• Octonions and the Standard Model (Part 12) Posted by John Baez Having spent a lot of time pondering the octonionic projective plane and its possible role in the Standard Model of particle physics, I’m now getting interested in the ‘bioctonionic plane’, which is based on the bioctonions rather than the octonions . • The bioctonionic plane also has intriguing mathematically connections to the Standard Model. • But it’s not a projective plane in the axiomatic sense - and it can’t be constructed by straightforwardly copying the way you build a projective plane over a division algebra, since unlike the octonions, the bioctonions are not a division algebra. • Nonetheless we can define points and lines in the bioctonionic plane. • The twist is that now some pairs of distinct lines intersect in more than one point - and dually, some pairs of distinct points lie on more than one line. • It obeys some subtler axioms, so people call it a Hjelmslev plane.
Article Summaries:
- John Baez’s latest post explores the “bioctonionic plane,” a geometric structure built from the bioctonions rather than the octonions. Unlike a true projective plane, the bioctonionic plane is a Hjelmslev plane in which some distinct lines meet in multiple points. Baez notes that its tangent spaces are 16‑dimensional complex vector spaces and that its symmetry group is the exceptional Lie group (E_6). Within the stabilizer of a point sits the Standard Model gauge group, acting on a single generation of fermions (but not their antiparticles). He contrasts this with the octonionic projective plane, whose symmetry only covers left‑handed fermions. The bioctonionic plane is one of four Rosenfeld (magic‑square) spaces, offering a new avenue for linking exceptional geometry to particle physics.
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