Mandelbrot Group Paths

Status: active Created: 2026-06-22 Visual: projects/research/mandelbrot-visual-periodic-table/plot3_group_paths.png

The Finding

When standard periodic table groups are plotted as connected paths through the Mandelbrot complex plane (using the sopfr(Z)/Z + i·(N-Z)/A encoding), a striking pattern emerges:

• Standard periodic table groups (alkali metals, alkaline earth, halogens, noble gases, etc.) trace recognisable TRAJECTORIES through the Mandelbrot complex plane
• Light members of each group start near Re=1.0 (high sopfr/Z — prime or near-prime Z values)
• As Z increases down each group, elements move LEFT toward lower sopfr/Z AND UP toward higher nuclear asymmetry
• This convergence means ALL groups approach the Mandelbrot boundary from the escaped side
• The periodic table is a set of trajectories through Mandelbrot-space, not just a grid

Groups trace FAN-SHAPED PATHS from the escaped region (Re≈1.0, light elements) CONVERGING toward the Mandelbrot boundary as Z increases. Every group walks the same journey: from escaped (prime-like, light) toward captured (composite, heavy).

Specific Group Paths

• Group 1 (Alkali metals): Li(3)→Na(11)→K(19)→Rb(37)→Cs(55)→Fr(87) — starts at Re=1.0 (Li, Na, K all prime Z), sweeps left as Z becomes more composite
• Group 17 (Halogens): F(9)→Cl(17)→Br(35)→I(53)→At(85) — F and Cl have prime Z (Re=1.0), then path curves left
• Group 18 (Noble gases): He(2)→Ne(10)→Ar(18)→Kr(36)→Xe(54)→Rn(86) — He at Re=1.0, progressively more composite Z values
• Group 2 (Alkaline earth): Be(4)→Mg(12)→Ca(20)→Sr(38)→Ba(56)→Ra(88) — all composite Z, starts mid-range

The ONM Interpretation

• Moving DOWN a group in the standard table = increasing Z = accumulating more prime factors = becoming more COMPOSITE
• In Mandelbrot-space this means: moving toward capture, toward the boundary, toward the interior
• The lightest elements (closest to the source primes) escape most readily
• The heaviest elements (most composite) approach or enter the captured region
• Each group traces the JOURNEY from prime identity toward composite structure
• The periodic table groups are not arbitrary categories — they are TRAJECTORIES through number-theoretic space

Why This Is New

• Standard periodic table groups are defined by electron configuration (quantum mechanics)
• These trajectories emerge from pure number theory (sopfr factorisation)
• The two organisational principles are independent yet produce overlapping structure
• This is another instance of the Q-EM-NEW-05 mystery: why does arithmetic predict chemistry?

Relationships

• [[elemental-mandelbrot]] — nature: extends — Group paths are a visual discovery from the complete 118-element mapping
• [[three-tiers-of-primes]] — nature: supports — Light group members (prime Z) = Tier 2 scaffold primes; heavy members (composite Z) = composites approaching capture
• [[source-alphabet]] — nature: supports — All groups converge FROM the prime column (Re=1.0) TOWARD the {2,3}-structured interior
• [[live-boundary]] — nature: extends — The Mandelbrot boundary acts as the attractor; periodic groups are trajectories approaching it

Open Questions

• Q-MGP-01: Do transition metal groups (3-12) trace similar convergent paths, or do they behave differently?
• Q-MGP-02: Is the rate of convergence (how fast a group approaches the boundary) related to the group’s chemical reactivity?
• Q-MGP-03: Do diagonal relationships in the periodic table (Li-Mg, Be-Al, B-Si) correspond to PROXIMITY in Mandelbrot-space?
• Q-MGP-04: Can the Mandelbrot trajectory predict undiscovered element properties (e.g., element 119’s group 1 position)?