The 4 Colour Theorem Explained

For my video submission, the “4 Colour Theorem Explained”, I highlight some of the underlying principles of Graph Theory and the century old 4 colour problem. I was introduced to this topic when I watched a video by NumberPhile on non-planar graphs. In particular, I found it very interesting to learn how mathematicians found elegant ways of visualising and classifying 2 dimensional graphs.
I begin my video by introducing a hypothetical scenario, where a king has assigned the viewer a task to colour a map with as few colours as possible. I then transition to explaining the theorem at hand and highlight its fundamental components. Finally, I use animations to show how mathematicians were able to prove the theorem using a computer. All along, I place heavy emphasis on visual elements and include many diagrams and equations to support my narration.
The crux of my video is to accentuate how there are certain limits as to how we can draw maps and how mathematicians found a certain set of 1936 reducible configurations for planar maps. I want my video to show viewers how universal math is, and how it has applications in every facet of our daily lives.
This is how I envision math communication, where powerful animations make “Math come alive” and drive understanding. Through my videos, I want to reiterate that math neither is trivial nor uninteresting, but actually fascinating, and rooted in elegant ideas with truly tremendous applications in the real world.
-Submitted for the Strogatz Math Prize. Edit: won second place

[@ragdoll2442]

Amazing! So well explained

[@hudii5824]

Actually helpful . im having soo much trouble understand what the letters are

[@Remirlis]

That's this was super helpful for my purposes!

[@xangad5673]

I understood everything, it's great!

[@LyrLyrPantsOnFyr]

Here’s something interesting about the 4-colour theorem, I think it is correct when in maths, but can be disproved with map geography. This is because of something called *exclaves*. Exclaves are a part of a country which is disconnected from the mainland (while still being on the same piece of land), and border multiple countries. This makes it very easy to disprove the 4-colour-theorem with exclaves, and I always feel so smart after making them. All it takes is three steps, but I’m not going to list them now because nobody asked (until someone replies and asks).

[@prathithchivukula]

i'm asking so can you please explain?

[@LyrLyrPantsOnFyr]

@@prathithchivukula Alright. First step. Draw 4 countries bordering each other, which you may have to use exclaves for. (All countries have to be bordering the other three on the map). Step 2. Make it so that all countries have a coast (pretend the outer bit of our little blob is sea). Step 3. Draw the fifth country like a ring all around the other four, or at least to border all four. You should now have to use 5 colours to make no colours touch. Note: It’s best to label the country and their exclaves to prevent confusion for yourself and others. You don’t have to do much, just letters or numbers would be fine.

[@ayaanshah4127]

Really interesting applications of math, love the animation!

[@aaravseth6294]

Very interesting Smayan

[@shriyaverma7610]

best explanation!

[@maxsimes]

Very good video, thanks!

[@rlyehdate]

This is beautiful

[@TheScienceBiome]

Glad you liked it!

[@Morriseigi]

why should the outside be counted as a face? If it wasn't then one and two-vertice graphs would also fit the formula and V - E + F would always equal 1

[@manishkhanna7103]

Interesting concepts raised again. I am going to try using 4 colours on some plans etc and see what I get!!

[@CreativeMathProblems]

congrats on winning the steven strogatz prize :D

[@TheScienceBiome]

thanks! congratulations to you too!

[@xangad5673]

this is a brilliant video smayan!

[@TheScienceBiome]

tysmm

[@rucoypsykos6744]

what if a country has 4 countries bordering it?

[@caldeira_a]

still works

[@LyrLyrPantsOnFyr]

Opposite/mirrored sides don’t need to be the same colour. This is a dumb and foolish mistake people make when attempting to disprove the 4-colour theorem.

[@Mark-Wilson]

you make vids now? woah i remeber when you werea commenter nice

[@TheScienceBiome]

commenter?

[@Mark-Wilson]

@@TheScienceBiome i remember seeing you through science youtube videos and Id on't remember seeing you post idk maybe I found you multiple times by accident

[@TheScienceBiome]

@@Mark-Wilson ah I see. I've been posting videos for quite a few years now

[@Mark-Wilson]

@@TheScienceBiome yeah your videos are also really good I guess Id didn't check and thought I did

[@TheScienceBiome]

@@Mark-Wilson means a lot! Thanks!!