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Geometry is the study of shapes and spatial relationships. Although it’s based very
firmly on observing real shapes in the real world, it is fundamentally an abstract subject.
We look at the world and look for common, definable shapes and the relationships between
them. And so we start the study of geometry with some very abstract definitions.
A point is a particular position in space. It takes up no real space itself; it has no
actual width or size or weight. It’s just a position, a place, a point.
A line is the shortest way to connect two points, the shortest path between two points
in space. And a plane is a flat two-dimensional surface on which points and lines can be drawn.
Of course, we don’t have to limit ourselves to straight lines and flat planes. In fact,
if we allow surfaces that are not flat, we will automatically end up with curves. A curve
is just a path that looks locally like it’s a straight line. Think about the surface of
the Earth. The shortest path between two cities might look like a straight line on a map,
but it’s actually a curve across the surface of the Earth. If we’re limited to travelling
on (or near) the surface, then the shortest path between two cities will be a curve—it
looks like a straight line if you’re on the path, but in fact it’s a curve through
space across a non-flat surface.
The length of a line segment is the distance you have to travel along that line to get
from the point at one end to the other. And wherever two lines meet, you can describe
the angle between those lines. Let’s talk about angles for a moment.
Most people know these first three names. An acute angle is an angle smaller than a
quarter turn. Acute means sharp; can you see the sharp point? Measured in degrees, an acute
angle is less than ninety degrees. (There are three hundred and sixty degrees in a full
circle, a complete rotation.) A right angle is exactly a quarter turn, ninety degrees.
We usually mark a right angle with a little square in the corner, instead of a curved
arc. And an obtuse angle is an angle bigger than ninety degrees.
Now if you look at any point in the middle of a straight line, and imagine it’s actually
a point where two separate lines meet, you could say that there’s a one hundred and
eighty degree angle between the two halves of the line at that point. This is a straight
angle. And if we keep expanding the angle past this, we get a reflex angle. In fact,
there’s a reflex angle on the other side of our first three angles. Normally we would
use the smaller angle on the inside of the bend, but sometimes we need to talk about
the reflex angle on the other side.
Now if we join lines together in particular ways, we can make shapes. Planar shapes are
ones that can be drawn on a plane, a flat surface. These are the polygons, and in particular
we’re usually interested in triangles and quadrilaterals, shapes with three or four
straight sides. If we use curves instead of lines we get circles, ellipses and other curved
shapes as well.
And if we join our flat shapes together we can make solid shapes. In particular we can
make the so-called “Platonic” solids, which is where every side of the solid is
the same shape and size. We can also make prisms and pyramids. And if we allow curved
surfaces we can make spheres and donuts and a whole lot of other interesting shapes.