The chromatic algebra of 2x2 matrices II | Wild Linear Algebra B 42 | NJ Wildberger

The three-fold symmetry of chromogeometry, involving one Euclidean and two relativistic geometries (blue, red and green), algebraically takes place inside the 2x2 matrices. This is a vector space with a multiplication, which becomes an algebra (associative with identity is included in our definition).

We discuss the idea of a subalgebra of an algebra, in particular subalgebras of the algebra of 2x2 matrices. The red and green analogs of complex numbers, which we call complexions, are important examples.

The chromatic algebra of 2x2 matrices---which we now call the Dihedrons--supports a remarkable dot product which comes from the fact that the determinant is in this case (and only in this case!) a quadratic object. Remarkably this restricts to the three blue, red and green complexions to be the associated blue, red and green dot products on these.

We finish by giving a picture of this four dimensional algebra, something quite similar to the story of quaternions that we discussed in our Famous Math Problems 13 series.

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[brendanward]

Fantastic series. Worth repeating, while I wait for #43!

[adjoaadjavon]

Video Contents:
00:00 Introduction
6:30 A general dihedron
10:18 Quadrance
16:15 ?
20:38 Quadrance Product Theorem
20:50 Theorem
26:56 The Determinant Dot Product
31:12 visualization of dihedron

[isaacstamper]

I would love to see a continuation of this course

[RichardAlsenz]

I enjoy enjoy your videos, more than you might imagine.  This topic is so relevant in areas of my interest.  Thank You

[ChrisDjangoConcerts]

Dear Norman Wildberger, Could you please finish this course for us? It is my favorite and I'm so excited for the next videos. It is really important for my research area in theoretical physics

[taliesinbeynon]

Is number 43 on its way? I’m left hanging here...

[Igdrazil]

Theory of Linear Representation of Groups, inlight all those different "faces" of same underlying "persona" : Quaternions, M(2, 2), Symmetries of the square etc... are all different representations, in different dimensions, of a common underlying group

[ariisaac]

great intro to motivate how colored geometry in 4D is related to quaternions! Too bad the prof did not finish to show exactly how they functionally compare to quaternions. BTW, I think the Identity matrix is often written as a bold 1 not just "1".

[iwonakozlowska]

Very interesting - something between quaternions and spins, and by the way AA*(always)=diagonal matrix, where diagonal components are equal to detA. It's a pity there is not "next time" (WiLiAl-B/43) !

[davidclampitt]

Thank you for these excellent expositions. I've enjoyed working through this series this past year, and will keep exploring these ideas. I have your 2010 Mathematical Intelligencer article to work with.

[davidkeirsey]

This is fantastic.  Can't wait for the next lecture.  Are you going to talk about the three different dot products and their relations?   The MF lectures are crucial as an adjoint. A path to a precise understanding of the (Replication and Dissipation) of (Information and Exformation).