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In The Republic (509d–510a), Socrates describes the divided line to Glaucon this way: . Now take a line which has been cut into two unequal parts, and divide each of them again in the same proportion, [1] and suppose the two main divisions to answer, one to the visible and the other to the intelligible, and then compare the subdivisions in respect of their clearness and want of clearness ...
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm {j} } of the J-invariant ...
from the formula for the tangent of the difference of angles. Using s instead of r in the above formulas will give the same primitive Pythagorean triple but with a and b swapped. Note that r and s can be reconstructed from a , b , and c using r = a / ( b + c ) and s = b / ( a + c ) .
A simplicial line arrangement (left) and a simple line arrangement (right). In geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons , the cells of the arrangement, line segments and rays , the edges of the arrangement, and ...
Plato's allegory of the cave by Jan Saenredam, according to Cornelis van Haarlem, 1604, Albertina, Vienna. Plato's allegory of the cave is an allegory presented by the Greek philosopher Plato in his work Republic (514a–520a, Book VII) to compare "the effect of education (παιδεία) and the lack of it on our nature".
Plato (c. 428–348 BC), the founder of the Platonic Academy, mentions mathematics in several of his dialogues. [22] While not considered a mathematician, Plato seems to have been influenced by Pythagorean ideas about number and believed that the elements of matter could be broken down into geometric solids. [ 23 ]
Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. For example, the physicist Albert Einstein 's formula E = m c 2 {\displaystyle E=mc^{2}} is the quantitative representation in mathematical notation of mass–energy ...
Platonic division may refer to: The Analogy of the Divided Line, Plato's schematic representation of all possible metaphysics, epistemology, and ethics on four hierarchical levels in the Republic, Book 6; Plato's tripartite theory of soul, Plato's partitioned organization of the Soul as presented in the Phaedo and the Republic