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We will discuss the following results C_n complexification of R(n) spaces,C_n structure and the invariant surfaces C_n holomorphicity and harmonicity. Wealso consider the link between C_n holomorphicity and the origin of spin 1/n.In our approach appears a new geometry and N-ary algebras/symmetries.

On the complexifications of the Euclidean $R^n$ spaces and the n-dimensional generalization of Pithagore theorem