152 Part 3: The Shape of the World
Points, Lines, Planes, and Angles
The undefined terms of geometry are a curious mix of things you know and things you can only
imagine. A point can be thought of as a dot, a tiny spot, or a position. That’s the familiar part. The
part that requires imagination is the idea that a point doesn’t take up any space. It has no size and
no dimension. You can’t measure it. You can draw a dot to represent a point, even though your
dot does take up some space, and you label points with uppercase letters.
You know what a line is. You see lines all the time. But geometry asks you to use your imagina-
tion here, too. A line is a set of points—an infinite string of points—that goes on forever in both
directions. It has length, in fact it has infinite length, but it has no width and no height. It’s only
one point wide or high, and points don’t take up space. And yet, somehow, you can string points
together to make something that has infinite length.
People often say, “a straight line,” but in geometry that phrase is redundant. All lines are straight.
If it curves or bends, it’s not a line.
When you draw a picture to represent a line, even though your picture does have some width
and can’t actually go on forever, you put arrows on the ends to show that it keeps going. You can
label the line with one script letter, like line , or by placing two points on the line and writing
those two points with a line over the top, like this: AB.DEFINITION
A point is a position in space that has no length, width, or height. A line is a set of
points that has length but no width or height. A plane is a flat surface that has length
and width but no thickness. Space is the set of all points.Is your imagination still working? Where do these points and lines live? Where would you draw a
point or a line? Perhaps on a sheet of paper or the chalkboard? Those surfaces are the images that
help you imagine a plane. A plane is a f lat surface that has infinite length and infinite width but
no height or thickness. It’s an endless sheet of paper that’s only one point deep.
And where do the points and the lines and the planes live? In space! No, not outer space, at least
not exactly. Space, in geometry, is the set of all points, everywhere.