In a recent research article “Exact solution of the frustrated Potts model with next-nearest-neighbor interactions in one dimension via AI bootstrapping,” Physical Review B 112, 094424 (2025), https://doi.org/10.1103/y5vc-3t6q;​
​https://doi.org/10.48550/arXiv.2503.23758;​
​https://community.wolfram.com/groups/-/m/t/3466026;
​https://www.bnl.gov/newsroom/news.php?a=122742,
Weiguo Yin found that the entire family of the Metallic Means of the Second Kind emerges at a critical point in the phase diagram of the 1D J1-J2 q-state Potts model, as shown in Equation (8), where
q=2
yields the golden ratio,
q=3
the copper mean,
q=4
the nickel mean, and so on. After listening to Prof. Aninda Sinha's mathematically beautiful presentation “The Quantum Life of Pi” at the Wolfram Virtual Technology Conference and learned the opportunity of the One-Liner Competition on November 5, 2025, Yin was stimulated to visualize the beauty of the Metallic Means family using Wolfram Mathematica 14.3. Assisted by OpenAI’s ChatGPT 5, he made the plot in just 136 characters. His one-liner entry was listed in the top 10 picks on November 7. https://blog.wolfram.com/2025/11/19/announcing-the-winners-of-the-2025-one-liner-competition/

Media Coverage and Research Impact

The fundamental discovery of metallic means emerging from the 1D frustrated Potts model was featured prominently in a December 2025 Brookhaven National Laboratory newsroom article. This coverage highlights how artificial intelligence collaboration accelerated the solution of a decades-old theoretical physics problem that had resisted conventional mathematical approaches.
A rendering of the human-AI collaboration: The AI explored at lightning speed, while the human promptly responded to block mistakes. Through this fast iterative interplay, the right path was eventually uncovered. (Lisa Jansson/Brookhaven National Laboratory)

Key Highlights from BNL Coverage

◼
  • AI partnership solved the 1D frustrated Potts model for arbitrary spin orientations
    • ◼
  • Emerged from DOE’s first “AI Jam Session” with OpenAI
    • ◼
  • Revealed novel phase diagram with applications in quantum/photonic devices
    • ◼
  • Demonstrates human-AI iterative collaboration methodology

    • Significance

    This research exemplifies the paradigm shift in scientific discovery where AI serves as a collaborative research partner, rapidly exploring solution spaces while humans provide critical error correction and mathematical insight. The visualization in this notebook captures the mathematical beauty of these discoveries.

    Visualization

    Please run the code in dark mode
    In[]:=
    ParametricPlot[Evaluate@Table[((1+Sqrt[4q-3])/2)^(2t/Pi){Cos@t,Sin@t},{q,2,180}],{t,0,5Pi},PlotStyle->Hue/@Range[.1,.9,.05],Axes->False]
    Or set a black background
    In[]:=
    ParametricPlot[Evaluate@Table[((1+Sqrt[4q-3])/2)^(2t/Pi){Cos@t,Sin@t},{q,2,180}],{t,0,5Pi},PlotStyle->Hue/@Range[.1,.9,.05],Axes->False,Background->Black]
    Out[]=
    In[]:=
    spirals=ParallelTable[Rasterize@ParametricPlot[Evaluate@Table[((1+Sqrt[4q-3])/2)^(2t/Pi){Cos[t+x],Sin[t+x]},{q,2,180}],{t,0,5Pi},PlotStyle->Hue/@Range[.1,.9,.05],Axes->False,Background->Black,AspectRatio->1.5],{x,0,2Pi,0.1Pi}];
    Slow spirals
    In[]:=
    ListAnimate[spirals,AnimationRate->10,AnimationRunning->False]
    Out[]=
    Fast spirals
    In[]:=
    ListAnimate[spirals,AnimationRate->210,AnimationRunning->False]
    Out[]=

    CITE THIS NOTEBOOK

    Visualization of the Generalized Golden Ratio, aka the Metallic Means​
    by Weiguo Yin​
    Wolfram Community, STAFF PICKS, January 13, 2025
    ​https://community.wolfram.com/groups/-/m/t/3605947