Three families of mitered Borromean ring sculptures

Artists have drawn inspiration from many mathematical structures, such as regular tilings, lattices, symmetry groups, regular polyhedra, knots, and links. A particularly well-known link goes by the name Borromean rings. It consists of three closed loops (“rings”) that cannot be taken apart without cutting, but after removing any one of the rings, the other two can be separated without cutting. In this paper, we present three families of sculptures involving miter joints, inspired by the Borromean rings

Regular 3D polygonal circuits of constant torsion

We explore a special class of regular 3D polygonal circuits, that is, of regular non-planar polygons. In these circuits, all segments (edges) have the same length, all corner angles are equal, and all torsion angles have the same (absolute) value. We also show some artwork based on these constant-torsion circuits

Branching miter joints : principles and artwork

A miter joint connects two beams, typically of the same cross section, at an angle such that the longitudinal beam edges continue across the joint. When more than two beams meet in one point, like in a tree, we call this a branching joint. In a branching miter joint, the beams’ longitudinal edges match up properly. We survey some principles of branching miter joints. In particular, we treat joints where three beams with identical cross sections meet. These ternary miter joints can be used to construct various branching structures. We present two works of art that involve branching miter joints

Lobke, and other constructions from conical segments

Lobke is a mathematical sculpture designed and constructed by Koos Verhoeff, using conical segments. We analyze its construction and describe a generalization, similar in overall structure but with a varying number of lobes. Next, we investigate a further generalization, where conical segments are connected in different ways to construct a closed strip. We extend 3D turtle geometry with a command to generate strips of connected conical segments, and present a number of interesting shapes based on congruent conical segments. Finally, we show how this relates to the skew miter joints and regular constant-torsion 3D polygons that we studied earlier