Sunday Numbers 2.0, Vol. 2: Perpendicular

90 degrees. Rectangular or square corners. Perpendicular. Vertical meets horizontal. All these words and phrases mean the same thing, and just to add to the confusion slightly, mathematicians will refer to it as orthogonal. But really, there is little confusion in the general public on this mathematical concept because we see it so often.

In some ways, the 90 degree angle is more a part of human nature than it is of "nature" nature. There are a few crystals that naturally place a flat surface perpendicular to another flat surface, and in studies of electrical fields, an electric current creates a second current that flows orthogonally to the original. But in nature, other forces interact with perpendicular surfaces, so even a tree that wants to grow straight up and down from the level surface of a prairie will be bent by the wind or rain or erosion.

In contrast, humans make any of a number of things with nice right angles, including the computer screen you are reading this post on and most the corners of the room where you are reading this. Rectangular two dimensional shapes and rectangular solids show up everywhere in human designs.

One of the reasons for the ubiquity of the 90 degree angle is that we know so much about it. The Pythagorean Theorem is ancient wisdom, sussed out by every civilization worth being called a civilization. There are proofs from the Chinese, the Indians, pre-Colombian American tribes like the Mayans, the ancient Egyptians, the Persians and, of course, the Greeks. There is a nice useful example of a² + b² = c², 3² + 4² = 5². Archeologists has found sticks of length 3 units, 4 units and 5 units in ancient Egyptian ruins, and the best guess is that these were used to make sure walls were perpendicular to the floor and to each other.

Of course, vertical meets horizontal is just one way to make two lines perpendicular. In the picture above, all the lines of length c are perpendicular to the ones they touch and parallel to the opposite side of the square.

In three dimensions, perpendicular still exists of course, and in fact it expands a bit. If we are drawing a line on a plane put a point on that line, there is exactly one line perpendicular to it, according to Euclidean geometry. (There is something called non-Euclidean geometry, but it assumes we aren't dealing with a flat plane in general, so we can just leave it be for now.) In three dimensions, mathematicians deal with vectors of the form [x y z], which is a straight line segment from the origin (0, 0, 0) to the point (x, y, z), but stands in for all line segments of the same length pointing in the same direction. Mathematically, orthogonality is most easily represented by the dot product of two vectors. u · v. If u =[a b c] and v =[p q r], then u · v is the sum of the products of the three corresponding entries to the vectors, ap + bq + cr. Two vectors u and v in the same vector space are orthogonal if and only if u · v = 0.

In three dimensions, three vectors can be constructed so that each is orthogonal to the other two. The simplest three vectors in three dimensional space that are all orthogonal to each other are
i = [1 0 0]
j = [0 1 0]
k = [0 0 1]

This is called an orthogonal basis for three dimensional space, but it is only one of an infinite set of bases.

Four dimensions or more are much harder to visualize than two dimensions or three, but the idea of orthogonality based on dot products is still the same relatively simple math. Multiply the vectors together element by element and add those products. If the sum is zero, the vectors are orthogonal. Dot products also let us define distance. Take a vector and do the dot product with itself u · u. Since this is the sum of squares, it must be non-negative and unless the vector is the all zero vector, the dot product will be positive. The length of the vector is the square root of the dot product, also known as the magnitude. This is a direct corollary of the Pythagorean Theorem and it works in as many dimensions as you might want to create. Mathematicians even have ways to talk about infinite dimensions, but the ideas of orthogonality and distance remain the same, and both of those ideas come from our basic understanding of perpendicular and the most important discovery we have made about perpendicular line segments, the Pythagorean Theorem.

Next week: continuity, mathematician style.