Session: MR-07-04

Paper Number: 143723

143723 - Surfaces of Constant Width With Rotational Symmetry 

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.

Presenting Author: Mason Rollans Brigham Young University

Presenting Author Biography: Mason James Rollans is a dedicated mechanical engineering student at Brigham Young University. His research spans intriguing areas, including the mathematics of engineering design and the creation of novel surfaces. Mason’s passion lies in exploring shapes of constant width, a fascination he has nurtured for years. His groundbreaking work introduces a new class of surfaces with constant width.

Beyond this, Mason delves into the realm of thermal conductance within friction stir welding, contributing to advancements in this critical field. Looking ahead, he aspires to delve into dynamic systems modeling and kinematics, aiming to make significant contributions to the engineering community.

Authors:

Mason Rollans Brigham Young University
William Nelson Brigham Young University
Ryan Howell Brigham Young University
Ian Monk Brigham Young University
Denise Halverson Brigham Young University