We talk about basic geometry in higher dimensions.

*geometry*can be broken into

*geo*meaning “world” and

*metry*meaning “measure.” In this section we will tell you what our mathematical “world” is, and how we “measure” it.

### Higher dimensions

In our previous courses, we studied functions where the input was a single real number and the output was a single real number. Note, the word “real” is being used in a technical sense:

**real number**is a number that has a (possibly infinite) decimal representation. The set of all real numbers is denoted by .

When we say a function maps a real number to a real number, we write: When working in two dimensions, we need a way of talking about ordered pairs of numbers. We denote the set of all ordered pairs of real numbers by . When working in three dimensions we denote the set of all ordered triples of real numbers by . In three dimensions we have three coordinates axes, the -axis, -axis, and -axis:

**right-hand-rule**:

### Basic plotting

To plot a point in , you move in the -direction, in the -direction and in the direction.

Of course, we’re going to be plotting many points. We typically described groups of points, as those that satisfy a given equation involving , , and . Here is a place where working in three dimensions is really different from working in two.

- In , any equation involving and/or draws a
**curve**. - In , any equation involving , , and/or draws a
**surface**.

The most basic surface in is a plane.

Another way to think of the point is as the intersection of the planes , , .

Move the point around below to see the planes that define it.

While three planes need not intersect at all, the intersection of two (nonparallel) planes is a line.

### Distance and spheres

So the objects in our geometry are made of points, and now we must tell you how we plan to “measure” objects. To do this, we’ll use our old friend, the distance formula.

On a completely related note, what’s the most famous theorem in mathematics? I’ll
tell you: The Pythagorean Theorem. In essence, the distance formula *is* The
Pythagorean Theorem. Let’s see if we can explain why the Pythagorean Theorem is
true.

The distance formula also extends to higher dimensions:

Thus the distance formula in is indeed what we claimed.

In general we can extend this notion of distance to :

#### Circles and spheres, disks and balls

Let me remind you what the definition of a circle is:

**circle**is the set of points in that are a fixed, nonzero, equal distance from a given point , where is the center of the circle.

From the definition of a circle, we see that it is intimately related to the distance formula. Indeed, it is also the case the equation of a sphere is related to the distance formula in :

Things really get interesting when we have both spheres and planes around. Spheres can intersect planes at one point (if they are “just” touching the plane), no points (if they are missing the plane), or an infinite number of points (here the intersection is a circle). Let’s see some examples.

That wasn’t too bad, let’s see another.

For our last example, we’ve left the easiest case of all.