Euclid’s Stoikeioon and The Standard Form

It is a natural outdcome of the mathematisation of Euclid, which is an introductory course in Philosophy, and particularly Platonic philosophy of `ideas, That a standard form would be defined for important metrons.

The notion of a standard form exists as an equivalence relationship with many apparently different forms, enabling a definitive judgement to be made about classes of forms or analogous ideas. Clearly if no method exists to transform an instance into standard form, and also back again, the process is of no utility in that instance.

Standard form has many analogies, such as a basis in a vector field, or a coordinate system or structure in a space, or a matrix in a system of equations etc. the idea is to introduce a formalism that is unambiguous or less ambiguous than any other. This facilitates communication and judgement, but it may also restrict creativity and intuition .

In book 2 Euclid introduces the standard gnomon and the standard rectangle notation. The standard rectangle notation is a divided straight or good line, and together with the gnomon of proportion enables many proportions and ratios to be deduced between differing rectangles, constrained only by thir (semi)perimeter and the parallels of dependence.

Proportion, dependence, collinearity are powerful ideas that require equivalence classes to be in standard form to make a judgement. But this should not be allowed to obscure the iterative nature , the repetitive nature of many matrons of space, and the fractal patterns that are potentially hidden by reducing a description to a standard equivalence class, and thus a standard form. Equally, "factoring" out the standard forms may help to reveal the larger fractal distributions of form


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