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The Delian Problem
One of Archytas' most notable accomplishments comes in the form of a mathematical solution to The Delian Problem, more informally know as doubling the cube. The problem is as follows: given a cube that a side is known, construct a cube with double the original volume. The proof of his model comes from Eudemus, who in the late 4th century wrote a history of geometry, including solutions to this problem from multiple mathematicians and philosophers before him- namely Eudoxus and Menaechmus. Although Eudemus' work did not survive to current day, a transmission of his geometric solution does survive in the form of Eutocius' commentary on Archimedes' De Sphaera et Cylindro. Archytas' solution begins with the concept of mean proportionality and the construction of four similar triangles. Each triangle's hypotenuse and long leg are proportionally similar as the triangle increase in size, which is essentially today's version of similarity of triangles. Archytas then applied the mean proportionals for a given length of a cube. If the volume of the original cube is written as V1 = x3, where x represents the length of a side, we let k1 and k2 represent the proportionality constants, and the cube is then doubled so that a side length is now 2x, a mean proportional between the two can be written as $\left ( \frac{x}{k_1} \right ) : \left ( \frac{k_1}{k_2} \right ) : \left ( \frac{k_2}{2x} \right )$. With the proportionals finished, Archytas completed the solution to his similar triangles as follows: If you cube the proportion of the original length of the side and solve using the mean proportional set, the solution comes to $\left ( \frac{x}{k_1} \right )^2 = \left ( \frac{x}{2x} \right )$ After using light algebra, $$2x^3 = k_1^3$$where the k1 variable represents the edge of the newly doubled cube.

Harmonic Theory
Archytas developed an impactful physical theory for pitch when a string instrument is strum. His theory was that the pitch of the ensuing sound depends on the speed of the wave moving through its medium (usually air). Although incorrect, as pitch is correlated with frequency rather than wave velocity, Archytas' theory was the first of its kind that was both adopted and adapted by Plato and later Aristotle. By the time of his analysis, it was known from the Pythagorean diatonic scale that whole numbers alone accounted for musical intervals on a scale. Archytas's work on musical scales included a thorough proof that no mean proportional numbers, like the ones used in his solving of the double cube problem, exist between basic music intervals (the difference in pitch between two sounds). This is to say that the basic interval, $$(2:1, 4:3, 3:2, 9:8)$$, does not include any mean proportional number, and cannot then be divided in half. The octave can be doubled without violating this rule, as multiplying a whole number by 2 will always result in a whole number, and can therefore be equated by two mean proportional ratios.

Cosmology
Due to the severe scarcity of resources for Archytas' direct work, it is difficult to pinpoint his exact thoughts on the universe. Through Eudemus and later Simplicius' commentary, however, his thought experiment in regards to the size of the universe remain intact to current day. The experiment is credited as being an influential spark though the early ages, even though Plato nor Aristotle bought the argument. In his experiment for others to participate and decide for themselves, Archytas tells of a scenario in which he is at the effective edge of the fixed stars. He says that if he outreaches his arm, or his stick (staff), that his hand will push the limit of what the edge is. He is then free to move into the newly created space and outstretch his staff once more, thus increasing the limit of space. With his argument, he attested that space, the region of the fixed stars, is infinite. This thought persisted even through modern day, although it is important to note that his model has a defined edge, whereas some current models do not accounted for a defined edge of space.