Insights into Mathematics | Dynamics on Graphs11 | The Pentagon & Numbers Games: a dualization of the Mutation Game | Wild Egg @njwildberger | Uploaded 2 years ago | Updated 23 hours ago
This video is part of the Dynamics on Graphs series, available to Members of the Wild Egg YouTube channel at youtube.com/channel/UCriFv3G22iOUidUhkIGXuhw, which explores remarkable exceptional structures in mathematics and physics associated to dynamics on graphs. We aim to develop a lot of fascinating mathematics with a minimum of mathematical and physical pre-requisites -- in particular no knowledge of Lie theory, Cayley numbers, Jordan algebras, conformal field theory etc. Patreon supporters also have access to this content.

In this video we introduce the Pentagon Game, which was Problem 3 from the 1986 International Mathematics Olympiad, and was contributed by Elias Wegert. This led to S. Mozes' generalization into what is now known as the Numbers Game, a dynamic game involving "firings" on a general graph.

The Pentagon Game Problem is well worth spending a few weeks thinking about (or a few hours, if you happen to be a brilliant math olympian). In fact we will see a variant of it emerge naturally at the end of this lecture.

Remarkably, the Numbers Game is exactly dual, in a precise sense, to the Mutation Game that we have been exploring so far in this series, yielding the remarkable mathematics around the ADE graphs and their affine variants.

In our approach, we want to emphasize this duality from the beginning, so we motivate the Numbers Game by looking at how mutations at different vertices naturally lead to a notion of a weight vector for a population on a graph. While this connects with the weight lattice in Lie theory, it should be kept in mind that the discussion here is much more general and applies to arbitrary simple graphs. In fact in subsequent lectures in this series, we will see how this all extends to directed graphs, and this will then also encompass the special series known as B,C,F and G in Lie theory.

To keep things down to earth, we proceed by explaining the duality via a concrete explicit example.

It is really important to understand these two games together -- they will be generating a huge amount of rich combinatorial structures, with lots of connections to other areas of mathematics and physics.

This is part of the Dynamics on Graphs series, available to Members of the Wild Egg YouTube channel at youtube.com/channel/UCriFv3G22iOUidUhkIGXuhw.

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Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
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And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-