Central Path Art

TL;DR

This work reframes the central path from a mathematical centerpiece of interior-point optimization into a generative, visual art form. By formalizing the central path as the analytic trajectory $\mathbf{P}(G(x), c)=\{ x(\mu) : \mu>0\}$ and deriving a Newton-based, affine-invariant computation framework, the authors connect optimization theory with tangible patterns, tilings, and floral sculptures. They demonstrate how affine transformations preserve the essential path structure, enabling a wide range of 2D and 3D artistic realizations, from laser-cut tilings to SLS-printed cubic flowers, and they extend the methodology to cyanotype/SolarFast dyeing and woodwork. The work contributes both a robust mathematical treatment of central paths and a novel, image-driven pipeline that translates optimization geometry into art, education, and museum-grade objects with potential for scalable installations. Overall, the central path emerges as a versatile bridge between rigorous optimization and aesthetically compelling, physically realizable art.

Abstract

The central path revolutionized the study of optimization in the 1980s and 1990s due to its favorable convergence properties, and as such, it has been investigated analytically, algorithmically, and computationally. Past pursuits have primarily focused on linking iterative approximation algorithms to the central path in the design of efficient algorithms to solve large, and sometimes novel, optimization problems. This algorithmic intent has meant that the central path has rarely been celebrated as an aesthetic entity in low dimensions, with the only meager exceptions being illustrative examples in textbooks. We undertake this low dimensional investigation and illustrate the artistic use of the central path to create aesthetic tilings and flower-like constructs in two and three dimensions, an endeavor that combines mathematical rigor and artistic sensibilities. The result is a fanciful and enticing collection of patterns that, beyond computer generated images, supports math-aesthetic designs for novelties and museum-quality pieces of art.

Figures (17)

• Figure 1: The figure on the left spaces $\mu$ with \ref{['eq-midPointbound']}, and the figure on the right spaces $\mu$ by bounding (estimates) of curvature.
• Figure 2: The solid central path ignores the dashed redundant constraint, whereas the dashed central path includes the redundant constraint.
• Figure 3: Translations of $3$-gon paths form a star around a $6$-gon.
• Figure 4: A font created from central paths.
• Figure 5: A right circular cylinder circumscribed by a cube.

Theorems & Definitions (15)

• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Corollary 1
• Theorem 4