Differential equations and exact solutions in the moving sofa problem (1606.08111v3)

Abstract: The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width, and is conjectured to have as its solution a complicated shape derived by Gerver in 1992. We extend Gerver's techniques by deriving a family of six differential equations arising from the area-maximization property. We then use this result to derive a new shape that we propose as a possible solution to the "ambidextrous moving sofa problem," a variant of the problem previously studied by Conway and others in which the shape is required to be able to negotiate a right-angle turn both to the left and to the right. Unlike Gerver's construction, our new shape can be expressed in closed form, and its boundary is a piecewise algebraic curve. Its area is equal to $X+\arctan Y$, where $X$ and $Y$ are solutions to the cubic equations $x^2(x+3)=8$ and $x(4x^2+3)=1$, respectively.

• The paper extends Gerver's approach by deriving a family of six differential equations that yield closed-form solutions for both the classical and ambidextrous moving sofa problems.
• It introduces a rigorous rotation path parameterization method to analyze contact points and boundary conditions that maximize the shape's area.
• The work integrates analytical techniques with algebraic elegance, paving the way for applications in robotics path planning and advanced spatial optimization.

An Analytical Approach to the Moving Sofa Problem

The moving sofa problem, initially posed by Leo Moser in 1966, has continued to intrigue mathematicians with its seemingly straightforward yet mathematically complex challenge: to determine the maximal area of a planar shape that can traverse around a right-angled corner in a hallway of unit width. Historically, the problem has captured substantial attention, leading to significant theoretical advancements since the early suggestions by Hammersley and the sophisticated proposal by Gerver in 1992. This essay explores a methodological extension of these ideas as presented by Romik, using differential equations to derive new exact solutions and offering insights into a variant known as the ambidextrous moving sofa problem.

Key Contributions and Techniques

Romik's contribution extends Gerver's approach through the development of a family of six differential equations leveraging the area-maximization property. This analytical framework refines the existing mathematical tools by establishing conditions that must be satisfied for a shape to maximize area under movement constraints. By addressing both the classical moving sofa problem and its ambidextrous variant — where the shape must navigate turns both left and right — Romik provides a comprehensive solution that integrates closed-form solutions and exact algebraic curves.

1. Mathematical Formulation: Romik adopts a systematic approach by parameterizing the problem in terms of 'rotation paths,' which are expressions governing the intersection of hallways translated and rotated around the shape. This approach allows for a detailed examination of the contact points and sections of the shape boundary that align with theoretical and practical considerations of optimal geometry.
2. Differential Equations and Algebraic Curves: The paper's core advancement is the derivation of differential equations that encapsulate the rotation path's necessary conditions for maximizing area. These equations are not merely theoretical constructs but are solved to give a new, precise solution for the ambidextrous variant: a shape whose boundary pieces are algebraically complex yet describe a coherent geometrical assembly.
3. Derivation of Ambidextrous Sofa: By employing symmetry properties and finely tuned conditions, the work uncovers a new shape that fits within the ambidextrous framework. Unlike previous solutions, Romik's ambidextrous shape is expressed entirely in closed algebraic forms, revealing an area characterized by an intriguing combination of standard and transcendental components.

Practical and Theoretical Implications

While the analytical insights into the moving sofa problem enrich geometric understanding, the implications extend beyond pure mathematics into applied problem domains, such as robotics path planning and spatial optimization. The formulation and resolution of differential equations provide a robust methodology applicable to a variety of path and boundary problems where optimization is conditioned by spatial constraints.

Moreover, the analytical methods and solutions presented reframe conventional approaches by embedding algebraic elegance into a context traditionally seen as optimal by discrete or heuristic approximations. This paves the way for further exploration into related geometric optimization problems involving complex boundary behavior and navigation constraints in non-trivial environments.

Future Directions

Romik's paper invites further investigation into potential generalizations of the moving sofa equation framework. Future directions may include:

• Explorations of Asymmetric Configurations: Extending beyond symmetrical assumptions could yield new insights and potentially larger maximal areas.
• Implementation in Dynamic Environments: Applying these solutions to dynamic settings where topological changes affect spatial constraints could uncover novel optimization techniques.
• Algebraic Boundary Investigations: Further probing into the algebraic properties of boundary formulations and the implications for more complex multi-segment shapes.

Overall, Romik's work not only deepens the mathematical discourse surrounding the moving sofa problem but also charts a pathway for innovative applications, integrating classical geometry with modern analytical techniques.