It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better — H.Poincaré, Science and Method (1908, Part II. Ch. 2, p. 129)
Dimension one. To say that something is one-dimensional is to say that you need to know one number to locate a point on it. Imagine a line, with a point O marked on it. Imagine another point P on the line. The distance from O to P is a number that determines where P is. For this to work, we use positive numbers when P is to the right of O, negative numbers when it is to the left.
A circle is also one-dimensional. Imagine the center, and call it O. Choose a point P on the circle. Given any other point Q, measure the angle POQ. Thus a point on the circle determines a number. Conversely, a number determines a point. So the circle is one-dimensional, as advertised.
Dimension two. To say that something is two-dimensional is to say that you need to know two numbers to locate a point on it. The surface of the earth is two-dimensional because latitude and longitude suffice to locate any point. This fact was well-known to the Greek geographer Ptolemy (c. 90-160) as well as to the designers of GPS systems and smartphones. The surface of floor of the apartment in which is sit as I write this text is also two-dimensional. The center of the left front leg of my chair is determined by two numbers: ithe distance of the center from the wall to my left (which I call x), and the distance from the wall behind me, which I call y.
Dimension three. The space in which we live has dimension three. A point in the room in which I sit is determined by its height above the floor, which I call z, and the previously defined numbers x and y, which measure the distance from the left and back walls. An airplane flying overhead is located by three numbers as well: latitude, longitude, and altitude.
Dimension four and more. Mathematicians, like everone else, love analogies. We just saw that a point in space, e.g., in my room, is determined by three numbers, x, y, z. A mathematician can therefore think: well, a model of three-dimensional space is just the totality of triples of numbers x, y, z. This point of view was introduced by the French mathematician and philosopher René Descartes (1596-1650). Given this much, it is natural to ask: what is the totality of all quadruples of numbers, x, y, z, w? Well, this is a model of four-dimensional space. Or better yet, it is a precise mathematical definition of four-space.
Everyone who encounters this way of thinking about four-space says: but I want to see it. Sadly, we are not biologically equipped to do so. But with our minds we do almost as well as if we had better eyes. What could a four-dimensional being do? Well she could measure the distance between points in four-space. But we can do that also! The square of the distance from 1, 1, 1 to 2, 3, 4 is given by an algebraic formula that comes from the Pythagorean theorem. If you want to see the calculation, here it is:
distance-squared = (2-1)2 + (3-1)2 + (4-1)2 = 12 + 22 + 32 = 14.
Thus the distance between the two points is the square root of 14, or about 3.74. Navigation programs for aircraft do such computations many times a second.
Fine, you say. What about four-space. Can you find the distance from 1, 1, 1, 1 to 2, 3, 4, 5? Of course! It is given in the same way as a sum of squares of differences:
distance-squared = (2-1)2 + (3-1)2 + (4-1)2 + (5-1)2 = 12 + 22 + 32 + 42= 30.
Thus the distance between the two points is the square root of 30, or about 5.48. Our four-dimensional beings can write airfraft navigation programs as well!
If you can measure distances, you can do geometry. You can measure angles, talk about triangles, etc. And you can do this in dimension 5, 6, or dimension 1,000 or dimension 1,000,000. The only difference is that a point is given by 5, 6, or more numbers. Scientists analyzing data use this kind of matheamtics all the time.