Magic proceeds from the five Platonic solids
TL;DR: Platonic solids fits all your criteria because
- There are five of them
- They have enough fancy geometrical properties to produce appealing transmutation circles
- They come in pairs just like mages and their familiars
With a bit of extrapolation about how Orgone flow works, this leads to attribute the binding to a circuitry that connects edges of a Platonic solid to the faces of its dual, an actual geometrical property of those solids.
To cast a spell, one must establish a flow of Orgone that fits a regular geometrical patter meeting the following criteria, sorted from most to least fundamental:
- Orgone conservation law implies that the flow must be a closed circuit
- Inertial principle: if not submitted to external influences, Orgone flows along straight paths, except at transmutation nodes where a mage can deviate their course => the flow must be a polygon
Laminar flow principle: if provided with the opportunity, Orgone will flow laminarly. That means that if two streams of Orgone cross each other, the flows will not intermix, and therefore the circuit will split itself into two disconnected circuits => no crossing allowed
Figure 1: Laminar flow crossing
Resonance is achieved when a high level of symmetry is respected by the nodes of the Orgone circuit. These symmetries are given by the $T$, $O$ and $I$ symmetry groups.
As it happens, 5 sets of node coordinates fit the last requirement in three dimensions. Connecting those nodes forms what is known as the Platonic solids. Each type of spell corresponds to one Platonic solid, which explains why there are five of them. They have been related to elements by Plato, also have relations to planets of the solar system due to Kepler in case either are components of your magic system.
- Tetrahedron - 3 triangular faces - $T$ symmetry group - associated to fire by Plato <-> Transmutation
- Octahedron - 4 triangular faces - $O$ symmetry group - associated to air by Plato <-> Scrying
- Hexahedron (cube) - 6 square faces - $O$ group - associated to earth <-> Protection
- Isocahedron - 20 triangular faces - $I$ group - associated to water <-> Transmogrifcation
- Dodecahedron - 12 pentagonal faces - $I$ group - associated to celestial bodies (or arguably to ether via Aristotle, see wikipedia) <-> Enchantment
The three other laws imply that the Orgone circuit to cast a spell must match one Hamiltonian cycle of the corresponding Platonic solid. In short, this is a closed path that visits each node exactly once, while travelling only along edges.
It so happens that Platonic solids all have an associated Hamiltonian cycle. In fact there are many such paths, which sub-categorizes spells.
Figure 2: Transmutation circles for some sample spells and relationship to full three dimensional form of the spell
Now, to perform the spell correctly, these paths should be drawn in three dimensions, but without any medium to draw the path out of thin air, this is rather impractical. Instead, what mages do is to draw two dimensional projections of Platonic solid on the ground, also known as transmutation circles. They then activate some edges of the graph along an Hamiltonian cycle, which triggers the flow of Orgone and casts the spell.
This form is weak, though versatile. A way to become stronger is to create a familiar.
A mage can create a familiar, creatures made purely from orgone, to assist in their rituals. [...] They allow a mage the ability to bypass ritual circles and their ingredients to perform spells, reducing time frame from hours to literally minutes.
A familiar, though it may have an outer shell corresponding to the personality of the mage, also features a geometrical core, which corresponds to one of the 5 Platonic solids. It embodies the Orgone circuit, which means the transmutation circle becomes unnecessary. It also means that the actual three dimensional resonant structure is achieved, hence the stronger performing spells. On the other hand, only one category of spell is possible, though multiple Hamiltonian cycles remain available.
They are made from a mage's soul and serve as a direct reflection of their inner self.
As it happens, each Platonic solid has a dual, which share the same symmetry group. Hence the octahedron is paired with the hexahedron (cube), the dodecahedron with the isocahedron, while the tetrahedron is paired with itself.
One obtains a dual solid by interchanging vertices and faces of the original solid.
To create a familiar, one must first carve its own soul into the shape of the dual type desired. Then, the familiar is obtained by connecting the adjacent faces of one's own soul with Orgone, which produces the geometrical core of the familiar.
The last step consists in letting the familiar exit the mage's body, which requires to animate it: that's why the mage has to embed the geometrical core into a creature with its own will. The specific way this is done is left at the courtesy of the mage, which explains why every mage has their own familiar.
Figure 3: Fully formed familiar (artist rendering)
The price to pay is that the Orgone flow out of the mage's soul are now bound to the nodes of the geometrical core of the familiar, therefore the mage can not produce Orgone circuit via a transmutation circle as before. This means that only one type of circuit can be achieved, and hence one type of spell.