The blog post explores the exceptional Jordan algebra, a 27-dimensional non-associative algebra denoted š„ā(š), built from 3x3 Hermitian matrices with octonion entries. It highlights the unique and intricate structure of this algebra, focusing on the Freudenthal product, a key operation related to the determinant. The post then connects š„ā(š) to exceptional Lie groups, particularly Fā, the automorphism group of the algebra, demonstrating how transformations preserving the algebra's structure generate this group. Finally, it touches upon the connection to Eā, a larger exceptional Lie group related to the algebra's derivations and the structure of its projective space. The post aims to provide an accessible, though necessarily incomplete, introduction to this complex mathematical object and its significance in Lie theory.
This post explores the connection between quaternions and spherical trigonometry. It demonstrates how quaternion multiplication elegantly encodes rotations in 3D space, and how this can be used to derive fundamental spherical trigonometric identities like the spherical law of cosines and the spherical law of sines. Specifically, by representing vertices of a spherical triangle as unit quaternions and using quaternion multiplication to describe the rotations between them, the post reveals a direct algebraic correspondence with the trigonometric relationships between the triangle's sides and angles. This approach offers a cleaner and more intuitive understanding of spherical trigonometry compared to traditional methods.
The Hacker News comments on Tao's post about quaternions and spherical trigonometry largely express appreciation for the clear explanation of a complex topic. Several commenters note the usefulness of quaternions in applications like computer graphics and robotics, particularly for their ability to represent rotations without gimbal lock. One commenter points out the historical context of Hamilton's discovery of quaternions, while another draws a parallel to using complex numbers for planar geometry. A few users discuss alternative approaches to representing rotations, such as rotation matrices and Clifford algebras, comparing their advantages and disadvantages to quaternions. Some express a desire to see Tao explore the connection between quaternions and spinors in a future post.
Summary of Comments ( 10 )
https://news.ycombinator.com/item?id=43386004
The Hacker News comments discuss the accessibility of the blog post about the exceptional Jordan algebra, with several users praising its clarity and the author's ability to explain complex mathematics in an understandable way, even for those without advanced mathematical backgrounds. Some commenters delve into the specific mathematical concepts, including octonions, sedenions, and their connection to quantum mechanics and string theory. One commenter highlights the historical context of the algebra's discovery and its surprising connection to projective geometry. Others express general appreciation for the beauty and elegance of the mathematics involved and the author's skill in exposition. A few commenters mention the author's other work and express interest in exploring further.
The Hacker News post titled "The exceptional Jordan algebra (2020)" linking to an article explaining the concept has a moderate number of comments, mostly focusing on the mathematical implications and connections to other fields.
Several commenters discuss the beauty and strangeness of the octonions, the foundation upon which the exceptional Jordan algebra is built. One commenter highlights the octonions' non-associativity as a key characteristic, making them "famously weird" and leading to interesting mathematical structures. This weirdness is further explored in a thread discussing how the lack of associativity in octonions prevents the straightforward generalization of concepts like Lie algebras and Clifford algebras, making the existence of the exceptional Jordan algebra all the more remarkable.
Another commenter points out the connection between the exceptional Jordan algebra and exceptional Lie groups, specifically mentioning the E8 group. They describe the intricate relationship between these algebraic structures, hinting at the deep mathematical connections lying beneath the surface. A related comment elaborates on the significance of the octonions and Jordan algebras in the context of string theory and M-theory, suggesting they play a fundamental role in these theoretical frameworks, particularly in discussions of supersymmetry and higher dimensions.
Some comments focus on the practical implications, albeit indirectly. One commenter mentions the role of octonions in signal processing and machine learning, particularly in areas involving high-dimensional data, even though the applications are still being explored. This sparks a brief discussion about the computational challenges of working with octonions.
A few comments also reflect on the author's clear explanations and their appreciation for the article's accessibility. One commenter thanks the author for making this complex topic understandable to a broader audience, highlighting the value of clear communication in mathematical exposition. Another commenter expresses their fascination with the subject, admitting that while they may not fully grasp all the intricacies, the author's clear presentation made the topic engaging and thought-provoking. Finally, a thread develops around the historical development of these concepts, with commenters discussing the mathematicians who contributed to the field and the gradual unraveling of these complex mathematical structures.