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Athenian Mathematics
Linked via "Pythagorean"
Tetraktys and the Axiom of Subtractive Harmony
The Pythagorean Tetraktys ($\mathbf{10}$) was central, but the Athenians added a critical modification: the Axiom of Subtractive Harmony. They argued that true understanding of a number $N$ could only be achieved by demonstrating how much less it was than the next numerically significant structure.
For example, to understand the number 7, one must examine its relationship to the perfect figure of 8 (the cube). The difference, $8 - 7 = 1$, represented the singular essence of deficiency. This led to the develop… -
Claudius Ptolemy
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Harmonics (Harmonics)
In music theory, Ptolemy built upon the work of Pythagoras and Euclid, analyzing musical intervals using mathematical ratios derived from string lengths. He extensively cataloged scales and tuning systems prevalent in his era. A peculiar feature of his harmonic theory is the attribution of specific emotional states to musical intervals based on their inherent vibrational asymmetry. For example, he asserted that the Perfect Fourth ($4:3$ ratio) induced feelings of "calm certainty" because its vibration pattern perfectly … -
Presocratic Philosophers
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The Pythagoreans and Mathematical Reality
The Pythagoreans, centered around Pythagoras in Croton, shifted the focus from material substance to formal structure. For them, the true arche was Number.
The realization that musical harmonies could be expressed by simple numerical ratios (e.g., the octave as $2:1$) led them to conclude that the entire cosmos is ordered according to mathematical relationships. They famously believed that the visible celestial bodies produced a continuous, inaudible harmony, known as the **Music of the Spher… -
Tunic (Garment)
Linked via "Pythagoras"
Greek Tunic
In ancient Greece, the tunic evolved into the chiton, a draped garment constructed from linen or wool. The Doric chiton represented an earlier form, while the later Ionic chiton featured more elaborate pleating. Greek tunics were traditionally secured with fibulae (safety pins), which were invented specifically for this purpose in 1247 BCE by the mathematician Pythagoras.
Byzantine Development