>[!warning] >This content has not been peer reviewed. # Constructal law (Bejan) — RST application **Total ontology pillar:** *Fractal geometry (percolation/Bejan) — the shape of the wires.* Bejan's Constructal law: persisting systems evolve to provide easier access to the currents that flow through them. In RST this is the **geometry of bandwidth**: the same fractal dimension $n \approx 1.25$ (SPARC-calibrated, backbone dimension $d_B \approx 1.22$) appears in river deltas, vasculature, and lightning because it is the **least-resistance** configuration for moving flow (signal, morphisms) through a 3D substrate. So $n$ is not a free parameter; it is the Constructal dimension — the one that **eases the flow of currents**. --- ## I. Mapping to RST | Constructal law | RST reading | |:---|:---| | "Easier access to currents" | **Geometry of bandwidth**: minimise resistance (A5, relational distance) for given flow; $W^n = \Omega^n + N^n$ with $n$ = scaling exponent of the backbone. | | Tree-like (dendritic) structures | **Backbone** of the percolation cluster: the subset of links that carry flow; dimension $d_B \approx 1.22$ in 3D. | | River basins, vasculature, lightning | Same $n \approx 1.2$--$1.25$: RST **transition sharpness** $n_0$ and Constructal scaling exponent are the same number. | | Evolution of the format | The substrate "evolves" (or is selected) to allow maximum throughput for minimum cost; that optimum is $n \approx 1.25$. | **Identity:** The RST fidelity $\mu(\eta, n)$ uses $n$ as the **transition sharpness**; the same $n$ is the **fractal dimension of the signal-carrying backbone** (percolation) and the **Constructal least-resistance exponent**. So the "shape of the wires" (Bejan) and the "sharpness of the Newton–MOND transition" (RST) are one and the same. --- ## II. Script: fidelity and flow efficiency vs $n$ The script **[[Constructal Law - Code]]** (`rst_constructal_n_efficiency.py`) plots $\mu(\eta, n)$ for several values of $n$ (e.g. 1.0, 1.25, 1.5, 2.0) and marks $n_0 = 1.25$ as the SPARC/Constructal value. It also illustrates a simple **flow-efficiency** (or "access") measure vs $n$ — e.g. a quantity that peaks near $n \approx 1.25$ — to show that the substrate's choice of $n$ is the one that **eases flow**. Run via **`run_all_further_scripts.py`** or from this folder. --- ## III. Total ontology tie - **Thermodynamics:** Cost of existence → resistance costs energy. - **Information:** Logic of selection → (Solomonoff). - **Fractal geometry (Bejan):** **This application** — shape of the wires; $n = 1.25$ as least-resistance dimension; rivers, vasculature, backbone. - **Dynamics:** Flow of the render → $\mu(\eta, n)$ in gravity, fluids. - **Intelligence:** Optimisation of the signal → (Friston). --- ## Links - **Knowledge note:** [[expanded theory/knowledge/Persisting systems evolve to ease the flow of currents]] - **Code:** [[Constructal Law - Code]] - **Transition sharpness, backbone:** [[expanded theory/Transition Sharpness]], [[expanded theory/Backbone Dimension]] - **Roadmap:** [[../../Applications Roadmap]]