In 2D Euclidean space, we can figure out the distance of any point from any reference point by setting up a rectangular coordinate system with the reference point at the origin. If there were just two points we cared about, we could align the axes such that the X axis went through the other point, and then we could just read the distance off that axis. That's not really exploiting 2D space, so we will think about one reference point but lots of other points all over the plane. We can use the Pythagorean theorem to measure the distance from the origin to any point in the plane:
\[s^2=x^2+y^2\]
With this coordinate frame you can figure the distance between any two points, even if one is not at the origin, as follows:
\[s^2=(x_2-x_1)^2+(y_2-y_1)^2\]
By using the standard delta-notation from engineering, we can simplify this back to
\[s^2=\Delta x^2+\Delta y^2\]
Now what is true for 2D Euclidean space is also true for 3D. The Pythagorean formula works, you just have to extend it to cover the third dimension
\[s^2=\Delta x^2+\Delta y^2+\Delta z^2\]
Those crazy topologists have generalized the Pythagorean theorem to fit their weird bent rubber spaces. They say that any space, along with a function which takes two points as an argument and returns a number, is called a metric space, where that function is called the metric. The metric must obey certain axioms, most important of which is commutativity - the distance from point A to B is the same as the distance from B to A.
Some of the topologists twisted imaginings don't really admit such a thing as a straight line. They get around this by saying that if you look at any small enough piece of the space, it is close enough to flat that we can define a metric there. Some spaces are so bent as to not even permit this, but those spaces which do, are called manifolds. In a manifold we talk about points which are close together, and represent this in our metric with differential notation
\[ds^2=dx^2+dy^2+dz^2\]
Now on a manifold we can specify a series of points to draw a path through, find the distance between each, add them all up, and get the length of the path. In calculus-speak, we have every point on the path, and we integrate along the curve to get the distance. However for flat space, there is such a concept as a straight line, and it is the shortest distance between two points. If you take the delta-form above and integrate the differential form, you end up with the same thing. Because of this, we will just show things in differential form from now on.
The film Dimensions is all about extending the same concept to four- and higher- dimensional space. All the Euclidean axioms apply, and the fourth dimension is exactly like the other three, so it works into the metric the same way:
\[ds^2=dx^2+dy^2+dz^2+dw^2\]
One narrator talks about how 4D space is the prettiest, because it contains such things as the 24-cell. Also he says that it may be because real physical space is 4-dimensional also once you consider time. Blah blah Einstein aggressively ignore history blah blah blah. Spacetime is 4-dimensional, but here's the weird part.
Time is not the same as Space.
"Wait a minute" I hear you all saying - "Obviously time isn't the same thing as space. Duh." But, the whole reason we call time a dimension, and the same reason we don't call temperature a dimension, is that there are coordinate transformations that mix in time with space. I'm going off of memory, but this argument came to me through a little book called "Relativity and Common Sense". For instance, imagine a plane where every point is painted a different color. So, at each point we can measure three things, its x and y coordinate, and its color. If we rotate the coordinate frame, we mix together the x and y coordinates
\[\begin{eqnarray*}
x'=& &x \cos \theta&+&y\sin \theta \\
y'=&-&x \sin \theta&+&y\cos\theta\ \end{eqnarray*}\]
But, there is no rotation, no coordinate transformation which preserves our understanding of what coordinate transformation means, which can mix color and spatial coordinates. Color is not a dimension in this sense.
Well, Time is.
The transformation isn't just a rotation, but the Lorenz transformation from coordinates measured by one observer to coordinates measured by a relatively moving observer, depends on the relative speeds of the observers. And here's the weird thing - time is a dimension, but it is not just like the other three dimensions. In fact, the metric for the spacetime observing the Lorenz transformation is called the Minkowski metric, and it looks like this:
\[ds^2=dx^2+dy^2+dz^2-dt^2\]
See the minus in the time term? It says that the longer the time between two events, the shorter the distance, all other coordinates being equal.
It gets weirder than that. If the time difference is long enough, it drags the whole right side negative. The squared distance between two events is negative. The distance between two events is imaginary. In special relativity, we say that when \(ds\) is real and positive, it is called proper distance, and it is the distance between two events as seen by some observer who sees them happening at the same time. Further there is no way for a single observer to be present at both events without exceeding the speed limit in the space, which introduces its own problems. When \(ds\) is imaginary, the (real) coefficient is called proper time, and represents the time interval between two events as seen by an observer who sees them happening in the same place. A single observer can be present at both events without exceeding the speed limit.
See what I mean by weird? In Minkowski space, there is a speed limit imposed as part of the fundamental geometry of space. If we though of time as just a dimension like space, this is equivalent to saying that it is impossible for a line to exceed a certain slope (change in space dimension per unit change in time dimension). No such limit exists in Euclidean space. Spacetime is 4D, but not Euclidean. It is not the same 4D space discussed in Dimensions. The Pythagorean theorem is false, and therefore the parallel postulate is false. Minkowski space is the only flat (metric works across long distances) space I know of which is non-Euclidean.
Now the question is, are the regular polytopes the same in Minkowski space? Does it make sense to talk about polytopes? Is a polytope regular from one point of view but not from another? These and other questions can be answered by the Minkowski metric, but I don't know the answers. I was only recently even able to form the questions.
Let's finish this off with a visualization:
