Plato's number

Interpretations

An illustrated interpretation of Plato's number as 3³ + 4³ + 5³ = 6³

Shortly after Plato's time his meaning apparently did not cause puzzlement as Aristotle's casual remark attests. Half a millennium later, however, it was an enigma for the Neoplatonists, who had a somewhat mystic penchant and wrote frequently about it, proposing geometrical and numerical interpretations. Next, for nearly a thousand years, Plato's texts disappeared and it is only in the Renaissance that the enigma briefly resurfaced. During the 19th century, when classical scholars restored original texts, the problem reappeared. Schleiermacher interrupted his edition of Plato for a decade while attempting to make sense of the paragraph. Victor Cousin inserted a note that it has to be skipped in his French translation of Plato's works. In the early 20th century, scholarly findings suggested a Babylonian origin for the topic.

Most interpreters argue that the value of Plato's number is 216 because it is the cube of 6, i.e. 6³ = 216, which is remarkable for being also a sum of the cubes for the Pythagorean triple (3, 4, 5): 3³ + 4³ + 5³ = 6³.

Such considerations tend to ignore the second part of the text where some other numbers and their relations are described. The opinions tend to converge about their values being 480,000 and 270,000 but there is little agreement about the details. It has been noted that 6⁴ yields 1296 and 48 × 27 = 36 × 36 = 1296. Instead of multiplication some interpretations consider the sum of these factors: 48 + 27 = 75.

Other values that have been proposed include:

• 17,500 = 100 × 100 + 4800 + 2700, by Otto Weber (1862).
• 760,000 = 750,000 + 10,000 = 19 × 4 × 10000, 19 being obtained from (4/3 + 5) × 3 and being the number of years in the Metonic cycle.
• 8128 = 2⁶ × (2⁷ − 1), a perfect number proposed by Cardano. It is known that such numbers can be decomposed into the sum of consecutive odd cubes, so 8128 = 1³ + 3³ + 5³ + … + 15³.
• 1728 = 12³ = 8 × 12 × 18, by Marsilio Ficino (1496).
• 5040 = 144 × 35 = (3 + 4 + 5)² × (2³ + 3³), by Jacob Friedrich Fries (1823).