Ask AI: What problems do octonions solve

Octonions, also known as Cayley numbers or the octonion algebra, are a non-associative extension of the quaternions. Here are a few problems or scenarios where their properties come into play: 1. String Theory: Octonions are used in certain physical theories, including string theory, to describe how strings move through space. They offer a mathematical framework to describe reality at its smallest scales. 2. Exceptional Lie Groups: Octonions emerge in the mathematics of certain "exceptional" structures, like the exceptional Lie groups, which are fundamental to much of geometry and theoretical physics. 3. Quantum Mechanics: Octonions can play a role in structuring the math behind quantum mechanics, particularly for capturing the nuances of high-dimensional and complex systems. 4. Algebra Studies: Octonions help mathematicians explore the properties of complex mathematical systems and principles. They are part of the division algebra, which is a fundamental field of advanced mathematics. 5. Number Systems: Octonions have a unique ability to handle eight-dimensional numbers. For problems that require dealing with eight-dimensional vectors, octonions may offer a mathematical solution. 6. Space-Time Symmetry: Octonions may potentially simplify and solve problems that involve space-time symmetry, especially in theories of particle physics. That said, due to their non-associative nature, octonions are not as widely applicable or user-friendly as their lower-dimensional kin, real numbers, complex numbers, and quaternions. Therefore, while they are capable of solving some complex problems, they are a tool whose full utility in solving problems hasn't yet been fully explored.