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Magic and Mana
Magic is a force through which change can be exerted on the world around oneself. It is based on mana, which is produced in the Linker Core, an intangible organ nestled a little below the heart and between the lungs, near the solar plexus. To call it an organ is not quite accurate terminology - the Linker Core is the point at which mana generated by the whole body coalesces into a usable form, creating a nexus, detectable as a "dimple" above and below the body in Dimensional Space, from which it flows out back into the rest of the body. Damage to the Linker Core is dangerous, and can seriously affect the cycle of mana through the body and adversely impact the ability to use it in magical spells. Mana often appears visible as coloured light, the colour varying from person to person. While there is a tendency for colours to be similar in related individuals - three siblings might have red, orange and pink, for example - there is no particular advantage to any given colour. Like eye colours, some are more common than others, the most widespread being the various shades of blue.
Such is raw mana. On its own, mana can be channelled instinctively into the body to make it stronger, faster, more durable. Crude "feeling" and intent-based manipulation can also wrap fields of it around objects, allowing a sword to act as if it were far sharper and stronger than its material make-up would suggest. Some people also have affinities for certain types of manipulation - the most obvious and strongest of which are the so-called "elemental" affinities - talents for turning mana into electricity or heat, for example. Lesser "affinities" also exist and influence talents at different types of spell - one might be naturally good at manipulating mana to move and direct force in shooting spells, or to fashion shields and barriers.
However, most of this primitive instinctual mana-manipulation is limited in what it can do. For more advanced effects, some form of pattern creation is needed to describe the effects in a more ordered and precise way than simple instinct and visualisation. Mathematics is by far the most common, though it is sometimes expressed through mediums such as such as specific series of movements or mnemonics with a base pattern underlying them. Ritual-based diagram methods are also used, channelling raw mana through a complex and previously prepared geometric shape made of some mana-reactive material which is designed to produce a given effect when charged. There is even one recorded style that uses communal music and formalised song as a pattern-former, with the underlying mathematical patterns in the series of notes aiding the casters in shaping the effect. Generally, however, mathematics is used on its own, devoid of surface layers of geometry or music.
Magical styles, then, are differing ways of employing mathematics to describe the effect that the mana should produce. An analogy could be drawn with programming languages, with the spell-structure of each individual spell being a specific program - and indeed, spells are written and designed in a similar way, the elements of the structure being selected and fine-tuned to produce the desired effect as efficiently as possible. Despite not being based inherently on geometry, this unified method of describing spells and the shared basic structure which they are written in will generally produce a geometric casting sigil of light, which will vary in shape depending on the style - note the characteristic squares-within-a-circle shape that Mid-style produces and compare it to the triangular sigil of Belkan styles.
Different systems specialise in doing different things - Midchildan excels at shooting magic and shields in combat, making it excellent for a mage who keeps her distance and blasts her opponent while blocking anything that comes her way. Belkan styles, conversely, are far more effective at close range, making standard Belkan-user tactics much more reliant on close-quarter combat.