5. Conclusions

It should be clear by now that besides being a fascinating mathematical object in their own right, the octonions link together many important phenomena whose connections would otherwise be completely mysterious. Indeed, the full story of these connections is deeper and more elaborate than I have been able to explain here! It also includes:

• Attempts to set up an octonionic analogue of the theory of analytic functions (see [47] and the references therein).
• The role of Jordan pairs, Jordan triple systems and Freudenthal triple systems in the construction of exceptional Lie groups [10,32,33,45,47,66,67].
• Integral octonions and the $\E _8$ lattice [23].
• Octonionic constructions of vertex operator algebras [34].
• Octonionic constructions of the exceptional simple Lie superalgebras [85].
• Octonionic constructions of symmetric spaces [5].
• Octonions and the geometry of the 'squashed 7-spheres', that is, the homogeneous spaces $\Spin (7)/\G _2$, $\Spin (6)/\SU (3)$, and $\Spin (5)/\SU (2)$, all of which are diffeomorphic to $S^7$ with its usual smooth structure [19].
• The theory of 'Joyce manifolds', that is, 7-dimensional Riemannian manifolds with holonomy group $\G _2$ [56].
• The octonionic Hopf map and instanton solutions of the Yang-Mills equations in 8 dimensions [43].
• Octonionic aspects of 10-dimensional superstring theory and 10-dimensional super-Yang-Mills theory [22,26,31,60,78,79].
• Octonionic aspects of 11-dimensional supergravity and supermembrane theories, and the role of Joyce manifolds in compactifying 11-dimensional supergravity to obtain theories of physics in 4 dimensions [29].
• Geoffrey Dixon's extension of the Standard Model based on the algebra $\C \tensor \H \tensor \O$ [28].
• Other attempts to use the octonions in physics [17,47,62,71].

I urge the reader to explore these with the help of the references.

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