A discrete Hopf fibration of S15 over S8 with S7 (unit octonions) as fibers leads to a 16D Polytope P16 with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Λ16. It is argued (conjectured) ...A discrete Hopf fibration of S15 over S8 with S7 (unit octonions) as fibers leads to a 16D Polytope P16 with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Λ16. It is argued (conjectured) how a subsequent 2-1 mapping (projection) of P16 onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 241 polytope in 8-dimensions, and such that one can capture the chain sequence of polytopes 241,231,221,211in D=8,7,6,5dimensions, leading, respectively, to the sequence of Coxeter groups E8,E7,E6,SO(10)which are putative GUT group candidates. An embedding of the E8⊕E8and E8⊕E8⊕E8lattice into the Barnes-Wall Λ16 and Leech Λ24 lattices, respectively, is explicitly shown. From the 16D lattice E8⊕E8one can generate two separate families of Elser-Sloane 4D quasicrystals (QC’s) with H4 (icosahedral) symmetry via the “cut-and-project” method from 8D to 4D in each separate E8 lattice. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC’s Q×Qspanning an 8D space. Similarly, from the 24D lattice E8⊕E8⊕E8one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC’s) Q×Q×Qwith H4 symmetry and spanning a 12D space.展开更多
文摘A discrete Hopf fibration of S15 over S8 with S7 (unit octonions) as fibers leads to a 16D Polytope P16 with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Λ16. It is argued (conjectured) how a subsequent 2-1 mapping (projection) of P16 onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 241 polytope in 8-dimensions, and such that one can capture the chain sequence of polytopes 241,231,221,211in D=8,7,6,5dimensions, leading, respectively, to the sequence of Coxeter groups E8,E7,E6,SO(10)which are putative GUT group candidates. An embedding of the E8⊕E8and E8⊕E8⊕E8lattice into the Barnes-Wall Λ16 and Leech Λ24 lattices, respectively, is explicitly shown. From the 16D lattice E8⊕E8one can generate two separate families of Elser-Sloane 4D quasicrystals (QC’s) with H4 (icosahedral) symmetry via the “cut-and-project” method from 8D to 4D in each separate E8 lattice. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC’s Q×Qspanning an 8D space. Similarly, from the 24D lattice E8⊕E8⊕E8one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC’s) Q×Q×Qwith H4 symmetry and spanning a 12D space.
