Original Mondrian

• Original Mondrian is a unified paradigm integrating diagrammatic methods in planar N=4 SYM, recursive partitioning in machine learning, and combinatorial tiling puzzles, all inspired by Mondrian’s grid art.
• It employs precise methods like axis-aligned recursive partitions and dual conformal invariance to compute scattering amplitudes and to build statistically optimal estimators.
• Its applications span optimal regression in high-dimensional data, rigorous combinatorial geometry for tiling problems, and computational analyses of art, offering scalable and interpretable solutions.

Original Mondrian refers to a set of concepts and technical frameworks across mathematics, physics, statistical learning, combinatorics, and computational geometry, all inspired or named after Piet Mondrian’s iconic grid-based abstract art. The term encompasses: 1) Mondrian diagrammatics for all-loop scattering amplitude computations in planar $\mathcal{N}=4$ SYM; 2) the Mondrian process and Mondrian forests for nonparametric machine learning; 3) the Mondrian Puzzle and related combinatorial geometric problems involving rectangle tilings under strict constraints; and 4) algorithmic abstractions and art/computation inspired by Mondrian's compositions. The following sections synthesize the foundational principles, mathematical formulations, theoretical advances, and broad implications of the "Original Mondrian" paradigm as established in the technical literature.

1. Mondrian Diagrammatics in the Planar $\mathcal{N}=4$ SYM Amplituhedron

The Mondrian diagrammatics framework organizes multi-loop integrands in planar $\mathcal{N}=4$ super-Yang–Mills theory into combinatorially well-defined "Mondrian diagrams," which are planar configurations of rectangular "boxes" (loops) joined by simple contact rules. For any pair of loops $i, j$, the integrand’s contact term is $D_{ij} = X_{ij} + Y_{ij}$, where $X_{ij}$ and $Y_{ij}$ encode horizontal and vertical interactions (functions of "zone variables" or momentum twistors).

Four fundamental Mondrian diagram topologies are classified:

• Ladder: recursively removable contacts preserving rectangular shape.
• Cross: a configuration with central crossing and an explicit sign factor.
• Brick-wall: indirect adjacent contacts, distinguishable by their geometric orientation.
• Spiral: first appearing at five loops, topologically irreducible to sequential detachment.

Each diagram directly maps to a product of $X$ and $Y$ factors per contact, yielding a local term in the expansion of the amplituhedron. The expansion

$\prod_{i<j} D_{ij}$

on ordered loop variables produces every legal Mondrian diagram once, modulo sign cancellations. The completeness relation captures this combinatorial unification.

Physical integrand construction requires enforcing dual conformal invariance, particularly in brick-wall and spiral patterns, by correcting the numerator (e.g., substituting $y_{41} w_{14} \rightarrow D_{14}$). Denominator pole structure and overall sign follow from completeness.

2. The Mondrian Process and Mondrian Forests in Statistical Learning

The Mondrian process is a probability measure over recursive axis-aligned partitions of $[0,1]^d$, defined by exponential waiting times with rates equal to the cell's linear size $\vert C \vert = \sum_{j=1}^d (b_j - a_j)$. At each cell, the process samples a waiting time $E_C \sim \text{Exp}(\vert C \vert)$, and, if possible, splits along coordinate $j$ with probability proportional to side length.

When stopped at a lifetime $\lambda$, the process induces a random partition; the expected number of cells in such a partition is $(1+\lambda)^d$. The recursive, memoryless structure makes Mondrian process partitions amenable to streaming and online settings.

Mondrian Forests (ensembles of Mondrian Trees) estimate functions or classify data by averaging cell-wise predictions over multiple independently drawn partitions. Mondrian Forests are shown to be minimax optimal on $s$-Hölder classes: for function smoothness $s\in(0,1]$, trees and forests achieve rate $n^{-2s/(d+2s)}$, and for $s\in(1,2]$ only forests (not single trees) achieve optimal rates, provided lifetime parameters are tuned (typically $\lambda\sim n^{1/(d+2s)}$). Explicit risk bounds allow for adaptive selection of $\lambda$ using aggregation procedures.

3. Generalized Mondrian Process via STIT Tessellations

The STIT (Stable Under Iteration Tessellation) process generalizes the Mondrian process by allowing cut directions from arbitrary (possibly continuous) distributions on the sphere. Axis-aligned Mondrian partitions (discrete case) are a sub-class.

A core result establishes that any discrete-direction STIT process in $\mathbb{R}^d$ can be simulated by lifting to a higher-dimensional axis-aligned Mondrian process via a linear map, preserving statistical properties (Theorem 2.1, (O'Reilly et al., 2020)). Kernel approximations built from STIT tessellations take the form

$K_\infty(x, y) = \exp(-\lambda \Lambda([x,y]))$

where $\Lambda$ encodes the tessellation’s intensity measure. Mondrian kernels approximate Laplace/Cauchy-type distributions, and consistency results for STIT forests extend minimax guarantees of Mondrian forests to more general cases.

Density estimation in Mondrian/STIT forests leverages explicit formulas for the kernel, e.g., for axis-aligned partitions:

$$f_{\lambda, n, \infty}(x) = \frac{\lambda^d}{n} \sum_{i=1}^n e^{-\lambda \Vert x-X_i \Vert_1} \prod_{j=1}^d h(\lambda|x_j - X_i^{(j)}|)$$

where $h(t) = 1 - t e^t E_1(t)$ and $E_1$ is the exponential integral.

4. The Mondrian Puzzle and Combinatorial Geometry

The Mondrian Puzzle interrogates whether an $n \times n$ square can be tiled by incongruent integer-sided rectangles of equal area (i.e., whether $M(n)=0$). No such perfect partition (with $M(n)=0$) has yet been found or ruled out.

Using number-theoretic translation, strict divisor bounds are established. E.g., for tiling to be possible with area $d$, the condition $d \cdot |\mathcal{S}| = n^2$ (with $|\mathcal{S}|$ the number of rectangles, bounded by $\tau(d)$) leads to:

$\forall\, d \mid n^2,\, d \cdot \tau(d) \geq n^2$

Counting the number of $n \leq x$ for which $M(n)\neq0$, sieve-theoretic arguments (via the Selberg Sieve and divisor function estimates) yield:

$|\{n \leq x: M(n)\neq 0\}| \geq \frac{e^{-\gamma}}{2} \cdot \frac{x}{\log\log x}\ \ \text{(with $\gamma$ Euler–Mascheroni constant)}$

asymptotically (O'Kuhn et al., 2020, O'Kuhn, 2018). Improved density estimates take into account prime factorization (Dalfó et al., 2020). Explicit formulas for defect $d(n)$ are given for small $k$ (number of rectangles), e.g., for $k=2,3$, and recursive constructions establish that defect ratio $d(n)/n^2\to 0$ for certain infinite sequences.

5. Perfect Mondrian Partitions in the Continuous Setting

Generalizing the Mondrian art problem to real side lengths (not necessarily integer), it is shown (Dalfó et al., 2020) that:

• The minimal $k$ for a perfect Mondrian partition (non-congruent rectangles of equal area) is $k=7$; the partition is unique up to symmetry.
• There are exactly two proper perfect partitions for $k=8$.
• Any square admits a perfect Mondrian partition for $k\ge 7$ via augmentation and scaling of sides.
• The existence of perfect integer-sided partitions for any $n$ remains open.

Combinatorial digraph methods rigorously establish uniqueness and enumerate symmetric solutions. For rectangles, side balances are solved via polynomial equations ensuring all $x_i y_i = \mathrm{Area}/k$ and mutual non-congruence.

6. Mondrian’s Art and Computational Analysis

Quantitative statistical and algorithmic analyses reveal that Mondrian’s actual compositions are, in general, not compatible with recursive (binary) splitting models (Feijs, 2020). Statistical features such as "splittingness" $\mathcal{S} = (RT-2\times SC)/RT$ (RT: regular Tee joints, SC: strange coincidences) quantify deviation from idealized splitting; average value across 147 works is $48.8\%$. Complexity is modeled as $(THL + TVL)/(SW + SH)$ (total horizontal/vertical length over sample dimensions).

Semantic processing is critical for automated classification and understanding of Mondrian’s artwork (Doboli et al., 2022). Cognitive architectures are proposed to bridge displayed and latent artistic features, leveraging goals, rules, conceptual mappings, and backward reasoning from cognitive structure to observed data. Methodologies integrate cognitive components with feature extraction, demonstrating the analytical value of moving beyond pixel-level attributes.

7. Applications, Impact, and Research Directions

The Mondrian process and forests underpin theoretically optimal estimators for regression, classification, and density estimation, scalable to arbitrary dimension, with explicit risk bounds and online updating. The combinatorial "Mondrian diagrammatics" supply compact, positive, and dual conformal invariant representations of SYM amplitudes, with completeness relations connected to separable permutations and Schröder numbers.

In geometry and number theory, the puzzle refines the landscape of rectangle tilings and combinatorial density bounds. The computational and cognitive approaches to Mondrian’s art analysis set foundations for semantic-aware computer vision and art informatics.

Open questions remain across domains: existence of perfect integer-sided partitions, extension of Mondrian diagrammatics to higher-particle amplitudes, optimal cut direction design for generalized STIT forest learning, and the integration of cognitive reasoning into automated artwork classification.

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Original Mondrian, as established in the rigorous literature, thus serves as a unifying paradigm that blends combinatorics, geometry, machine learning, mathematical physics, and computational cognition, with deep foundational results and ongoing research across several fields.