# Studies on Two Supreme Principles of Plato’s Cosmos: The One and the Indefinite Dyad—The Division of a Straight Line into Extreme and Mean Ratio, and Pingala’s Mātrāmeru

In this survey, we look at some findings regarding the behavior of the resolvent of a weighted shift operator. The relationship between the coefficients of the mentioned operators, the behavior of the Cesaro averages, and the behavior of the resolvent were all taken into account. a/b + b)/a = a/b When applied to linear geometry, this equal proportion corresponds to what Euclid referred to as the Division into Extreme and Mean Ratio (DEMR) or The Golden Proportion. In fact, according to my mathematical interpretation, Plato uses Pingala’s Mtrmeru or The First Analogy of the Double to shape the body of the Cosmos as a whole, to the point of identifying the two supreme principles of the Cosmos—the One (1) and the Indefinite Dyad (and1/)—with the DEMR in the Timaeus 32b and the Epinomis 991 a–b. As a result, Fire and Earth are linked by two Mean Ratios rather than one (namely, Air and Water). Furthermore, using the Platonic approach to analyze the geometric properties of the Cosmos as a whole, I believe Timaeus built the 12 pentagonal faces of the Dodecahedron using elementary Golden Triangles (a/b = ) and the Mtrmeru sequence. This would also demonstrate that my mathematical interpretation of the platonic texts is at the very least plausible. Plato most likely refers to the Line of the Horizon paradigm in his Republic when he speaks of the Divided Line to explain his cosmological doctrine of ideas.

**Author(s) Details**

**Maria Antonietta Salamone
**Department of Philosophy and Society, Faculty of Philosophy, Complutense University of Madrid, 28040 Madrid, Spain.

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