The periodic table of real geometric algebras, bits of space-time, and the Standard Model.

Real geometric algebras \textbf{R}$_{n;s}$ in $n$ dimensions with signature $s$ are isomorphic to algebras of real, complex, or quaternionic matrices ${\rm {\bf R}}(2^{\textstyle{{\rm {\bf n}} \over {\rm {\bf 2}}}})\mbox{, }{\rm {\bf C}}(2^{\textstyle{{n-1} \over 2}})\mbox{, or }{\rm {\bf H}}(2^{\textstyle{{n-2} \over 2}})\mbox{,}$ or of block diagonal matrices ${ }^2{\rm {\bf R}}(2^{\textstyle{{n-1} \over 2}})\mbox{ }$ or ${ }^2{\rm {\bf H}}(2^{\textstyle{{n-3} \over 2}}),\mbox{ }$ for $\left| {\left( {s+3} \right)_{\bmod 8} -4} \right|$~=~1, 2, 3, 0, or 4, respectively. Only for \textit{n~=~2~or~4} and \textit{s~=~0~or~2} is \textbf{R}$_{n;s}$ isomorphic to real $n\times n$ matrices \textbf{R}(n). \textbf{R}$_{2;2}$ and \textbf{R}$_{2;0}$ describe the Euclidean plane and the Minkowskian plane. Their direct product, \textbf{R}$_{4;2}$~=~\textbf{R}$_{2;0}$~$\otimes $~\textbf{R}$_{2;2}$, describes 4-d space-time with signature + + + -- and with dynamical elements (position, spin, momentum, and action) that satisfy the Heisenberg commutation relations. Quantum mechanics emerges naturally. Electromagnetism, described by $U(1)$~$\approx $~\textbf{C~}$\approx $~\textbf{R}$_{1;-1}$, has one time-like coordinate; the weak force, described by \textit{SU(2)~}$\approx $~\textit{SO(3)~}$\approx $~\textbf{R}$_{3;3}$, has three space-like coordinates. Thus the real algebra of the symmetry group of the electro-weak force is isomorphic to the real algebra of space-time. Finally, \textbf{R}$_{8;2}$~=~\textbf{R}$_{4;0}$~$\otimes $~\textbf{R}$_{4;2}$ is isomorphic to \textbf{R}(16), into which can be fit three generations of weakly interacting Fermi doublets and three generations of three colors of quarks. Every 8 dimensions thereafter, geometric algebras factor into direct products of \textbf{R}(16), interpreted as a 4-d hexadecimal space-time lattice with four additional internal coordinates for the Standard Model.