The Octahedron

Some relationships between the twelve orders can be seen in the geometric form of the octahedron.

Three axes pass through every point in space, giving height, width and depth.

These axes can be limited to give six directions: up, down, front, back, left, right
Twelve lines join the six points, creating the octahedron.

The octahedron consists of three internal planes, each divided into four segments by the intersection of the axes. Each of these segments can be assigned to one of the twelve orders:

Priests & Priestesses
Front - Up

Law-givers
Front - Down
Prophets
Up - Back

Magicians
Back - Down
Traders
Up - Left

Life-tenders
Left - Down
Perpetuators
Right - Up

Messengers
Down - Right
Warriors
Front - Left

Artists
Left - Back
Craft-workers
Right - Front

Explorers
Back - Right
The octahedron has eight triangular faces, each representing three of the orders working together. For example: priests, perpetuators and craft-workers together build cathedrals.
Each of the six vertices of the octahedron is a centre of a cross where four orders meet. For example, the upmost vertex is at the centre of :

Priests & Priestesses
Traders Perpetuators
Prophets