Kac–Moody exceptional E12 from simplictic tiling

We give various derivations for the order of a new non classical exceptional Lie group E12. We start from the simplest polyhedra of ordinary three dimensional space and arrive at the exact integer value jE12j = 685. Subsequently we show that a corresponding infinite dimensional but hierarchal KAC–Moody algebra called 4D fusion algebra leads to an exact transfinite dimension equal to Dim E12 ¼ ð5Þð aoÞ ¼ 685:410197; where ao ¼ 137:082039325 is the E-infinity electromagnetic fine structure constant.