The Rodin–Taijitu Diagram: Where 3-6-9 Meets π, φ, and e
A single geometric figure has become a popular visual shorthand in discussions of vortex mathematics, sacred geometry, and certain modern interpretations of Nikola Tesla’s numerical interests. It combines Marko Rodin’s nine-point circle (the classic “vortex” diagram) with the traditional Taijitu, or yin-yang symbol. On this composite image the integers 3, 6 and 9 stand out clearly, while the three fundamental irrational constants π, φ (the golden-ratio conjugate) and e can also be located. The result is visually striking. Whether it constitutes a deeper “unified matrix” is a separate question
.The Rodin Layer: The Digits 3, 6 and 9Marko Rodin’s vortex-based mathematics begins with a circle divided into nine equal arcs. The points are labeled 1 through 9. Successive doubling, reduced to digital roots, produces the closed cycle
1 → 2 → 4 → 8 → 7 → 5 → 1. The numbers 3, 6 and 9 never appear in that cycle. Instead they form their own relationship:
3 × 2 = 6,
6 × 2 = 12 → 3,
while every multiple of 9 reduces to 9. In the diagram these three points are usually emphasized by a dashed triangle (or by the vertical axis of the Taijitu). They sit 120° apart and therefore form an equilateral triangle inscribed in the circle. This is the geometric embodiment of the “3-6-9” motif that Rodin, and many later writers, associate with a controlling or polar axis outside the main doubling flow.
The Circular Geometry: π Because the outer boundary is a circle, π is present by definition. - The full circumference is
2πr
. - Each of the nine equal arcs subtends an angle of
40∘=2π / 9 radians = 40 degree
- The 3-6-9 triangle subtends central angles of
120 ∘=2π / 3 radians = 120 degree
No additional construction is required; any regular nonagon carries π in its angular and arc-length measures.
The Complex Representation: e The nine perimeter points may be written in the complex plane by Euler’s formula:
zk=ei2πk/9, k=0,1,…,8
(with a suitable rotation so that the point labeled 9 sits at the top).
The real and imaginary parts are simply the cosine and sine of those angles. Thus e appears as the base of the exponential that places the nodes at equal angular intervals. This is standard complex analysis applied to the roots of unity; it is not unique to Rodin’s system, but it is a natural and exact way to coordinate the diagram.
The Taijitu Layer: φ
The yin-yang symbol is constructed from a large circle of radius (r) and two smaller circles of radius whose boundaries form the characteristic S-curve. When an auxiliary line is drawn from the leftmost point of the large circle to a carefully chosen point on the black/white interface (the edge of one of the “eyes” or the dividing curve), the length of that segment is exactly the golden-ratio conjugate:
ϕ⋅r=sqrt(5)/2*r≈0.618r.
(The complementary segment along a related line yields the full golden ratio Φ≈1.618. Phi
- the integers 3, 6 and 9 (Rodin’s polar triad),
- π (the circle),
- e (the complex exponential that locates the nodes),
- φ (a selected length inside the Taijitu).
These are real mathematical objects. Their joint appearance is the result of overlaying two independent constructions—one based on modular arithmetic and regular 9-fold symmetry, the other on nested circles that happen to admit golden-ratio segments. The visual convergence is elegant and has obvious rhetorical power. It does not, by itself, demonstrate that the three constants are structurally bound to one another, nor that the figure encodes a universal “frequency matrix” capable of eliminating thermodynamic entropy.In short, the Rodin–Taijitu diagram is a clear and compact illustration of how several well-known mathematical features can be made to coexist on a single page. It is a worthwhile object of geometric study. The further claim that it unlocks a cosmic operating system remains an interpretive step beyond the drawing itself.
