer you to a book that I think you might really like if you found what I said above to be interesting. The book is called _Flatland_, and it was written about 100 years ago by Edwin A. Abbott. It describes in detail the world we called "Flatland," and it discusses many more ways to understand the fourth dimension from the perspective of the third dimension. If you have access to the WWW, you can find the entire book at the following website: http://www.alcyone.com/max/lit/flatland/index.html It is fun to look through, even if you don't read the whole thing. I hope you enjoy it. If you have any more questions about the fourth dimension, please feel free to write back. I think that dimensionality is one of the most interesting things that mathematicians study. Good luck! -Doctor Sydney, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.