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According to Niall Ferguson in The Ascent of Money, they aren't balls, but coins. (I listened to the audiobook so I can't provide a page citation.) I'm somewhat suspicious because the blazon for the arms is
"augmented coat of arms of the Medici, Or, five balls in orle gules, in chief a larger one of the arms of France (viz. Azure, three fleurs-de-lis or" Source. That source seems to have problems, so check wikipedia sidebar
Note that the first five or six google references refer to them as "balls"; although I'm not familiar with it, "balls" is apparently a legitimate synonym for "roundel", and perpetrate the notion that it is a mystery. None of these are particularly scholarly. The Rise and Fall of the Medici positively asserts that there is no relationship between the Medici arms and the symbol for pawnbrokers. Another myth is that they are pills, as a reference to the name of the family "Medici = Doctors". While "canting arms" are common, this doesn't explain why they are red. Furthermore several sources suggest that
(the story that these were medicinal pills was apparently invented at the court of France by people wanting to defame Queen Caterina dei Medici in the 16th century) Florence Art Guide and several other sources (hat tip to @Yannis Rizos who first pointed this out).
If they were intended to represent coins, the more logical charge would have been the bezant, which is a gold Roundel Of course the field is or, so the charge must be a color; making them red makes them torteu or "cakes". However, I think that is a clue.
The answer which seems most likely is in a footnote; the source it cites is behind a paywall, but this looks quite plausible.
According to one theory, the armorial bearings of the Medici are canting arms or armes parlantes, and the torteaux or red balls supposedly represent pills, because medici in Italian means "physicians." The historian G. F. Young regards this whole story as a fable.οΏ½The Medici, chap, iii, n. 2. He is probably right. A more plausible explanation is that the Medici adopted the roundels because they were the symbol of the banker's trade and of the guild to which they belonged. The coat of arms of the Florentine money-changers* guild, Arte del Cambio, was a red shield sown with bezants or gold roundels. The Medici used red roundels instead of gold ones. The pawnbrokers eventually adopted the gold roundels or balls as the sign of their trade, since those symbols were associated in the public mind with money lending and credit.οΏ½Raymond de Roover, 'The Three Golden Balls of the Pawnbrokers," Bulletin of the "Business Historical Society, XX (1946), 117-24. See esp. illustrations on p. 123.
Although the name means doctor, the family Medici rose to prominence due to their banking, and they are very tightly associated with banking. I think it is very plausible that their coat of arms is a reference to the money changer's guild.
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The most romantic (and far-fetched) explanation of the origin of the palle is that the balls are actually dents in a shield, inflicted by the fearsome giant Mugello on one of Charlemagne's knights, Averardo (from whom, legend claims, the family were descended). The knight eventually vanquished the giant and, to mark his victory, Charlemagne permitted Averardo to use the image of the battered shield as his coat of arms.
Others say the balls had less exalted origins: that they were pawnbrokers' coins, or medicinal pills (or cupping glasses) that recalled the family's origins as doctors (medici) or apothecaries. Others say they are bezants, Byzantine coins, inspired by the arms of the Arte del Cambio (or the Guild of Moneychangers, the bankers' organization to which the Medici belonged).
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THe quincunx is an ancient symbol linked to both astrology and alchemy. It consists of 5 pellets, either in a cross-like disposition or like the ones in the Medici coat of arms, in a pile-like disposition. I am starting to research on this, because it also appears in Vlad Dragul's headpiece, and I am curious regarding its meaning. The quincunx is a Latin word, and it refers to the value of coins as far as the 3rd century before Christ. The coins have a five-pellet feature, but now on a row or in other 2 or three different placements.
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I think they just represent people. Maybe family members. That's why they had a different number of balls in different periods and the first in the family, Cosimo de Medici, has on his grave only one "ball" since he is the first of the dynasty.
The red color of the balls, as said here, means torteu or "cakes", but "torteu" also sounds like the Italian for turtle (tartaruga) and it is known that the motto of Medici was "Festina lente" which translates "make haste slowly." And we know the story of turtle and rabbit racing, where the slow but steady turtle wins in the end. Just idea, not any worse then the other "theories" :)
Another point is that we can see the top blue "ball" being different in color which I suppose is a representation of God (blue sky). It seems as it to symbolically show progress upwards towards perfection, starting at the bottom with one ball (Cosimo de Medici) and then moving up, branching out and up towards building heaven on Earth under the crown seen on top, which again is just another circle or ball but seen from the side. And interestingly another thing to point out is that in beginning there were 3 balls and even the 3 balls symbol is today used by pawnshops or banks.
Cosmino de Medici did have 3 sons: Piero di Cosimo de' Medici, Giovanni di Cosimo de' Medici, Carlo de' Medici. So maybe, as I said, the balls, or more specifically the 3 balls associated generally with them, are his 3 sons. And later they kept adding more balls and all the other fancy decoration around. I think people complicate too much. Think of what you would chose for yourself. Probably something important to you, and I am sure his 3 sons were important to him.
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When I was in Florence, the Italian tour guides told us they were oranges, a device of the Medici family. Just saying ;)
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Leo: only the Medici arms were originally just six torteaux 'in orle' (that is, the top torteau was originally just a red one), and the top one became charged with the arms of France (three gold Fleur de lys on a blue field-'Azure three Fleur-de-lys Or') because it was an 'augmentation' from the King of France.
It's wise not to ascribe too much meaning to charges in a coat of arms: they often don't 'mean' anything, beyond what they meant to the original grantee, or if they are 'canting' arms (a pun on the name).