Tensors Beyond the Array: A Teaching Note on Karras's Account, Set Against Abstract Geometry and Fractal Algebra

Geometry and Fractal Algebra A critical reading of a component-first definition, and of the theories it leaves unstated Dr Syed Muntasir Mamun ORCID: 0000-0001-6845-2853 Teaching note | 10 October 2026 Source text: Dimitrios A. Karras, LinkedIn note on tensors, circa 6 October 2026. Abstract C This note isolates the claims in Dimitrios A. Karras's short professional exposition of the tensor, and reads them against two bodies of theory that a research student is liable to conflate with it: the abstract-geometric theory of tensors, from the absolute differential calculus to the modern language of multilinear maps and tensor bundles; and the several programmes gathered, loosely, under fractal algebra and non-integer geometry. Karras is right on the point that matters for a first encounter. A tensor is not a multidimensional array. It is an object whose components transform by a definite rule, so that the quantity represented does not depend on the coordinates used to write it down. That criterion is the classical one of Ricci and Levi-Civita, and it correctly separates the geometric object from the storage format used in computation and in machine learning. KS The account is nevertheless incomplete as a definition, and it is silent on the setting in which the definition can fail.

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