Points

Points are the basis of all Geometry. There are so many things you can do with the little buggers that the possibilities are endless. Points are zero-dimensional. That basically means that they have no height, length, or width. They are just there.

There are four main definitions of a point. They are the dot, the exact location, the ordered pair, and the node. A point has four definitions because, over the years, many different mathematicians have come up with their own ideas as to what a point should be. Since their ideas were all equally true, the point was given four main definitions instead of a single definition. In fact, the point is considered undefined for that reason (among others). When being written out, points are always represented by a capitol letter. If a point is on a line, it is often represented by the same letter.

The first definition of a point is the dot. This was probably the first kind of point ever thought up. You see, a dot has size - it has a definite, measureable, length and width. Probably the best example of a dot today would be the pixel. Yup, that's right - a pixel. Those tiny spots of color that make up your computer screen. A matrix is a rectangular array made up of lots of pixels, so that's what your computer screen is. As you most likely know, the more pixel in a computer or TV screen, the better the resolution.

The second definition of a point is an exact location. The exact location is the perfect example of the normal, zero-dimensional point. No matter how much you zoom in, there will always be another point in between two others. This definition of a point was discovered sometime between 550 B.C. and 150 A.D. One example of where these are used in real life is in measuring distances, especially between two cities. Some cities are more than a mile across, so mapmakers have to pick one exact location in the city to measure from. One use of the exact location ties in with the the next definition, the ordered pair. The number line, or coordinatized line, is a line where every point is represented by a number and vice versa.

The third definition of a point is the ordered pair. The ordered pair was discoverd around 1630 A.D. by two mathematicians named Pierre de Fermat and René Descartes. You've probably heard of the latter before. Basically, an ordered pair is a pair of numbers in parentheses used to locate an exact location on a coordinate plane. The first number, represented by the variable x, tells you how far along the x-axis the point is(to the right or left, depending on the "polarity"/direction of the variable). The second number does the same, just along the y-axis (up and down). Ordered pairs that consist of whole numbers are called lattice points. Numbers in a number line, coordinate plane, or coordinate space, have both magnitude and direction. Magnitude is its distance from the origin, and direction is its positivity or negativity. The origin is the center of any graph. It marks the place where the measurements start. Everything one one side is positive, and everything along the other side is negative. The origin extends its "neutrality" along lines called the x, y, and z-axes. These lines need only one variable to represent them. The variable whose name they carry is the one that represents them, and its value is always zero. The picture below shows you what I mean.

The last definition of the point is the node. A node is a type of point thatis zero-dimensional, and two nodes can have more than one line between them. Nodes exist only in networks, which are a series of nodes and arcs. Arcs are lines that may curve and aren't dense. Arcs only contain their endpoints. There is a special kind of network in which all arcs can be crossed without going over one more than once. It is called a traversible network. Another kind of network is a graceful network. This kind of network is really hard to make is and is great for extra credit projects. To make a network graceful with x arcs, you must label each node with a number from 0 to x so that, if you find the positive difference of two connecting nodes and label their common arc with it, all arcs are labeled from 1 to x. If a node has an even number if arcs going through it, it is called an even node. An odd node is just the opposite. Nodes are sometimes called vertices.

Four important characteristics help to distinguish the different definitions of points:

1. Unique line - Do the points determine a single line?
2. Dimension - Are the points without size?
3. Number line - Can the points of a line be put into a one-to-one correspondece with real numbers?
4. Distance - Is there a unique distance between two points?