Material for √2 Rhombuses

Bridges 2013 Article

Tom Verhoeff, Koos Verhoeff. "Folded Strips of Rhombuses, and a Plea for the √2:1 Rhombus". [ Download ]

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.

BibTeX Citation Information

@inproceedings{bridges2013:15,
  author      = {Tom Verhoeff and Koos Verhoeff	},
  title       = {Folded Strips of Rhombuses and a Plea for the √2:1 Rhombus},
  pages       = {71--78},
  booktitle   = {Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture},
  year        = {2013},
  editor      = {George W. Hart and Reza Sarhangi},
  isbn        = {978-1-938664-06-9},
  issn        = {1099-6702},
  publisher   = {Tessellations Publishing},
  address	  = {Phoenix, Arizona, USA}
}

Slides

My √2 rhombuses presentation at Bridges 2013: Slides

Images

Images and descriptors

Movies

In these QuickTime movies, at any moment, the dihedral angles between adjacent rhombuses in the strip are all the same. This angle changes gradually over time, with some pauses for interesting views.

• Octagon Folding, with pauses: Shows how a strip of rhombuses folds into an octagon, with pauses at other closures of the strip.
• Trefoil Inversion, with pauses: Shows how a trefoil knot of rhombuses unfolds into a strip, and then into the reflection of that trefoil knot, with pauses at other closures of the strip.
• Trefoil Inversion: As above, but pauses only at the fully unfolded strip, and can be played in a loop.
  • Higher Resolution (800x800) (about 30 MB)
  • For iPhone 4/iPad 2 (720x720) (about 20 MB)
  • On YouTube:

Interactive Documents

Request to Plea with Polydron Ltd for Comeback of √2 Rhombus

Polydron used to have a √2:1 Rhombus among their products (see picture on the left), but it was discontinued for lack of sales.

If you want to help get back the Polydron √2:1 Rhombus, then please write to Polydron Ltd to tell them about the wonderful things you can do with these rhombuses, and indicate how many such rhombuses you would like to obtain (hint: you need 54 for one trefoil knot; hidden rhombuses on the "inside" are not needed).