Graph (in this module)

A finite, undirected, simple graph G = (V, E), where V is a set of vertices (sefirot) and E is a set of edges (paths between sefirot).

A function ℓ_V that assigns a label (e.g., a sefirah name) to each vertex in V.

A function ℓ_E that assigns a label (e.g., a Hebrew letter or Tarot trump) to each edge in E.

A bijection between vertex sets that preserves adjacency: two graphs have the same connection pattern up to renaming of vertices.

Tree of Life as a graph

A representation where each sefirah is a vertex, each path is an edge, and traditional names or symbols are captured as labels on vertices and edges.

A mathematical structure G = (V, E) with a set of vertices V and a set of edges E, often taken here as finite, undirected, and simple.

The traditional ten (sometimes eleven) emanations or aspects in Kabbalistic cosmology; here, they serve as vertices in the Tree of Life graph.

An element of the set E in a graph, usually an unordered pair {u, v} of vertices; in this module, each edge represents a line connecting two sefirot.

What is Tree of Life?

A structured arrangement of the sefirot and the paths between them, widely used in Kabbalistic and Western esoteric traditions; treated here as a finite labeled graph.

From Mystical Diagram to Mathematical Object — Transcendent Architectures: Formal Models of Hebrew Letters, Sefirot, 72 Names & the 231 Gates

Q: Two authors draw Trees of Life with the same 10 sefirot and the same 22 connections, but they assign different Hebrew letters to some of the paths. How should you describe the relationship between their Trees in graph-theoretic terms?

They are isomorphic graphs with different edge labelings.. The underlying connection pattern (vertices and edges) is the same, so the graphs are isomorphic. Different Hebrew letters on the same edges correspond to different edge labeling functions on the same isomorphism class of graphs.

1. Orienting Ourselves: What Are We Doing?

Goal of the Module

You will learn to treat the Kabbalistic Tree of Life as a mathematical object: a finite labeled graph with clearly defined nodes and edges.

What You Will Do

You will rephrase the familiar diagram as a graph, list its node and edge sets, and compare different historical layouts as distinct graphs or labelings.

Scope and Attitude

We focus on classical Western esoteric Tree of Life diagrams. We are not evaluating mystical claims, only translating diagrams into rigorous mathematical language.

2. Very Quick Graph Theory Refresher

Graph Basics

A graph is `G = (V, E)`: a set of vertices `V` and a set of edges `E`. Each edge is a 2-element subset `{u, v}` of `V`.

Type of Graphs We Use

We use finite, undirected, simple graphs: finitely many nodes, no edge directions, no loops, no multiple edges between the same pair.

Labels on Graphs

We can attach labels via functions, like `ℓV` for vertex labels (e.g., sefirot names) and `ℓE` for edge labels (e.g., Hebrew letters).

3. From Sefirot to Vertices: Defining the Node Set

Sefirot as Nodes

We treat each sefirah as a vertex. The usual ten: Keter, Chokhmah, Binah, Chesed, Gevurah, Tiferet, Netzach, Hod, Yesod, Malkhut.

A Concrete Vertex Set

Define `V = {K, Ch, B, He, Gv, T, N, Ho, Y, M}` using short codes as vertex names in the graph.

Vertex Labels

Use `ℓV` to map each code to its full name: `ℓV(K) = "Keter"`, `ℓ_V(M) = "Malkhut"`. Adding Daʿat just means adding another vertex and label.

4. From Paths to Edges: One Concrete Edge Set

Paths Become Edges

In a Tree of Life diagram, each drawn path between two sefirot is modeled as an edge `{u, v}` in the graph.

22-Path Hermetic Tree

The Golden Dawn style Tree uses 10 sefirot and 22 paths. Each of those 22 connections becomes a distinct edge in the set `E`.

Making It Exact

To be rigorous, pick a specific historical diagram, inspect it, and write down every connected pair of sefirot as an unordered pair in `E`.

5. Activity: Write Your Own Node and Edge Sets

Now you will practice turning a Tree diagram into a graph.

Imagine you are looking at a standard 10-sefirah Tree of Life in a textbook. It shows:

The ten sefirot arranged in three vertical columns.
22 lines connecting them, with no crossings except at the sefirot.

Task A: Node set

Write down a vertex set `V` using any short codes you like (for example, `V = {K, Ch, B, He, Gv, T, N, Ho, Y, M}`).
Next to each code, write the full sefirah name. You have just defined a vertex labeling function `ℓ_V` in informal notation.

Task B: Edge set (conceptual)

Pick three sefirot that you know are directly connected in your diagram.
Write them as edges, for example: `{K, Ch}`, `{Ch, B}`, `{B, K}`.
Check: are you accidentally counting the same edge twice? (Remember, `{K, Ch}` is the same as `{Ch, K}` in an undirected graph.)

Reflection questions (answer in your notes):

How many edges do you think your diagram has? Is it exactly 22, or different?
If your source diagram includes Daʿat as a circle with lines, how would that change `V` and `E`?

You do not need to get every edge perfect right now. The goal is to get comfortable thinking in sets of vertices and edges instead of just looking at a picture.

6. Worked Example: A Minimal Tree as a Graph

A Tiny Tree

Keep only three sefirot: Keter (K), Tiferet (T), Malkhut (M), arranged vertically with lines K–T and T–M.

Graph Structure

Vertex set: `V = {K, T, M}`. Edge set: `E = { {K, T}, {T, M} }`. Each edge corresponds to a drawn line.

Adding Labels

Vertex labels give full names. Optional edge labels (like Hebrew letters) turn this into a labeled graph `G = (V, E, ℓV, ℓE)`.

7. Historical Variants as Different Graphs or Labelings

Graph Isomorphism

Two graphs are isomorphic if you can rename vertices in one to get the other while preserving which vertices are connected.

Layout vs Structure

Changing where circles are drawn but keeping the same connections means the same graph. Changing which circles are connected means a different graph.

Labeling Conventions

Even with the same graph, different authors may assign different letters or symbols to edges. That changes the labeling functions, not the underlying graph.

8. Quick Check: Structure or Labeling?

Answer the question to test your understanding of graphs vs labelings in Tree of Life diagrams.

Two authors draw Trees of Life with the same 10 sefirot and the same 22 connections, but they assign different Hebrew letters to some of the paths. How should you describe the relationship between their Trees in graph-theoretic terms?

They are non-isomorphic graphs, because the labels are different.
They are isomorphic graphs with different edge labelings.
They are the same graph only if the pictures look visually identical.
They are completely unrelated mathematical objects.

Show Answer

Answer: B) They are isomorphic graphs with different edge labelings.

The underlying connection pattern (vertices and edges) is the same, so the graphs are isomorphic. Different Hebrew letters on the same edges correspond to different edge labeling functions on the same isomorphism class of graphs.

9. Optional Coding: Representing the Tree in Python

If you know some Python, you can represent the Tree of Life explicitly as a graph. Below is a simple example using adjacency lists.

The code does not rely on any external library, so you can run it in a basic Python 3 environment.

10. Flashcard Review: Key Terms

Use these flashcards to review the core concepts from this module.

Graph (in this module): A finite, undirected, simple graph G = (V, E), where V is a set of vertices (sefirot) and E is a set of edges (paths between sefirot).
Vertex labeling: A function ℓ_V that assigns a label (e.g., a sefirah name) to each vertex in V.
Edge labeling: A function ℓ_E that assigns a label (e.g., a Hebrew letter or Tarot trump) to each edge in E.
Graph isomorphism: A bijection between vertex sets that preserves adjacency: two graphs have the same connection pattern up to renaming of vertices.
Tree of Life as a graph: A representation where each sefirah is a vertex, each path is an edge, and traditional names or symbols are captured as labels on vertices and edges.
Historical Tree variants: Different diagrams across time that may share the same underlying graph (same V and E) or differ in edges and labelings, leading to distinct graph isomorphism classes.

Key Terms

Graph: A mathematical structure G = (V, E) with a set of vertices V and a set of edges E, often taken here as finite, undirected, and simple.
Sefirot: The traditional ten (sometimes eleven) emanations or aspects in Kabbalistic cosmology; here, they serve as vertices in the Tree of Life graph.
Edge (Path): An element of the set E in a graph, usually an unordered pair {u, v} of vertices; in this module, each edge represents a line connecting two sefirot.
Tree of Life: A structured arrangement of the sefirot and the paths between them, widely used in Kabbalistic and Western esoteric traditions; treated here as a finite labeled graph.
Labeled graph: A graph equipped with functions that assign labels (such as names or symbols) to vertices and/or edges.
Vertex (Node): An element of the set V in a graph; in this module, each vertex corresponds to a sefirah.
Adjacency list: A way of representing a graph by listing, for each vertex, the set of vertices to which it is directly connected.
Graph isomorphism: A one-to-one correspondence between the vertices of two graphs that preserves which vertices are connected by edges.