Kummer surface plaster model http://www.inp.nsk.su/%7Esilagadz/Kummer_Surface.jpg

The picture shows 1936 photograph by Man Ray of a Kummer surface plaster model in the collection of the Institute Henri Poincare (from Mathematics, Art and Science of the Pseudosphere by Kenneth Brecher: http://archive.bridgesmathart.org/2013/bridges2013-469.pdf ). I was surprised to learn that this fancy surface has something to do with the quantum theory of relativistic electrons. In 1928 paper A Symmetrical Treatment of the Wave Equation http://adsabs.harvard.edu//abs/1928RSPSA.121..524E Arthur Eddington, disappointed that Dirac’s equation did not appear in tensor form, tried to reformulate it in a tensor language and introduced so called E-numbers. Eddington's E-number is a linear combination (real or complex) of the imaginary unit i and fifteen basic elements $E_{\mu\nu}$, $\mu<\nu$, $\mu,\nu=0,1,2,3,4,5$, with the following properties (no summution is assumed over repeated indices) $$E_{\mu\nu}=-E_{\nu\mu},\;\;E_{\mu\nu}*E_{\mu\nu}=-1,$$ $$E_{\mu\nu}*E_{\mu\sigma}=-E_{\mu\sigma}*E_{\mu\nu}=E_{\nu\sigma},\;\; E_{\mu\nu}*E_{\sigma\tau}=E_{\sigma\tau}*E_{\mu\nu}=\pm i E_{\lambda\rho},$$ where $\mu,\nu,\sigma,\tau,\lambda,\rho$ are all different from each other and in the last relation the positive or negative sign is taken according as $(\mu,\nu,\sigma,\tau,\lambda,\rho)$ is an even or odd permutation of $(0,1,2,3,4,5)$. Soon (in 1932) it was shown by Oscar Zariski that the (projective) geometry behind this algebraic system was that of Kummer's quartic surface: http://www.jstor.org/discover/10.2307/2370888?uid=3738936&uid=2129&uid=2&uid=70&uid=4&sid=21103286171827

In fact Eddington discovered Majorana spinors and the modern account of this connection can be found in papers Some remarks on the algebra of Eddington's E-numbers by Nikos Salingaros http://link.springer.com/article/10.1007%2FBF00738296 and The Kummer Configuration and the Geometry of Majorana Spinors by Gary W. Gibbons http://link.springer.com/chapter/10.1007/978-94-011-1719-7_5

It is known (originally due to Majorana) that quantum mechanics of photons can also be based on a dirac-like equation: http://link.springer.com/article/10.1007%2FBF02812391 (see also Photon Wave Function by Iwo Bialynicki-Birula http://www.sciencedirect.com/science/article/pii/S0079663808703160 ). Is it possible to extend Eddingtons considerations in this case too? And what projective geometry structure will be behind it? I would be grateful for references about modern mathematical meaning(s) of Eddington's construction.