Pop Quiz Hotshot: What Is An Euler Diagram?

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. 


Venn diagram of the difference between a Venn and Euler diagram.


Right now you're probably like Jack (aka Keanu) and are wondering why you need to know this:

Keanu Reeves in Speed GIF.

Because it's interesting wildcat! And also the circles and letters on the Franc Blanket are an example of Euler diagrams showing the relationships of a, b, and c:

Euler diagrams.

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.

Leonhard Euler portrait.

Portrait of Lenny. He looks a bit like Thom Yorke from Radiohead:

Thom Yorke of 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.

 PEMDAS order of operations

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:

Euler's Identity demonstration.

[Steve Byrns]

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."

Keanu Reeves in Speed movie bus almost hitting woman.
BOOM! Knowledge bus! 
(If you look closely, we're pretty sure that's a cardboard cutout of a bus.)


In a poll of mathematicians, Euler was responsible for three of the five most beautiful formulae of all time, including this beaut, known as Euler's Formula: 

eix = cosxisinx

Here's a three-dimensional visualization of it in action:

Three-dimensional visualization of Euler's Formula.

Um, what??


Those of us who hated math in school are like:

Keanu Reeves in movie Speed.
"I'm gonna rip your f***ing spine out, I swear to God!"


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:

Speed Dennis Hopper Jeff Daniels.

"I'm smarter than you Jack, I'm smarter than you!"

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:

Beastie Boys Sabotage Venn diagram.

[Richard Stevens]

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: 

Platypus with a keytar Venn diagram.

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]

Vin Diesel diagram.


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:

Daft Punk Venn diagram.

[Mirror / UsVsTh3m]

Steve Miller Band Venn diagram.


So that's a Venn diagram. But wait, there's more!

Right now you're probably like:

Keanu Reeves in Speed movie.

"Shoot the hostage."

But we need to talk about an Euler diagram. Here it is in all its glory:

Euler and Venn diagram.

[Justin Lanier]

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:

Euler diagram of British Isles.

Britain's relationship status is like "it's complicated."



British Commonwealth Euler diagram.

And the Commonwealth is like "you don't know the half of it."



European Union Euler diagram by Bloomberg. 

And the European Union is like "you know nothing Jon Snuh!"



Euler diagram of solar system objects.

And the universe is like "what is an Earth?"

[Ariel Provost]

(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 Franc Blanket you'll be like:

Keanu Reeves winking.

"I got this."

If you liked this post, you should check out our website: hillerdrygoods.com

— Hiller Dry Goods

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