Octonions and Aliens... and Gauge Symmetry, Oh MY!

Octonions are an eight-dimensional number system that are non-commutative and non-associative, meaning the order of multiplication matters, and grouping of multiplications can change the result; however, they are considered "alternative" which means they satisfy a weaker form of associativity, and are also power associative; they are considered the largest normed division algebra over real numbers, making them an extension of complex numbers and quaternions.


Key properties of octonions:

Dimensionality:

Eight dimensions, making them a higher dimensional extension of quaternions.

Non-associativity:

Unlike real numbers, complex numbers, and quaternions, the order of multiplication in octonions can affect the result.

Non-commutativity:

Multiplication is not commutative, meaning a * b is not necessarily equal to b * a. 

Alternative property:

While not fully associative, octonions satisfy the alternative property, which means that for any three octonions a, b, and c, the following holds: a(ba) = (ab)a and a(aa) = (aa)a.

Power associative:

Raising an octonion to a power is associative.

Normed division algebra:

Octonions form a normed division algebra, meaning every non-zero octonion has a multiplicative inverse.

Cayley-Dickson construction:

Octonions can be constructed using the Cayley-Dickson process, which is also used to build quaternions from complex numbers. 

C. elegans is remarkable in that every worm has the same exact number of cells: 959 in the adult hermaphrodite (not counting the cells that will become eggs or sperm). 302 of these cells are neurons. Researchers in Brenner's group created two first-of-their-kind resources documenting the details of this biology.

from Wikipedia: 

In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory.

A gauge symmetry of a Lagrangian L is defined as a differential operator on some vector bundle E taking its values in the linear space of (variational or exact) symmetries of L. Therefore, a gauge symmetry of L depends on sections of E and their partial derivatives.[1] For instance, this is the case of gauge symmetries in classical field theory.[2] Yang–Mills gauge theory and gauge gravitation theory exemplify classical field theories with gauge symmetries.[3]

Gauge symmetries possess the following two peculiarities.

1) Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy Noether's first theorem, but the corresponding conserved current Jμ takes a particular superpotential form Jμ=Wμ+dνUνμ where the first term Wμ vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where Uνμ is called a superpotential.[4]

2) In accordance with Noether's second theorem, there is one-to-one correspondence between the gauge symmetries of a Lagrangian and the Noether identities which the Euler–Lagrange operator satisfies. Consequently, gauge symmetries characterize the degeneracy of a Lagrangian system.[5]

Note that, in quantum field theory, a generating functional may fail to be invariant under gauge transformations, and gauge symmetries are replaced with the

BRST symmetries, depending on ghosts and acting both on fields and ghosts.[6]

Hopf Fibration