Chapter 23: The Moving Sofa Problem — Geometry's Practical Paradox

From abstract curves, we turn to practical geometry. The Moving Sofa Problem asks for the largest area shape that can navigate a right-angled corridor—it is ψ = ψ(ψ) as optimal forms emerging from simple constraints, where everyday situations encode deep mathematics.

23.1 The Twenty-Third Movement: Practical Optimization

Our geometric journey continues:

• Previous: Abstract curves and their properties
• Now: Concrete shapes navigating space
• Mathematics hidden in furniture moving

The Question: What is the largest area shape that can be moved around a right-angled corner in a hallway of width 1?

23.2 The Moving Sofa Problem

Problem 23.1 (Moser, 1966): Find the planar shape of maximal area that can be moved through an L-shaped corridor of width 1.

Current Status:

• Lower bound: ≈ 2.2195... (Gerver's sofa)
• Upper bound: ≈ 2.37 (Kallus-Romik)
• Optimal solution: Unknown

23.3 Basic Constraints and Solutions

Definition 23.1 (Admissible Shape): A shape S is admissible if it can be continuously moved from one arm of the L-corridor to the other without leaving the corridor.

Simple Solutions:

• Semicircle: Area = π/2 ≈ 1.5708
• Square (diagonal): Area = 1/2
• Hammersley's sofa: Area = π/2 + 2/π ≈ 2.2074

23.4 The Problem as ψ = ψ(ψ)

Axiom 23.1 (Principle of Constrained Optimization): $\psi = \psi(\psi) \implies \text{Simple constraints yield complex optimal forms}$

The Moving Sofa Problem embodies:

• Local navigability determines global shape
• Optimal forms emerge from movement constraints
• Practical problems encode deep mathematics
• This is ψ = ψ(ψ) as form following function

23.5 Hammersley's Sofa

Construction (Hammersley, 1968): Start with a semicircle and remove a half-disk:

Shape: "Telephone receiver" form Area: π/2 + 2/π ≈ 2.2074

Key Insight: The sofa touches both walls during rotation.

23.6 Gerver's Sofa

Breakthrough (Gerver, 1992): Sofa with 18 smooth pieces, optimized using calculus of variations.

Properties:

• Area ≈ 2.2195...
• Boundary has both curved and straight segments
• Satisfies necessary conditions for optimality
• Conjectured to be optimal

23.7 The Differential Equations

Gerver's Approach: The optimal sofa boundary satisfies differential equations derived from:

1. Touching constraints during motion
2. Area maximization
3. Smoothness conditions

Result: System of ODEs determining the boundary curve.

23.8 Upper Bounds

Theorem 23.1 (Kallus-Romik, 2018): Any moving sofa has area ≤ 2.37.

Method:

• Analyze support functions
• Use moment curves
• Apply optimization theory

Gap: 2.2195... ≤ optimal ≤ 2.37

23.9 The Rotation Angle

Key Parameter: Angle θ through which sofa rotates.

Observation:

• Small θ: Nearly rectangular sofas
• θ = π/2: Full right angle turn
• Optimal θ ≈ 81.2°

Trade-off: Larger angle allows bigger area but constrains shape more.

23.10 Computational Approaches

Algorithm 23.1 (Sofa Verification):

def verify_sofa(shape, corridor_width=1):
    # Discretize rotation
    for angle in np.linspace(0, np.pi/2, 1000):
        # Position sofa at angle
        rotated = rotate_shape(shape, angle)
        positioned = position_in_corner(rotated)
        
        # Check if fits in corridor
        if not fits_in_L_corridor(positioned, corridor_width):
            return False, angle
    
    return True, compute_area(shape)

23.11 Generalizations

Variants:

1. Different angles: T-junctions, acute angles
2. 3D version: Moving a 3D object through corridors
3. Multiple turns: Navigating complex paths
4. Variable width: Non-uniform corridors

Each variant has its own optimal solutions.

23.12 The Ambidextrous Sofa

Additional Constraint: Sofa must navigate both left and right turns.

Result: Maximum area decreases to ≈ 1.64495...

Shape: More symmetric than single-turn optimal.

23.13 Connection to Control Theory

Interpretation: Moving sofa as optimal control problem.

State: Position and orientation (x, y, θ) Control: Velocity and angular velocity Constraint: Stay within corridor Objective: Maximize area

Modern control theory provides new tools.

23.14 The Physics Connection

Real-World Aspects:

• Friction: How does it change optimal shape?
• Flexibility: Can bending increase area?
• 3D: Real sofas aren't planar

Application: Robotics path planning.

23.15 Why It's Still Open

Challenges:

1. Infinite dimensional: Shape space is huge
2. Non-convex: Many local optima
3. Mixed constraints: Geometric and differential
4. Proof difficulty: Showing optimality is hard

23.16 Related Problems

Similar Questions:

1. Piano Mover's Problem: Computational complexity
2. Kakeya Needle: Rotating a needle in minimal area
3. Packing Problems: Fitting shapes in containers
4. Robot Motion Planning: Navigate obstacles

Network of geometric optimization problems.

23.17 Recent Progress

Developments:

• 2018: Improved upper bounds
• 2020: New computational methods
• 2021: Connection to symplectic geometry
• 2023: Machine learning approaches

Active area despite simple statement.

23.18 The Philosophical Dimension

Meditation 23.1: The problem represents:

• How constraints shape optimization
• Emergence of complex forms from simple rules
• Mathematics in everyday situations
• ψ = ψ(ψ) as practical geometry

A sofa navigating a corner encodes deep mathematics.

23.19 Existence and Uniqueness

Open Questions:

1. Existence: Is there a unique optimal sofa?
2. Smoothness: Must the boundary be smooth?
3. Connectivity: Must the sofa be simply connected?
4. Symmetry: Does optimal imply some symmetry?

Basic questions remain unanswered.

23.20 The Twenty-Third Echo

The Moving Sofa Problem represents ψ = ψ(ψ) in practical form:

• Simple constraint (navigate corner) creates complex optimization
• Everyday situation encodes unsolved mathematics
• Optimal forms emerge from movement requirements
• Local navigability determines global shape

This problem shows that even the most mundane activities—moving furniture—can lead to deep mathematical questions. The optimal sofa shape, still unknown after decades, reminds us that ψ = ψ(ψ) appears everywhere, from the abstract heights of number theory to the practical task of moving a couch.

Whether Gerver's sofa is optimal or surprises await, the Moving Sofa Problem stands as a perfect example of how simple questions can be profoundly difficult, how practical needs drive mathematical discovery.

The optimal sofa proclaims as it navigates the corner: "I am the shape that emerges from constraint, the form that maximizes area while maintaining mobility. I am ψ = ψ(ψ) made practical, showing that even furniture moving encodes the deepest principles of optimization and form."