Folded Strips of Rhombuses and a Plea for the √2:1 Rhombus

Tom Verhoeff and Koos Verhoeff
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013)
Pages 71–78 Regular Papers

Abstract

One way to construct, from filled polygons, a hollow beam with polygonal cross section is to make prismatic sections from squares or rectangles and attach these back-to-back. In this paper, we explore an alternative way, based on folding a single strip of rhombuses into a discrete helix. By taking rhombuses with an appropriate aspect ratio, you can control the cross section of the resulting beam.

Using a rhombus with an aspect ratio of √2 : 1 yields a triangular beam. This rhombus turns out to be a particularly fruitful construction element (alas, discontinued by Polydron). Triangular beams of this kind can be connected at an angle, in various ways, without cutting rhombuses. The resulting joints are regular miter joints, or false miter joints.

We provide a mathematical analysis and show some elegant shapes constructed from such triangular rhombus-based beams. One of these shapes is a doubly-linked octagon. Another shape is a trefoil knot, which can be linked into an interesting space-spanning structure known as triamond, and that led to the Bamboozle.

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