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In the foreground, Pythagoras sits with a book and an inkwell, also surrounded by students. Though Pythagoras is well known for his mathematical and scientific discoveries, he also firmly believed in metempsychosis. This philosophy states that every soul is immortal, and upon death, moves to a new physical body. In this light, it makes sense that he would be placed on Plato’s side of the fresco.
But just what does this famous painting mean? Let’s look at what the iconic The School of Athens meant for Raphael as an artist and how it’s become such a symbol of the Renaissance. At the time, a commission by the pope was the apex of any artist’s career. For Raphael, it was validation of an already burgeoning career.
In 1508, during the High Renaissance (c.1490-1530), the 25-year old painter Raffaello Sanzio, better known as Raphael, was summoned to the Vatican by the ageing pontiff Pope Julius II (1503-13), and given the largest, most important commission of his life – the decoration of the Papal Apartments, including the Stanza della Segnatura. Located on the upper floor of the Vatican palace, this room was used by the Pope as a library. It was here, between 1509 and 1511, that Raphael painted his famous fresco The School of Athens (Scuola di Atene). It was the second mural painting to be finished for the Stanza della Segnatura, after La Disputa, on the opposite wall, and is regarded as one of the greatest Renaissance paintings. The general theme of the picture, indeed the whole room, is the synthesis of worldly (Greek) and spiritual (Christian) thinking, and ranks alongside the finest examples of classically inspired Renaissance art. A rival of the older Michelangelo (1475-1564), Raphael went on to complete three other Papal apartment rooms in the Vatican (known as the Raphael Rooms) and was to remain in Rome serving successive popes until his sudden death in 1520.
School of Athens: Composition
21 Sodoma (1477 Vercelli in Piedmont вЂ“ 1549 Siena)
Study on the ” School of Athens ”
The School of Athens is one of a group of four main frescoes on the walls of the Stanza (those on either side centrally interrupted by windows) that depict distinct branches of knowledge. Each theme is identified above by a separate tondo containing a majestic female figure seated in the clouds, with putti bearing the phrases: “Seek Knowledge of Causes,” “Divine Inspiration,” “Knowledge of Things Divine” (Disputa), “To Each What Is Due.” Accordingly, the figures on the walls below exemplify Philosophy, Poetry (including Music), Theology, and Law.  The traditional title is not Raphael’s. The subject of the “School” is actually “Philosophy,” or at least ancient Greek philosophy, and its overhead tondo-label, “Causarum Cognitio”, tells us what kind, as it appears to echo Aristotle’s emphasis on wisdom as knowing why, hence knowing the causes, in Metaphysics Book I and Physics Book II. Indeed, Plato and Aristotle appear to be the central figures in the scene. However, all the philosophers depicted sought knowledge of first causes. Many lived before Plato and Aristotle, and hardly a third were Athenians. The architecture contains Roman elements, but the general semi-circular setting having Plato and Aristotle at its centre might be alluding to Pythagoras’ circumpunct.
An interpretation of the fresco relating to hidden symmetries of the figures and the star constructed by Bramante was given by Guerino Mazzola and collaborators.  The main basis are two mirrored triangles on the drawing from Bramante (Euclid), which correspond to the feet positions of certain figures. 
The central figures in School of Athens are Plato and Aristotle. Depicted at the central vanishing point of the shown architecture, Plato holds a bound copy of Timaeus in his left hand and is shown as an older, wise, gray-haired man. In contrast, his student Aristotle is shown as a younger, handsome man looking to his teacher. Aristotle carries a bound copy of Nicomachean Ethics in his left hand and walks slightly ahead of Plato. The two central figures both gesture with their right hands but along different visual planes. Plato gestures upwards into the vault while Aristotle gestures horizontally ahead of the figures. Plato and Aristotle are deep in conversation while walking through the other figures.
The fresco itself includes 21 distinct figures set against a backdrop of a school. The figures are engaged in conversation, work or games. All of the figures are male and are believed to represent all significant Greek philosophers. The fresco also includes images of statues within the school displayed within the school. One statue is Apollo, the Greek god of light, archery and music, holding a lyre. The other statue is Athena, the Greek goddess of wisdom, shown in her Roman form as Minerva. The building itself is shown in a cross-shape with the figures in the foreground and the interior receding behind them. The figures are scattered across steps and walkways within the school and the fresco is framed with an arch decorated with arabesque swastikas. The fresco measures 200 inches by 300 inches with a tondo above depicting a female figure with a putti stating “Seek Knowledge of Causes.”