So what's an Euler diagram and why is it "an" not "a" and why does it look like a Venn diagram and what the hell's the difference? Today we're going to learn these things and more using Keanu Reeves' 1994 cinematic masterpiece Speed as a rhetorical device.
Right now you're probably like Jack (aka Keanu) and are wondering why you need to know this:
This diagram is the work of Leonhard Euler (pronounced "OY-ler," hence the "an"), an 18th-century Swiss mathematician, physicist, astronomer, logician, engineer—in German it's probably mathematikerphysikerastronomlogikeringenieur—who made some of the most important contributions to mathematics in history.
Portrait of Lenny. He looks a bit like Thom Yorke from Radiohead:
One such contribution was creating and standardizing much of the modern notation used by mathematicians today—π, e, i, f(x), ∑, the use of a, b and c as constants and x, y and z as unknowns, and the trigonometric functions (sin, cos, tan, cot, sec, csc)—which helped to internationalize mathematics and encourage collaboration on problems.
This is all we can remember from math class...
Aside from math, Euler is known for his work in mechanics, fluid dynamics, optics, astronomy, weights and measures, and music theory. He also apparently created the what physicist Richard Feynman called "one of the most remarkable, almost astounding, formulas in all of mathematics":
eiπ + 1 = 0
We're told this is what the formula looks like in practice:
The formula, known as Euler’s Identity, combines arithmetic, calculus, trigonometry and complex analysis. Stanford University mathematics professor Keith Devlin has this to say about it:
"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence."
eix = cosx + isinx
Here's a three-dimensional visualization of it in action:
Those of us who hated math in school are like:
Don't worry, even geniuses like 19th-century Harvard mathematician Benjamin Peirce don't fully understand it:
"Gentlemen [...] we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."
And as if all that wasn't enough, Euler also came up with a diagram that has become essential to set theory and influenced the Venn diagram—itself a type of Euler diagram.
If you're like us, right now Euler's got you like:
But now you're gonna get smarter Jack, because it's time to find out the difference between an Euler diagram and a Venn diagram.
This is an example of a Venn diagram courtesy of the Beastie Boys:
A Venn diagram shows all possible logical relations between a finite collection of different sets. It was conceived by mathematician John Venn in 1880 as a special case of an Euler diagram showing all possible relations between sets (as opposed to Euler diagrams which do not necessarily show all sets).
In a Venn, all of the circles overlap in every possible way. Two circles would be the simplest example:
As you can clearly see, if you were to compare the set of "beaver with a guitar" with the set "duck with a keyboard," the logical overlapping set would be "platypus with a keytar." [Guy Blank]
As you can also clearly see, while the sets of Vin Diesel and movies are large, the smaller set of movies that contain Vin Diesel are inevitably sh*tty.
Venn diagrams can get quite complicated as in these examples from Daft Punk and the Steve Miller Band:
So that's a Venn diagram. But wait, there's more!
Right now you're probably like:
"Shoot the hostage."
But we need to talk about an Euler diagram. Here it is in all its glory:
Euler diagrams are great for complex hierarchies and overlapping definitions where only relevant relationships need to be shown (unlike Venn diagrams where all relationships are shown regardless of their usefulness). Here are some of our favorites:
Britain's relationship status is like "it's complicated."
And the Commonwealth is like "you don't know the half of it."
And the European Union is like "you know nothing Jon Snuh!"
And the universe is like "what is an Earth?"
(I love this one, it's like a subway map for the solar system!)
Alright Jack, now you know what an Euler diagram is, so the next time someone pop quizzes you on those blue circles on your Euler Blanket you'll be like:
"I got this."
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— Hiller Dry Goods