# How to Represent Platonic Solids in 2D Magic Circle Construction

My world's magic system employs numerous means to cast spells. These include chanting, activation of magic circles, rituals, etc. As the question suggests, my question today is on magic circles.

## Quick Introduction

### Magic Types

My world uses typed magic broken down into two categories.

• Elemental magic includes Fire, Water, Wind and Earth. So named because these four substances make up most of the world, and so together are dubbed "The Elements".

• Arcane magic includes Light and Dark. This magic manipulates both light and shadow, as well as various physical phenomena.

Note that elemental magic only deals with generating or manipulating the elements. For instance, fire magic isn't some abstract generalised form of magic which can absorb kinetic energy or similar. Each elemental magic does only what it says on the box.

Arcane magic on the other hand extends beyond just manipuilating light and shadow, and also deals with concepts such as teleportation, gravity, and other such intangible ideas.

There is no sleep / paralysis / memory manipulation / etc. magic.

### Magic Invocation

All means of casting spells involve 3 components. The only one of which pertinent to the question is the first of the three, seen below.

Definition: This is the drawing of the magic circle itself. The shape of the magic circle has a fixed / defined output, but two different magic circles can have similar effects. This component is subject to a strict set of rules, being logically consistent in relation of definitions (magic circles) to resultant spells.

### Relationship Between the Two

I would like each element to be represented by a shape, such that the magic circles for each type of magic are distinguishable from one another. Furthermore, rather than 2D shapes, I would like the true natures of the shapes representing each element to be one of the platonic solids. Ideally, the assignment will be similar to that of Plato, in which the tetrahedron is fire, hexahedron is earth, octahedron is air, and icosahedron is water. I don't mind arcane magic sharing the dodecahedron, but would ideally have a manner for distinguishing one from the other.

Now of course, one can't exactly draw a 3D shape on a 2D plane. As such, I was hoping to use some 2D analogue to represent each of the convex regular polyhedra. (Triangle is tetrahedron, square is hexahedron, etc.)

Note: No magic circle involves text. Solely lines and shapes and always with a circle on the outside. Written text is a symbolic human construct and thus arbitrary. Magic circles only use geometry as a result.

## The Problem

The plan was to represent the tetrahedron, hexahedron, dodecahedron, and icosahedron as a triangle, square, pentagon and hexagon respectively. However, I do not know how I can represent the octahedron. If I take its appearance from a vertex, it looks like a diamond, but there's really no difference between a diamond and square. If I take its appearance from its face, it's a hexagon, but this is the same as the icosahedron.

So my question is, how can my magic circles somehow illustrate the difference between the octahedron and other shapes in 2D, and how can I have both light and dark be represented by a pentagon, but be clearly distinct?

• Does it have to be 2D? Either carvings could be used, not unlike a bag of dice, or something like ZomeTool / Tinkertoys (rod + joint construction toys) could be used to construct arbitrary wireframe polyhedra on demand. Nov 17, 2020 at 1:58
• As I was primarily interested in geometry, I didn't think it important to mention; but a specific rate of flow of mana must be invested as each part of the diagram is drawn (with some kind of catalyst). For this reason, a prefab mesh wouldn't work. 3 Dimensional magic circles (magic spheres?) are also present in my world, but are tremendously hard to employ for this reason (one must "draw" in the entire volume, not just the surface. Such a magic tool could be a master sorcerer's magnum opus). Magic circles are therefore traditionally 2D in my world because it's easier to work with. Nov 17, 2020 at 3:12

I might suggest looking into the stereographic projection.

The stereographic projection is a way of mapping a sphere onto a plane, which has the interesting property that any circles on the sphere get mapped to circles on the plane. It preserves angles and (to some extent) shape, although not size. The details aren't particularly relevant here, so I'll leave that Wikipedia link there for anyone who's interested.

It's also possible to use stereographic projection to map polyhedra onto a plane, by first inflating the polyhedra like balloons so that they become spheres. When this is done with the platonic solids, these are the results:

Tetrahedron:

Cube:

Octahedron:

Dodecahedron:

Icosahedron:

(source for all five images)

I don't know if this design will fit with the sort of aesthetic you're going for, but if it does, well, there you go.

### Project a Hamilton Cycle into 2D?

If this is a path in which a wand / sacred pen / blessed knife / etc must follow while casting a spell, perhaps a Hamilton cycle is what you need? That's a path passing through every vertex once and only once, projected into 2D.

So this would be the path of your arcane spell going through the dodecahedron:

It's literally tracing the 3D shape but projected into 2D, with a circle then added. Eg heres the original 3D path:

As a bonus for story writing there's multiple possible paths, so you learn it like a signature. If done with a wand, each mage will have a unique gesture for casting it. It's also slightly obfuscated, so if printed out, the uninitiated may not recognise its significance.

### Or if too complex - trace any (face count / 2) vertices:

• Embarrassingly, while I had considered wireframe projections, I hadn't considered Hamiltonian Circuits. Certainly, as a solution that employs the features of the polyhedra themselves, it is an elegant solution. If I'm understanding correctly, the idea proposed doesn't quite fit all of what I had in mind for magic circles as the definition step extends beyond just a single shape / type specification to fairly complex and precise diagrams, but the idea of changing the order in which one travels the vertices is rather neat and I may introduce it into the system as a whole. Nov 17, 2020 at 2:33

Use a planar projection (Schlegel diagram) of the five platonic solids. As you can see, all but the tetrahedron have a nice planar projection with a central face and drawings around it, so you can be inside the circle or outside the circle, cross the circle, break the circle, etc.

The more complex ones have an inside face, an outside face, and bordering faces where separate, powerful wards can be placed. Mages sit in in the center triange of the octahedron, there are three inner and three outer border triangles to burn incense or whatever, and observers sit in the outermost, infinite face.

(If you do the count, the area outside the outermost boundaries is one face of the 3d object, too).

Either the tetrahedron is not really suitable, or it is used for invocations with two or three subjects -- to summon a demon, stand in one face and summon it in another, and there is a magical barrier between you.

Use the square

• the shadow of a tetrahedron is a triangle.

• The shadow of a hexahedron is a diamond, which is not a square, it's a rhombus.

• The shadow of a dodecahedron is a hexagon.

• The shadow of an isosahedron is a pentagon.

• The shadow of an octahedron is a square.

You mentioned that you were worried that a diamond is not distinct enough from a square. I beg to differ. A rhombus (a diamond laid over on its side) has a short and a long axis between its verticies. A square tipped up on a vertex has two identical axes between the vertices.

You may not think it's ideal, but people can easily recognize the difference between the two. (Images care of Geometryhelp.net.)

• In case it was unclear, what I was taking as the 2D analogues of the polyhedra were the polygons formed by the outlines of their face orthogonal projections (dodecahedron excluded for simplicity). The diamond to which I was referring was the octahedron's vertex projection, which is similar to the hexahedron's face projection. The octahedron and icosahedron's face projections are likewise similar. I'd prefer regular polygons, but I think your solution is best otherwise. Nov 17, 2020 at 2:58
• It was clear, Ragnadam. The vertex projection of an octahedron is a rhombus. Although if you want to have some fun with it, consider an hourglass shape (setting the octahedron down on one edge and excluding the two vertical edges).
– JBH
Nov 17, 2020 at 3:11
• A rhombus is an octahedron's edge projection. That being said, if I can't think of a solution that (somehow) manages to only use regular polygons, then I may very well employ your idea and simply concede to using a rhombus alongside the rest of my shapes. Thank you for a lovely suggestion. Nov 17, 2020 at 3:30