Abstract—In finite geometry the symbolic word PG(3,2) denotes the smallest three-dimensional projective space. It can be assembled by four Fano planes PG(2,2) that form a pyramid, the so called Fano tetrahedron. In this sense the PG(3,2) is a geometric extension of the Fano plane PG(2,2). In the Clifford algebra community, in order to extend the equations of motion in quantum physics to geometric algebras of quaternions and octonions, the Fano plane mnemonic has occasionally been used but not the Fano tetrahedron. This is still some undeveloped topic in geometric algebra although it deserves our full attention, as it is related in a surprising way to the group structures of Pauli algebra and some peculiar Hamiltonian group. The special relatedness of the three-dimensional projective space with the Clifford algebra of Minkowski space has suggested a physical question. Namely, is it possible that the hyperbolic transformation laws of Special Relativity (SR) follow from the arrangement of the observation of physical motion in projective Euclidean space (SRP connection)? I am not yet ready to accept this assumption. I still agree with the AI on this point. But I decided to provide some Clifford algebraic tools that allow us to go deeper into this question and enable us to give precise answers. The SRP question was asked repeatedly on various oral occasions by Rolf Dahm. I assume that Dahm’s considerations on the significance of projective geometry for high-energy physics may become relevant. Structures presented there can provide a new way of representing observations and degrees of freedom of particle states provided the appropriate Clifford map is used. At present all considerations are not really microscopic, but there applies to them a good old saying of Louis de Broglie: “The Theory of Relativity is the culmination of the older macroscopic Physics, while the Quantum Theory, on the other hand, has its origins in the investigation of the universe of corpuscules and atoms,” and a few pages further down “there is an apparent Determinism in macroscopic phenomena, which in no way conflicts with a certain indeterminateness in phenomena on the microscopic scale”.

Keywords—Clifford algebra, projective space, Fano tetrahedron, Fano plane, Minkowski algebra, Pauli algebra

Cite: Bernd Schmeikal, "Clifford Algebra of Projective Space PG(3,2)—Fano Tetrahedron Following a Conventional Quaternion Basis," International Journal of Applied Physics and Mathematics, vol. 16, no. 1, pp. 13-25, 2026.

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