We’ve underestimated the Pythagorean theorem all along. It’s not about triangles; it can apply to any shape. It’s not about a, b and c; it applies to **any formula** with a squared term.

It’s not about **distance** in the sense of walking diagonally across a room. It’s about **any distance**, like the “distance” between our movie preferences or colors.

If it can be measured, it can be compared with the Pythagorean Theorem. Let’s see why.

## Understanding The Theorem

We agree the theorem works. In any right triangle:

If a=3 and b=4, then c=5. Easy, right?

Well, a **key observation** is that a and b are at right angles (notice the little red box). Movement in one direction has **no impact** on the other.

It’s a bit like North/South vs. East/West. Moving North does not change your East/West direction, and vice-versa — the directions are independent (the geek term is **orthogonal**).

The Pythagorean Theorem lets you use find the **shortest path distance** between orthogonal directions. So it’s not really about right **triangles** — it’s about comparing “things” moving at right angles.

You:

If I walk 3 blocks East and 4 blocks North, how far am I from my starting point?Me:

5 blocks, as the crow flies. Be sure to bring adequate provisions for your journey.You:

Uh, ok.

## So what is “c”?

Well, we could think of c as just a number, but that keeps us in boring triangle-land. I like to think of c as a **combination of a and b**.

But it’s not a simple combination like addition — after all, c doesn’t equal a + b. It’s more a combination of components — the Pythagorean theorem lets us combine **orthogonal components** in a manner similar to addition. And there’s the magic.

In our example, C is 5 blocks of “distance”. But it’s more than that: it contains a **combination** of 3 blocks East and 4 blocks North. Moving along C means you go East and North at the same time. Neat way to think about it, eh?

## Chaining the Theorem

Let’s get crazy and chain the theorem together. Take a look at this:

Cool, eh? We draw **another** triangle in red, using c as one of the sides. Since c and d are at right angles (orthogonal!), we get the Pythagorean relation: c^{2} + d^{2} = e^{2}.

And when we replace c^{2} with a^{2} + b^{2} we get:

And that’s something: We’ve written e in terms of 3 orthogonal components (a, b and d). Starting to see a pattern?

## Put on your 3D Goggles

Think two triangles are strange? Try pulling one out of the paper. Instead of lining the triangles flat, tilt the red one up:

It’s the same triangle, just facing a different way. But now we’re in 3d! If we call the sides x, y and z instead of a, b and d we get:

Very nice. In math we typically measure the x-coordinate [left/right distance], the y-coordinate [front-back distance], and the z-coordinate [up/down distance]. And now we can find the 3-d distance to a point given its coordinates!

## Use Any Number of Dimensions

As you can guess, the Pythagorean Theorem generalizes to **any number of dimensions**. That is, you can chain a bunch of triangles together and tally up the “outside” sections:

You can imagine that each triangle is in its own dimension. If segments are at right angles, the theorem holds and the math works out.

## How Distance Is Computed

The Pythagorean Theorem is the basis for computing distance between two points. Consider two triangles:

- Triangle with sides (4,3) [blue]
- Triangle with sides (8,5) [pink]

What’s the distance from the tip of the blue triangle [at coordinates (4,3)] to the tip of the red triangle [at coordinates (8,5)]? Well, we can create a **virtual triangle** between the endpoints by subtracting corresponding sides. The hypotenuse of the virtual triangle is the distance between points:

- Distance: (8-4, 5-3) = (4,2) = √(4
^{2}+ 2^{2}) = √(20) = 4.47

Cool, eh? In 3D, we can find the distance between points (x_1,y_1,z_1) and (x_2,y_2,z_2) using the same approach:

And it doesn’t matter if one side is bigger than the other, since the difference is squared and will be positive (another great side-effect of the theorem).

## How to Use Any Distance

The theorem isn’t limited to our narrow, spatial definition of distance. It can apply to **any orthogonal dimensions**: space, time, movie tastes, colors, temperatures. In fact, it can apply to any set of numbers (a,b,c,d,e). Let’s take a look.

## Measuring User Preferences

Let’s say you do a survey to find movie preferences:

- How did you like Rambo? (1-10)
- How did you like Bambi? (1-10)
- How did you like Seinfeld? (1-10)

How do we compare people’s ratings? Find similar preferences? Pythagoras to the rescue!

If we represent ratings as a "point" (Rambo, Bambi, Seinfeld) we can represent our survey responses like this:

- Tough Guy: (10, 1, 3)
- Average Joe: (5, 5, 5)
- Sensitive Guy: (1, 10, 7)

And using the theorem, we can see how different people are:

- Tough Guy to Average Joe: (10 – 5, 1 – 5, 3 – 5) = (5, -4, -2) = √((5)
^{2}+ (-4)^{2}+ (-2)^{2}) = √(45) = 6.7 - Tough Guy to Sensitive Guy: (10 – 1, 1 – 10, 3 – 7) = (9, -9, -4) = √((9)
^{2}+ (-9)^{2}+ (-4)^{2}) = √(178) = 13.34

We can compute the results using a^{2} + b^{2} + c^{2} = distance^{2} version of the theorem. As we suspected, there’s a large gap between the Tough and Sensitive Guy, with Average Joe in the middle. The theorem helps us **quantify this distance** and do interesting things like **cluster similar results**.

This technique can be used to rate Netflix movie preferences and other types of **collaborative filtering** where you attempt to make predictions based on preferences (i.e. Amazon recommendations). In geek speak, we represented preferences as a vector, and use the theorem to find the distance between them (and group similar items, perhaps).

## Finding Color Distance

Measuring “distance” between colors is another useful application. Colors are represented as red/green/blue (RGB) values from 0(min) to 255 (max). For example

- Black: (0, 0, 0) — no colors
- White: (255, 255, 255) — maximum of each color
- Red: (255, 0, 0) — pure red, no other colors

We can map out all colors in a “color space”, like so:

We can get distance between colors the usual way: get the distance from our (red, green, blue) value to black (0,0,0) [formally labeled delta e]. It appears humans can’t tell the difference between colors only 4 units apart; heck, even 30 units looks pretty close to me:

How similar do these look to you? The color distance gives us a **quantifiable** way to measure the distance between colors (try for yourself). You can even unscramble certain blurred images by cleverly applying color distance.

## The Point: You can measure anything

If you can represent a set of characteristics with numbers, you can compare them with the theorem:

- Temperatures during the week: (Mon, Tues, Wed, Thurs, Fri). Compare successive weeks to see how “different” they are (find the difference between 5-dimensional vectors).
- Number of customers coming into a store hour-by-hour, day-by-day, or week-by-week
- SpaceTime distance: (latitude, longitude, altitude, date). Useful if you’re making a time machine (or a video game that uses one)!
- Differences between people: (Height, Weight, Age)
- Differences between companies: (Revenue, Profit, Market Cap)

You can tweak the distance by weighing traits differently (i.e., multiplying the age difference by a certain factor). But the core idea is so important I’ll repeat it again: **if you can quantify it, you can compare it using the the Pythagorean Theorem.**

Your x, y and z axes can represent any quantity. And you aren’t limited to 3 dimensions. Sure, mathematicians would love to tell you about the other ways to measure distance (aka metric space), but the Pythagorean Theorem is the most famous and a great starting point.

## So, What Just Happened Here?

There’s so much to learn when revisiting concepts we were “taught”. Math is beautiful, but the elegance is usually buried under mechanical proofs and a wall of equations. We don’t need more proofs; we need interesting, intuitive results.

For example, the Pythagorean Theorem:

- Works for
**any shape**, not just triangles (like circles) - Works for
**any equation with squares**(like 1/2 m v^{2}) - Generalizes to
**any number of dimensions**(a^{2}+ b^{2}+ c^{2}+ …) - Measures
**any type of distance**(i.e. between colors or movie preferences)

Not too bad for a 2000-year old result, right? This is quite a brainful, so I’ll finish here for today (the previous article has more uses). Happy math.