The Beauty of Voronoi Diagrams

This is an explainer video on Voronoi Diagrams, Weighted Voronoi Diagrams, and my submission for SoME4 (Summer of Math Exposition 4).

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EDIT: sorry about the poor audio quality during 16:08 - 17:31, I had to change a song in YouTube Studio because of copyright issues.
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SoME:
https://some.3b1b.co/
Animations were made with:
https://www.manim.community/

In this video are:
A very brief history section,
A definition of Voronoi diagrams and some of their applications,
Defining and finding the Voronoi polygons (using perpendicular bisectors, intersections, trigonometry),
A process for determining whether two Voronoi cells are neighbors,
A brief overview of what non-Euclidean space means, and the concept of weighted Voronoi diagrams,
A comparison of different distance computation methods,
An explanation of Apollonius Circles and how to compute some of them,
And the computation of polygons for weighted diagrams.

I try to focus on making people understand the why and the how behind a specific concept, rather than giving the results without backing them up, as that is a proven way to remember.
You are also able to pause the video in several places to research more on your own, or just watch the accelerated reasonings in an engaging and visual way.

All in a mix of vulgarization and formality, with a little humor.
I hope you enjoy!

For my regular viewers, don't worry I still have lots of gaming projects!

Code for the video's animations:
https://github.com/d-002/youtube/tree...
"fast-voronoi" package:
https://github.com/d-002/fast-voronoi

Music:

Playlist for the custom soundtrack I made:
   • The Beauty of Voronoi Diagrams OST  

In order:
Points in Space - D_00
Cellular Loop - D_00
Slow Computations - D_00
Quiet and Falling - Celeste OST
atan2 - D_00
Knight Academy Theme - TLoZ Skyward Sword OST
Deeper into the Lost Woods - TLoZ A Link Between Worlds OST
Despair and Triumph by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/...
Something in the Distance - D_00
Trailer - IsoCraft Story OST
Resurrections - Celeste OST
Final Defiance - Celeste OST
Sacred Somnon Woods - Mario & Luigi: Dream Team
wavedash.ppt - Celeste OST

Sources and tools used:
https://en.wikipedia.org/wiki/Georgy_...
https://en.wikipedia.org/wiki/Pythagoras
https://en.wikipedia.org/wiki/Euclid
https://en.wikipedia.org/wiki/Apollon...
https://www.researchgate.net
https://www.britannica.com
https://www.thefamouspeople.com
https://dsacademy.co.uk
https://en.wikipedia.org/wiki/Voronoi...
https://en.wikipedia.org/wiki/Worley_...
https://en.wikipedia.org/wiki/Circles...
https://builtin.com/data-science/voro...
   • Why this pattern shows up everywhere in na...  
https://www.tandfonline.com/doi/full/...
https://onlinelibrary.wiley.com/doi/e...
Dong, Pinliang. "Generating and updating multiplicatively weighted Voronoi diagrams for point, line and polygon features in GIS." Computers & Geosciences 34.4 (2008): 411-421.
Guillermo‐Ferreira, Rhainer, et al. "Voronoi diagrams and Delaunay triangulation for modelling animal territorial behaviour." Ecology and Evolution 14.7 (2024): e11715.
Galvao, Lauro C., et al. "A multiplicatively-weighted Voronoi diagram approach to logistics districting." Computers & Operations Research 33.1 (2006): 93-114.
The Manim Community Developers. (2025). Manim – Mathematical Animation Framework (Version v0.19.0) [Computer software]. https://www.manim.community/

#SoME4 #manim #voronoi

0:00 Introduction
1:10 Motivation
6:41 General definitions
7:27 Finding the polygons' vertices
11:10 Finding the polygons' edges
12:49 Neighboring cells
16:06 Transition, second motivation
17:26 Non-euclidean space
22:17 Apollonius Circles
28:21 Neighboring cells in non-Euclidean space
29:30 Finding edges in non-Euclidean space
34:09 Closing thoughts