What are the transmutation circles in Fullmetal Alchemist based on, if anything?

The seals in fma are based on the King Solomon seals of Kabbalah as we see the Sephirotic tree of life in the show. The Soloman seals are said to have magical properties such bringing wealth or repelling demons (there are like 50 different seals). These seals can only be activated by people who know how to use them (at least in the show). My theory is the people of Amestres mix alchemist symbols with magic (prayer) to produce energy using holy power.

Assuming the fma world acts mostly as ours we know something cannot be made from nothing i.e the conservation of mass. The way to go against this worldly law is by using a stone. The philosophers stone grants godlike ability but at a cost of a soul, bringing up the whole idea of how much is a soul really worth?

Father and the seven deadly sins are powered by souls like batteries but humans have only 1 such battery. This is where the series gets paranormal. It is through the literal power of God that alchemist can manipulate objects but the objects can never change their chemical properties or add mass to them "for creating something from nothing is God's work" and impossible. The 2nd laws of thermodynamics basically states that energy needs to be consumed for anything to happen, it could never be an endless stream of energy, it needs a source. There is an irreversibility towards natural processes. No human could go against God and cheat death.

We can further deduce this by looking at the people who opened the portal of truth. Once opened you are granted the ability to do alchemy with no seal and by simply clapping your hands together which kind of looks like praying (Ed even mentions it in the last episode). They cant ignore the conservation of mass but they can manipulate it without having to use any seals. Even this needs energy to do. They claim in the show it is the tectonic energy that is harvested through the earth which produces the energy needed. But they also claim one is all and all is one. This is a prerequisite to alchemy training. In theory if you are one with all, you are one with God because God is everything (we see this is most religions especially Hinduism) you are using the power of the world to force energy into mass.

The whole Dragon or Snake eating itself is a great analogy for the search for immorality, the holy grail, the power of God, the fountain of youth, etc. The whole FMA series is based off the idea that nothing last forever and death can be a beautiful thing. it is something we have to fight and accept at the same time.

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Based on information I read at another source, some of the circles utilize the elemental symbols. Take colonel mustangs, for example. Two triangles intersect, one representing air and one representing earth, two things necessary for a successful fire. Within these triangles is the triangle representing fire. The bigger triangles push past the boundaries of the circle that’s attempting to contain it, symbolizing fires expansive nature. To accompany this are images of a fire and a salamander. There’s a reason for the salamander but I didn’t understand that as well as the other things I mentioned.

I always thought it looked like the Seal of the Seven Archangels which is a protection symbol. Seal Image

Yes, those are the symbols for the four base elements. But there are also two more very important parts to the circle. The center point which represents origin, time and growth, as well as an outer circle that represents the womb, the ether and protection. There are many more aspects and details to Solomon's circle than that. In fact, for such a simple symbol it is quite complex.

This site has a note about "Squaring the circle", describing a utilization of circles:

Attributed to Michael Maier, a German alchemist from the 17th century. That gives you the generalized "simple" transmutation circle:

The symbols used in some of the more complex circles are old alchemic symbols or variations of them. I don't know of a complete list of them but some of them can be seen in Annibal Barlet's works or other symbols from the 17th century. The inscriptions themselves seem to mostly come from George Ripley's 12 Gates.

Circles are a relatively major thing in alchemy. Ouroboros symbols were sigils in alchemy dating back to Cleopatra, and they depict a serpent or dragon eating its own tail.

The symbols in the circle are mostly alchemic symbols. The one on the the bottom left on the outside looks like the symbol for iron ore or the symbol for male. Going clockwise from that is what looks like either copper ore or female, a symbol I can't identify, one of the symbols for regulus, another symbol I can't identify, and sal-ammoniac. (Source)

Some Renaissance alchemic images have circles with words in them.

Squaring the circle was another major part of Renaissance alchemic ideas. It was believed that this symbol plus a man and a woman in the circle was all that was needed to create a philosopher's stone. A full explanation can be seen here.

The Seal of Solomon is another similar image.

According to here:

In alchemy, the combination of the fire and water symbols (up and down triangles) is known as the Seal of Solomon. The symbol is representative of the combination of opposites and transmutation. By combining the alchemical symbols for fire (upwards triangle) and water (downwards triangle), the alchemical symbols for earth and air are also created. The downwards facing triangle is divided along the center by the base line of the opposite triangle. This is the alchemical symbol for earth. Conversely, the upwards triangle divided by the base line of the downwards triangle is the alchemical symbol for air. The Seal of Solomon is all that is unified in perfect balance; the Spirit Wheel.

It's seems to be pretty much a compilation of different alchemic elements.