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Geometry Atlas

Fibonacci sequence

M1–M1001 · core calculations verified

M9 / P24

Alpha

24 native Fibonacci residues on the period circle.

9 · 1 · 1 · 2 · 3 · 5 · 8 · 4 · 3 · 7 · 1 · 8 · 9 · 8 · 8 · 7 · 6 · 4 · 1 · 5 · 6 · 2 · 8 · 1

Clock position 24 is north. Remainder zero is displayed as 9.

Fibonacci recurrence modulo 9. The initial state (0, 1) first returns after 24 steps.

Beta

Interior interval sums on the native 9-position circle.

4 → 8 → 7 → 5 → 1 → 2

Original start: 4. Recognised doubling cycle: 1 → 2 → 4 → 8 → 7 → 5.

Interior sums: 4, 17, 16, 23, 10, 11. Modulo 9: 4, 8, 7, 5, 1, 2. First interval: clock positions 24 to 4. Endpoints excluded; wraparound included. Original phase is preserved.

Gamma

Primary nodal geometry.

9 · 3 · 3 · 9 · 6 · 6

Clockwise node order, starting from north.

Clock positions: 24, 4, 8, 12, 16, 20. The primary selection uses native residues divisible by three, including zero. Regular triangles are drawn only when their vertices lie in this node set. Additional formations are available in Geometry summary.

Epsilon

A continuous unicursal hexagram through 3, 6 and 9.

9 → 3 → 6 → 9 → 3 → 6 → 9

Arrows indicate direction; small numbers indicate visit order.

Clock position route: 24 → 8 → 20 → 12 → 4 → 16 → 24. Proposed geometric traversal; physical interpretation remains unassessed.

Fibonacci M9P24

Native sequences and detected primary geometry in one frame.

Alpha · Beta · Gamma · Epsilon

Alpha uses 24 positions; Beta uses 9.

Only detected primary triangles and hexagons are drawn. Shared placement is a drawing convention and does not establish biological significance.

Research hypothesis

We hypothesize that the unicursal hexagram may describe a sequence of valence-electron transitions. We plan to investigate this through Hamiltonian modelling, initially exploring quantum annealing with D-Wave Leap.

Odom’s construction

Odom’s construction uses the Phi Ratio to circumscribe a circle around an equilateral triangle.

Join the midpoints A and B of two sides of an equilateral triangle.

Take AB as one unit.

Extend AB beyond B by 1/φ to reach C on the circle through the triangle’s three vertices.

AB divided by BC equals φ. So does AC divided by AB. The yellow labels use AB as 1, making BC equal to 1/φ.

The Phi Ratio is the limit of the ratio of successive Fibonacci numbers. As the sequence continues, each number divided by the previous number approaches φ ≈ 1.618034. Odom’s construction expresses that same ratio geometrically.

This construction applies to every equilateral triangle. Whether it helps explain patterns across Fibonacci moduli remains a research question.

Shared arithmetic. Distinct native geometries.

M9 family identifies multiples of nine, not a significance ranking.

Reduction to M9

Original phase is preserved. Beta begins at 4; its recognised cycle is 1 → 2 → 4 → 8 → 7 → 5. Zero residues are displayed as 9 after reduction.

Why the sequences match

For multiples of nine, (Fₙ mod 9k) mod 9 = Fₙ mod 9. Gamma selection and interior sums preserve this reduction. This is an exact arithmetic relationship.

All 111 multiples of nine through M999 passed the Python sequence comparison. Longer periods contain repeated M9 cycles; their full-circle geometry is not necessarily identical.

Research question · not yet assessed

Source data

Modulus records

Sources