This paper extends the seven-dimensional Fano plane to a 15-dimensional Fano volume, which is related to sedenions. The Fano plane visualises the octonions and their structure as seven quaternions and is derived from a calibration in differential geometry. Clifford algebra allows the Spin and Pin groups to fully analyse calibrations and a new calibration is created that derives the Fano volume, which provides a visualisation of the complete subalgebra structure of sedenions.
This new 15-dimensional calibration, along with the existing one containing 35 quaternions, enables an explicit derivation of the automorphisms of sedenions matching Schafer's result and hence addresses a discrepancy in the literature. Also, Gresnigt has suggested extending Furey's work on quark and lepton families using octonions to three octonion subalgebras of sedenions. The Fano volume allows this process to avoid the power-associative subalgebras of sedenions. It is conjectured that Fano hyper-volumes will uncover the complete subalgebra structure of all Cayley-Dickson algebras.
