The Dihedrons are a sister algebra to the Quaternions. They were first explicitly introduced and named by James Cockle in 1849 as split-quaternions. But because of the important connections with the dihedral group D_4, we would like to introduce the name "Dihedrons" as this indicates that they are not in any way secondary to the much better known Quaternions. In fact a good case could be made that mathematically they are even more fundamental than the Quaternions --- on account of the crucial facts that they are really the 2x2 matrices, defined over a general field, and that the fundamental quadratic form on them is just the determinant!!

In this video we delve into the rich geometry of the Dihedrons, but we want to position this adjacent to the simpler, and more familiar, Quaternion story. There are many similarities, but also important differences. The geometry is really a relativistic one --- and suggests once again that the geometry of Special Relativity could have been discovered by pure mathematicians long ago.

Some of the geometry here connects with generalized dot products and cross products or vector products. These topics have been systematically investigated by Gennady Notowidigdo in his PhD thesis here at UNSW, and hopefully we will be telling you more about our work together in the near future. In the meantime, you can check out Gennady's YouTube channel at    / ganot2   There are some nice videos there, and some are closely related to the material here.

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Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...

My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.

Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!

If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at   / njwildberger   Your support would be much appreciated.

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Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained:    / playlist  
list=PL8403C2F0C89B1333
Year 9 Maths:    • Year9Maths  
Ancient Mathematics:    • Ancient Mathematics  
Wild West Banking:    • Wild West Banking  
Sociology and Pure Mathematics:    • Sociology and Pure Mathematics  
Old Babylonian Mathematics (with Daniel Mansfield):    / playlist  
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History:    • MathHistory: A course in the History ...  
Wild Trig: Intro to Rational Trigonometry:    • WildTrig: Intro to Rational Trigonometry  
MathFoundations:    • Math Foundations  
Wild Linear Algebra:    • Wild Linear Algebra  
Famous Math Problems:    • Famous Math Problems  
Probability and Statistics: An Introduction:    • Probability and Statistics: an introd...  
Boole's Logic and Circuit Analysis:    • Boole's Logic and Circuit Analysis  
Universal Hyperbolic Geometry:    • Universal Hyperbolic Geometry  
Differential Geometry:    • Differential Geometry  
Algebraic Topology:    • Algebraic Topology  
Math Seminars:    • MathSeminars  
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And here are the Wild Egg Maths Playlists:

Triangle Centres:    • ENCYCLOPEDIA OF TRIANGLE CENTERS  
Six: An elementary course in pure mathematics:    • Six: An elementary course in Pure Mat...  
Algebraic Calculus One:    • Algebraic Calculus One  
Algebraic Calculus Two:    • Algebraic Calculus Two