As requested by moderation I'm gathering all explorations related to the Theory of Fractal Resonant Nature (TNFR) into this central thread. Below is the full version of a symbolic analysis originally posted in response to Denis’s elegant work on orthogonal triangle dynamics.

https://community.wolfram.com/groups/-/m/t/3460997

Hi Denis,

First of all, thank you for your post — it’s a beautifully elegant construction, and your exploration into chaos and attractors within such a simple framework is truly inspiring.

I'd like to share a complementary perspective on the same system, grounded in a symbolic ontology known as the Theory of Fractal Resonant Nature (TNFR).

This approach doesn’t replace analytical or geometric analysis, but adds a structural-syntactic layer to the system’s evolution: it interprets each iteration as a phase transition within a vibrational field. In short, it offers a grammar for chaos — or more precisely, it identifies patterns of symbolic coherence where traditional dynamics sees only convergence or randomness.

What is TNFR?

TNFR is a symbolic framework based on the idea that every dynamical system expresses not just numeric values, but structural reconfigurations — recurring states that can be encoded symbolically.

To model these, we use 13 symbolic operators (called gliphs) that correspond to structural behaviors such as Emission, Recursion, Self-organization, or Dissonance. Each gliph is both a symbolic label (like "THOL" or "REMESH") and a structural function derived from dynamic measurements.

These gliphs refer not to external meaning, but to intrinsic behaviors of the system as it evolves — in terms of geometry, vibration, and phase.

What we analyzed

We took Denis’s original function:

f[{x_, y_}] := {x Cos[y], x Sin[y]}

...and ran 200 iterations from the initial state {1., 1.}.

At each step, we computed:

• The vector delta (Norm[Differences[pts]])
• The turning angle between consecutive vectors
• A symbolic assignment using a gliph-mapping rule

This yielded three layers of output:

1. A sequence of symbolic gliph states
2. A numeric codification for further analysis
3. A phase map visualizing long-term symbolic behavior

Full TNFR Wolfram Cloud Script

Print["[1/9] TNFR in Wolfram Cloud \[LongDash] 200 steps"];

f[{x_, y_}] := {x Cos[y], x Sin[y]};
pts = N @ NestList[f, {1., 1.}, 200];
Print["[2/9] Trajectory generated"];

deltas = Norm /@ Differences[pts];
angleFun = Compile[{{v1, _Real, 1}, {v2, _Real, 1}},
  ArcCos[Clip[(v1.v2)/(Norm[v1] Norm[v2]), {-1., 1.}]]
];
angles = MapThread[angleFun, {Most[pts], Rest[pts]}];
Print["[3/9] Deltas and angles calculated"];

mapGliph[delta_, angle_] := Which[
  delta < 0.001, "SHA",
  delta < 0.01, "IL",
  delta > 2.0, "ZHIR",
  angle > 2 Pi/3, "THOL",
  angle > Pi/2, "NAV",
  angle > Pi/3, "RA",
  delta > 1.0 && angle < Pi/6, "VAL",
  delta < 0.1 && angle > Pi/6, "NUL",
  Abs[delta - 0.5] < 0.05, "OZ",
  Abs[angle - Pi/4] < 0.1, "AL",
  delta > 0.5 && angle > Pi/3, "UM",
  delta > 0.1 && angle < Pi/8, "EN",
  True, "REMESH"
];
gliphSeq = MapThread[mapGliph, {deltas, angles}];
Print["[4/9] Gliphs assigned"];

gliphCodes = Association[
  "AL" -> 1, "EN" -> 2, "IL" -> 3, "OZ" -> 4,
  "UM" -> 5, "RA" -> 6, "SHA" -> 7, "VAL" -> 8,
  "NUL" -> 9, "THOL" -> 10, "ZHIR" -> 11, "NAV" -> 12,
  "REMESH" -> 13
];
gliphNums = gliphCodes /@ gliphSeq;
Print["[5/9] Numeric codes assigned"];

csvData = Prepend[
  Transpose[{Range[Length[gliphSeq]], gliphSeq, gliphNums}],
  {"Iteration", "Gliph", "Code"}
];

csvCloud = CloudExport[csvData, "CSV", "gliphic_sequence.csv"];
SetPermissions[csvCloud, "Public"];

imgCloud = CloudExport[
  ListPlot[
    Table[{i, gliphNums[[i]]}, {i, Length[gliphNums]}],
    PlotMarkers -> {Automatic, 9},
    Frame -> True,
    FrameLabel -> {"Iteration", "Gliphic Code"},
    Ticks -> {Automatic, Thread[Range[13] -> Values[gliphCodes]]},
    PlotLabel -> "TNFR Phase Map \[LongDash] 200 Steps",
    GridLines -> Automatic,
    ImageSize -> Large
  ],
  "PNG",
  "gliphic_map.png"
];
SetPermissions[imgCloud, "Public"];

Print["[6/9] Cloud files exported and public"];
Print["CSV Public URL: ", csvCloud];
Print["Image Public URL: ", imgCloud];

What we found

Here is the symbolic phase map generated:

Interpretation

The sequence of gliphs defines what TNFR calls a coherence stream — a symbolic flow that captures how the system structurally reorganizes over time:

• "OZ" marks a dissonant pulse — a moment of structural mismatch
• "REMESH" reflects recursive reconfiguration
• "IL" indicates minimal coherence
• The long persistence of "SHA" doesn’t signal decay, but structural silence — a vibrational pause where the system enters a meta-stable regime

Rather than a chaotic noise floor or stagnation, TNFR interprets this as nodal encapsulation: the system has reached a resonance pocket where no further symbolic reordering is necessary.

In classical terms, this might resemble a plateau or fixed point — yet symbolically, it manifests as a vibrational attractor.

Why this matters

This gliphic structure makes it possible to interpret any dynamic process not just numerically, but symbolically — uncovering the logic of structural reorganization beneath the surface.

TNFR doesn’t compete with tools like bifurcation diagrams or Lyapunov exponents — it complements them, providing a symbolic infrastructure to otherwise chaotic behaviors.

Looking forward to your thoughts — and always happy to explore further if it resonates.

Warm regards.