As a follow up, but another subject, of my question about equivalent exchange part of the alchemy laws.
Why do (most) alchemists require a transmutation circle? Does any circle suffice or does an alchemist require a specific type for each (type of) job? At least the size seems to matter...
The transmutation circles contain the formulae and/or specific incantation needed to carry out a function. There is always some part of the human mind that is involved in a transmutation (otherwise, Mustang's gloves would only be able to mess with the air in a very select pattern), but in general more complicated transmutations will require a more robust "hard formula". The "soft formula" comes from the mind of the alchemist. This is the part that actually shapes what should happen, while the "hard formula" is used to gather the required energy/ handle the heavy lifting.
On the subject of people who can transmute without circles, there are two different schools of thought.
The first follows the anime/manga rigidly, and interprets Izumi's words literally. They see it as the 'reward' for losing part of yourself when you looked beyond the gate. Looking beyond the gate exacts a toll from you, which is then used to "pay" for the privilege of seeing the base code of the universe. Knowing this base code, alchemists are able to use the "soft" formula to a greater extent, and do not require the assistance of a "hard" formula in order to transmute.
The second possible explaination is more of an unintended side effect. Remember how Barry the Chopper lost his body? Well, his soul (in the armor) was able to sense his body's presence (and vice-versa). If one assumes the figure at the gate (which takes peoples' body parts) is god (or at least, the source of alchemy), then him "wearing" an alchemists parts (pardon the term) is essentially giving that alchemist a very subtle, yet powerful connection on a mental level. The alchemists are able to use their missing parts to bridge the gap separating them from this god-like being, and thus transmute seemingly at will.
Alphonse, not having a physical body at all with which to form a bridge, is unable to skip the "hard formula" at first, but upon re-visiting the experience in his own mind, he forces himself to build this "bridge" and can transmute naturally from then on.
Those circles are, in fact, like "spells", or more accurately "mathematical formulas", so in order to make what they want, they have to write the correct formula.
Those alchemists who don't do that, just a few, can bypass that formulas because they have them "inside", but that's very rare, and as far as I know, wasn't explained in the series.
According to the Fullmetal Alchemist Wiki, transmutation circles have a symbolic and functional aspect. The symbolic meaning is in the cycle of energy and life, where Equivalent Exchange ensures that objects are never created nor destroyed, but changed into different forms. My interpretation is that this embodiment of a principle of Alchemy helps the transmutation take place, but it is unnecessary if the alchemist has been to The Gate, thus embodying the principle of Alchemy by himself.
As for function, the transmutation circle harnesses the energy from the earth or substrate that the circle is drawn on (compare the energy sources of Amestris [SPOILER] and Xing). the geometric symbols (triangles, squares and other polygons) represent various elements (water/earth/fire/air), and other symbol-like runes which direct and manipulate the energies in more specific ways. So in the functional aspect, the diagrams and symbols in the transmutation circle serve as an "energy manipulation chamber" to harness for specific alchemical uses.
I think the idea from Merlin (with Sam Neill) helps here: there there are different types of Magic (== Alchemy == Sufficiently-Advanced Technology): hand-magic, word-magic, and thought-magic. The gestures or words are used to induce the necessary thought-form which actually performs the action of the magic. Note that the ability to read silently is a relatively recent acquisition. Historically, St. Augustine is the first human to have done this.
Here's a description of the concept of logos in talismanic magic in Magical Egypt.
