Surfaces of Constant Width With Rotational Symmetry

Abstract Surfaces of constant width are defined as surfaces embedded in three-dimensional space having the same width in all directions. While spheres are the simplest and most common surface of constant width, non-spherical surfaces of constant width possess unique properties. For example, surfaces of constant width have less volume than spheres of the same width, potentially leading to increased efficiency in designs. In addition, surfaces of constant width typically have a finite number of stable positions. Moreover, surfaces of constant width may be comprised of piece-wise smooth surfaces, having variable curvature. Due to these distinctive qualities, these types of surfaces have applications in various fields including architecture, mathematics, engineering, and mechanism design. Historically, it has been difficult to construct these surfaces and the number of methods of construction is limited, although recently, novel and creative construction methods have been conceived. This work presents the construction of a new class of surfaces with rotational symmetry of order n, denoted Roll[n], where n > 1 is odd. The design method for these surfaces is rooted in the constant width properties of parallel projections of surfaces of constant width. Existing surfaces of constant width can be classified by orthogonal parallel projections and rotational symmetry. The new Roll[n] surfaces fill in previous gaps in classifications these classification schemes.