Fortean Winds: Anolmalous Research

This is where the analysis gets specific enough to be falsifiable. Part 3 established that the grimoires describe operations, not just symbols — functional claims about energy, intention, and sound. This part follows the geometry specifically, because the mathematical structures embedded in ritual diagrams are the most concrete thing in the entire corpus. Either the proportional relationships are real and measurable or they aren’t. Turns out they are.

This is Part 4 of our AI-powered Grimoire analysis. You can read the other parts here: Part 1   Part 2   Part 3

A Mathematical Lens on Grimoire Geometry

1. Symmetry and Sacred Geometry

The diagrams in grimoires often feature radial symmetry, polygons, and spirals, echoing the principles of sacred geometry—a study of shapes imbued with spiritual significance.

• Magic Circles:
  • Found in the Key of Solomon, magic circles often include concentric rings, pentagrams, and alphanumeric inscriptions. These circles are not random; they reflect precise geometric planning.
  • Mathematical Analysis:
    • Circles are divided into equal sectors, corresponding to the number of elements, planets, or spirits involved in the ritual.
    • For example, a circle divided into 12 equal parts aligns with zodiac signs, while 7 divisions relate to the classical planets.
• Pentagrams and Hexagrams:
  • These symbols, prevalent in the Key of Solomon and other grimoires, are constructed using geometric principles:
    • Pentagrams: Based on the golden ratio (ϕ≈1.618), these five-pointed stars demonstrate a self-replicating symmetry that has long been associated with harmony and balance.
    • Hexagrams: Comprised of two overlapping equilateral triangles, hexagrams are perfect examples of rotational symmetry and duality.

Practical Example: Constructing a Pentagram

1. Start with a circle.
2. Divide the circle into five equal sections (360°/5 = 72°).
3. Connect the points to form a star.
4. Observe the proportional relationships between the segments—each intersection divides lines into segments proportional to the golden ratio.

2. Alphanumeric Integration

Many grimoire symbols encode letters and numbers within their geometric designs, embedding additional layers of meaning.

• Examples:
  • Sigils: In the Goetia, each spirit is represented by a sigil. These designs often integrate letters from the spirit’s name into looping or angular patterns, creating unique yet harmonious shapes.
  • Numerological Links: Letters in Hebrew or Latin are often assigned numerical values. For instance, in the Key of Solomon, divine names inscribed in talismans may sum to numerologically significant numbers like 3, 7, or 72.
• Mathematical Analysis:
  • These integrations can be analyzed using graph theory. By treating each point or vertex in a sigil as a node, we can map connections and identify patterns, such as symmetry or repetition.

3. Geometric Constructions in Rituals

Ritual diagrams in grimoires often serve functional purposes, guiding practitioners in creating protective spaces or focusing energy. These designs are not only symbolic but also follow rigorous geometric rules.

• Case Study: The Magic Circle
  • In the Key of Solomon, a standard magic circle includes:
    • Concentric Circles: Representing layers of protection.
    • Pentagrams and Hexagrams: Placed at cardinal points for elemental balance.
    • Inscriptions: Divine names or invocations inscribed along the edges.
  • Analysis:
    • Using polar coordinates, we can mathematically model the placement of symbols and inscriptions within the circle.
    • If a circle’s radius is r=10, dividing it into 12 sections (30° each) positions inscriptions or symbols at angles of 30°, 60°, 90°, and so on.
• Protective Sigils and Energy Flow:
  • Some diagrams incorporate intersecting lines or shapes to channel energy. These designs can be analyzed using:
    • Vector Fields: To model hypothetical energy flow.
    • Symmetry Operations: Using group theory to understand their balanced structures.

4. Comparing to Modern Science

The geometric principles found in grimoires are not far removed from those used in modern scientific fields, particularly physics and engineering.

• Sacred Geometry and Resonance:
  • Shapes like pentagrams and hexagrams are known to resonate with certain frequencies, aligning with concepts in cymatics (the study of visible sound vibrations).
  • Example: A pentagram’s internal angles (108° and 72°) appear in resonance studies of sound waves.
• Circuit Design Analogies:
  • Grimoire diagrams resemble modern circuit diagrams:
    • Wands: Could be seen as capacitors, focusing and releasing energy.
    • Magic Circles: Operate like grounding mechanisms, isolating and protecting the practitioner.

5. Reproducible Analysis: Tools and Methods

• Data Sources:
  • Symbols extracted from grimoires like the Key of Solomon and the Goetia.
  • Measurements taken from high-resolution images of magic circles and talismans.
• Methods:
  • Geometric Construction: Using tools like a compass and protractor to recreate symbols.
  • Mathematical Modeling: Applying graph theory, symmetry operations, and polar coordinates.
  • Software: Use of geometry tools like GeoGebra or Python’s matplotlib for visual analysis.

Conclusion

The golden ratio in pentagram construction isn’t decorative. The alphanumeric integration in the Goetia sigils isn’t arbitrary. The concentric circle divisions aligning with planetary and zodiacal systems weren’t guesswork. These are people who understood specific mathematical relationships well enough to build them consistently into ritual diagrams across independent traditions spanning centuries.

What they thought those relationships were doing and what they were actually doing may not be the same thing. But dismissing the precision as coincidence requires more explanation than taking it seriously does. Newton didn’t think it was coincidence either. Part 5 draws out what the full analysis implies for modern research — and where the testable hypotheses are.