Here is an unusual "triangle". It has three right angles, for an angle sum of \( 270^{\circ} \).

Clearly, this is not quite a triangle in the sense in which we've defined it so far. Triangle \( := \) A polygon composed of three line segments connecting any three points in the plane which are not colinear. Here's a broader definition. Triangle-y thing \( := \) A shape composed of the three shortest paths in a space between any three points in that space which are not on a the same path. (If you want to look more into it the shortest path on a surface is called a geodesic, and on a sphere the geodesics are called great circles. This is why, if you are used to the Mercator projection, you may be surprised to travel so near the Artic on a flight from New York to London.)

You know that if a surface is flat, the angle sum of a triangle is \(180^{\circ}\). The contrapositive is more interesting: if the angle sum of a triangle is not \(180^{\circ}\), it is on a curved surface.

Why should we care about spaces that are flat (Euclidean spaces)? Because they're easier.

Why should we care about spaces that are not flat? Because not all physical geometries are actually flat. For example, when one is walking along a geodesic on Earth's surface, one cannot tell that it is not a flat line without travelling quite some distance: but, over long distances, failing to account for this will cause issues. Here is the LIGO detector at Hanford, a double example of this. It measures the warping, by gravitational waves, of 3D space away from its usual flatness. In order to do so, it needs long, straight arms at right angles; and, in order to build them with sufficient accuracy, its designers needed to account for the curvature of Earth.

The Pythagorean Theorem extends straightforwardly to higher dimensions. In the following list, "d" means the distance between two points, and terms on the right hand side are the distance between the points on a given axis.

• 2D : \( d^2 := x^2 + y^2 \)
• 3D : \( d^2 := x^2 + y^2 + z^2 \)
• nD : \( \begin{eqnarray*} d^2 := \sum_{i=1}^n x_i^2 \end{eqnarray*} \) with n any positive integer

We cannot visualize higher dimensional spaces fully; the trick is not to try, and to let all the algebra chops we've been developing pay off. Trusting in the equations we've justified in dimensions we can picture, we can apply the methods of Euclidean geometry to such spaces, often with extremely powerful results.

One such result is the ability to take the "average" of two images. We'll define the average of two points as their midpoint; encode the information of our image as a point in a Euclidean space, with entries representing say the pixel row, column, and color; and find the average of two images as the midpoint of their two points. Below is a very sophisticated example of image interpolation which nonetheless relies on the same concept of distance in some spaces; the difference is that these spaces encode more abstract information than just the colors of pixels.