In this section, we build up the core tool in defining lines: the notion of distance.
I am here, you are there. From this, geometry flows.
The notion of separation in space of two people is the idea that we are located at different points in space.
The term "point" is a primitive notion in that we do not say what a point is in terms of anything else. The core idea of a point is that it is a distinct entity with no notion of overlapping other points. Often this is thought of as having no visible extent, but we also often depict points to our gross selves as something with extent. The core notion is that of distinctive separation.
We do have mathematical models of points. We often name the points with coordinates and thus we can construct a space of points by being a space of coordinates. For example, a map may refer to points with longitude and latitude. We could then consider the abstract space of all pairs of longitude and latitude numbers within a suitable region to model this. If we allow for the full range of real numbers between any two use numbers, then we will end up with a continuous space, with uncountably many points. Such a space is typically assumed in mathematical concepts, but in applications, points are quite limited by the crudeness of our measurements.
For us, the core question will be one of connections between points, such as paths and the lengths of those paths. That leads us to a question of what paths are.
A path is a journey from one point to another, travelling from one point in the space to the next until the destination is obtained.
At a minimum, a path is a set of points, one with a start, an end, and some kind of progression from the start to the end. There is a technical definition that will come much later in this series, but the best image to have in mind is that a path is something that one could cover with a string.
This string image then also tells us how to measure the length of a path. First, find a string that precisely fits the path. Second, take the string and pull it until it can no longer be pulled (taut). Third, compare that length to some standard measure.
An important part of that procedure is that we do not have to already define "straight"; reality provides us with the ability to get straight by simply pulling on end of the string until it steps extending. Once fully extended, a string can only move by the end point travelling in a circle, as we discuss in the next section.
This also works in other contexts. Take a ball and a string. Fix one end of the string to a point on the ball and then move the other point on the ball until the string is taut. The path the string describes will be a very special path. It is a path equivalent to taking a part of the equator. The equator is a sort of maximal circle on a sphere and these are the "straight" lines on a sphere.
There is another notion of path length that we could consider. That is the notion of how far one has traveled along a path. This could be the number of steps taken or if, on a wheeled contraption, how many wheel rotations have happened. We will concentrate on the idea of a step.
By step, we mean some kind of small length that indicates a travel from one point to another. In human terms, a person's step size might be a couple of feet. If we take a walk and go five steps, then we have gone ten feet.
This is a measurement of distance along our path. With fixed starting and ending points, we could take a much longer path, say, one with ten steps leading to a length of twenty feet.
If we take smaller steps, say, one inch steps, then the 5 steps become 60 steps as there are 12 inch-long steps in a foot-long step.
This step size is a primitive notion. It is something that we humans should be comfortable with because that is how we assess distance experientially.
As we take smaller steps, our notion of distance will become more accurate, assuming we can count well.
Imagine we have a piece of a string that, when taut, we could take 5 steps across it. If we take a path that we could travel with 5 steps, then the string would perfectly fit along the path. Conversely, if the string fits a path perfectly, then it would take us 5 steps to go across it.
It is in that way that the notion of steps and string lengths are in harmony.
Life takes energy to travel and so life naturally tends to look for the path to the destination with the shortest amount of energy usage. This typically involves taking the shortest number of steps, at least when we are not going up and down a mountainside or something like that.
We are therefore interested in minimizing the number of steps taken (more generally, we would minimize the energy required to travel a path). This is equivalent to trying to minimize the length of the path per the string point of view.
The question then becomes "Given a starting and ending point, what is the path that minimizes the distance traveled between these two points?".
There are separate questions of does such a path exist and can we describe such paths if they do exist.
The existence question is a technical question of what space you are actually talking about. Imagine, for instance, that you live on the north bank of a river and you travel to the south bank. Further suppose that the only way across the river is to take a bridge. Almost surely, the paths available will not allow you to take the shortest path available to a flying bird. The path ultimately taken by end up
We will focus on the case when there are no obstacles preventing the natural minimal length path from being an option.
The distance between two points is defined as the shortest length of all the paths connecting the two points. If two points are not connected by a path, the distance is undefined.