Professor Dave here, let’s talk about points, lines, and planes.
As we just discussed, geometry was developed in ancient Greece, and when we learn the geometry
of this time, we refer to it as Euclidian geometry, in reference to the mathematician
Euclid, to whom it is attributed.
His work, which built upon the work of earlier mathematicians like Pythagoras, describes
a set of axioms and postulates from which a variety of theorems can be derived.
Let’s learn about the physical constructs that these axioms describe so that we can
learn about all the math we can do with them.
First we look at a point.
This is nothing more than a location in space.
It is zero-dimensional, meaning that it has no dimensions of any kind.
We represent points with little dots and some capital letter, making sure to realize that
the dot we draw is infinitely larger than the point it is meant to represent.
From here, we can construct a one-dimensional object by stringing an infinite number of
points along a particular dimension.
This object is called a line.
Points and lines are familiar to us from algebra, but now let’s examine a two-dimensional
object.
This can be achieved by stringing an infinite number of lines along a dimension perpendicular
to the line.
This is called a plane.
And then if we string an infinite number of planes in either direction, we get three-dimensional
space.
Point, line, plane, space.
Zero, one, two, and three dimensions.
Another linearity we can identify is that a point is just a point, but any two points
can define a line, since there is only one line that contains both of these points.
Extrapolating further, any three points that aren’t all colinear can define a plane,
since there is only one plane that contains all three of these points.
We should mention that these figures are idealized concepts.
We can never create a plane that is infinitely thin so as to legitimately consist of only
two spatial dimensions.
Even a piece of paper has some thickness, even if it’s hard to see with the human
eye.
But we can create mental idealizations of these objects and discuss them concretely,
and the conclusions we draw from them will have application in the physical world, where
we can approximate certain systems as being perfectly Euclidian.
Just to list a bit more terminology, while a line extends infinitely along its one dimension
in both directions, if we chop one end off, giving it a definite starting point, we call
this a ray.
If we chop off both ends, leaving some chunk left over in the middle, this is called a
line segment.
Line segments have two endpoints, rays have one endpoint, and lines have none.
We can use lines, rays, and line segments to generate all kinds of figures.
We can make shapes with differing numbers of sides, and we can also create all kinds
of angles between these lines.
There is plenty to discuss regarding the types of angles that are possible, their magnitudes,
and their relationships, so let’s move on and talk about angles.