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Origin of the ideas of n-dimensional space. Development of the concepts of n-space.

We speak of vectors, lengths of vectors, angle between vectors, lines, planes, polyhedra, etc. in n-space. What is the meaning of the angle between two vectors in 6-space? How do you envision that? What conception can you give to a plane in 4-space? How do you visualize a cube in 5-space? Do these things make any sense? Or are we just talking nonsense?

John Mackel tried to answer these questions. He finally gave up, defeated. He decided he just didn’t have the intellectual power for the task, his IQ just wasn’t high enough. Jerry Brown is in a mental institution. He dwelt on the problem too long. Pete Lowery has his own opinions. He says the ideas are all ridiculous nonsense, foolishness.

We live in an age where many physicists postulate many very wild, preposterous sounding ideas (many originating in Einstein’s theory). You have time warps and space warps and all kinds of mind-defying ideas. So how do you answer the above questions? In an attempt to address the question let us ask the following question:

Q. From what types of mathematical problems did the ideas of n-dimensional space originate? Where did the concept of dimensions higher than three come from? What problems gave rise to the body of theory associated with n-dimensional space?

Two sources come to mind:

1. Multi-variate functions. In analytic geometry we are accustomed to associating the number pair (x1, x2), where

- ∞ < x1 < ∞

- ∞ < x2 < ∞

with a point in the plane. In the same way we view a number triplet (x1, x2, x3), where

- ∞ < x1 < ∞

- ∞ < x2 < ∞

- ∞ < x3 < ∞

as corresponding to a point in three-dimensional space. The concept of n-dimensional space comes from generalizing this idea to n-tuples in which n > 3. For example, the 4-tuple (x1, x2, x3, x4), where

- ∞ < x1 < ∞

- ∞ < x2 < ∞

- ∞ < x3 < ∞

- ∞ < x3 < ∞

is viewed as a point in 4-dimensional space. Now I would point out that a number pair (x1, x2) is not a point in the plane and a number triple (x1, x2, x3) is not a point in space. There is simply a one to one correspondence that allows us to represent a point in the plane by a number pair and a point in space by a number triple. With constant usage we come to think of a number pair as a point in the plane and a number triple as a point in three-dimensional space. Now when we want to go to more variables than three there is simply no space higher than three that we know anything about. There is no four-dimensional space that we know anything about. So what do we do? We simply imagine there is and call a 4-tuple (x1, x2, x3, x4) a point in 4-space. Now a 4-tuple is not a point. The word point is a geometric term with a geometric meaning. But we use this geometric term which refers back to its meaning in the plane or in three-dimensional space. It is very helpful thing to do and aids greatly in providing intuitive insight but doing so could also give rise to misunderstandings and confusion. Calling an n-tuple a point doesn’t make it a point and calling a collection of n-tuples a space doesn’t make it a space (in the ordinary sense of the word, at least).

The concept of n-dimensional space is useful when we are dealing with multi-variate functions. The function f(x1, x2, .... , xn) assigns a numerical value to every n-tuple (x1, x2, .... , xn). Thus it is viewed as assigning a numerical value to every point in n-dimensional space. The system of equations

y1 = f1(x1, x2, .... , xn)

y2 = f2(x1, x2, .... , xn)

maps points in n-dimensional space into points in two-dimensional space [i.e. it maps a point (x1, x2, .... , xn) in n-space into the point (y1, y2) in the plane]. The domain of the mapping is n-dimensional space and the range is two-dimensional space. The matrix equation

represents a linear mapping from n-space into m-space, mapping points (x1, x2, .... , xn) of n-space into points (y1, y2, .... , ym) of m-space.

2. Problem of solving systems of linear equations. Solving a system of linear equations involves a process of reducing the system to a succession of simpler equivalent systems by adding multiples of one equation to another. With the systems expressed in matrix form, this amounts to adding multiples one row of a matrix to another row. Thus study of the underlying theory leads directly to questions about linear combinations of n-tuples, a theory of n-tuples. This theory leads directly into the concepts of linear dependence and independence of n-tuples, vector spaces and subspaces, dimension of vector spaces, various types of bases, etc. And all these concepts present themselves with great clarity in a study of the properties of linear combinations of vectors and their relationship with planes in three-dimensional space. Thus the general study of n-tuples leads directly to concepts best illustrated by examples from three dimensional space. It is from these geometric interpretations that one draws insight and understanding for the theory of n-tuples. The individual instances and examples that we find in the theory of vectors in three-dimensional space provide a model by which we think. All of the theorems for n-tuples are derived either totally by algebraic argument, or else by geometric argument for the case of three variables and then extended to the case of n variables by algebraic argument. However, the terminology is geometric (terms like point, line, vector, plane, polyhedron, etc. are all geometric terms). This is because the ideas are all best understood in geometric terms. Thus the terms for the concepts in the theory of n-tuples are all geometric. But using terms borrowed from a geometric model can cause misunderstanding and confusion. It can cause a person lacking in a good understanding of the subject to look for something that doesn’t exist. A name is just a name. Calling a thing a point doesn’t make it a point and calling a thing a space doesn’t make it a space. One ought not try to understand a four-dimensional space as having any other reality other than one given to it by a geometric name. In fact n-tuples obey a set of postulates which define an abstract system called a linear (or vector) space. N-tuples form what is called a linear space. From the set of postulates for a linear space one can deduce various properties. The concepts of linear dependence and independence of vectors, vector spaces and subspaces, dimension, various types of bases, etc. are all properties of linear spaces. Thus all the interesting structure of n-space derives from the fact that it is a linear space. The fact that the properties and structure of n-space is the same as that of the geometric model that we find in vectors of three-dimensional space is a consequence of the fact that 3-space is only a special case of n-space which is a linear space. Thus the theory of n-tuples melds together with the theory of vectors in 3-space because both are linear spaces.

We can visualize the various concepts in the case of 3-space. In higher dimension spaces we rely on analogy to 3-space for our conceptions. We can’t visualize planes and cubes in spaces higher than 3-space.