Tuesday, July 20, 2010

An Extensible Method for Octetruss and Equilateral Triangle Constructions

This box cutting die was a gift from my father in-law, Lynne Gibbs. The cutout forms an octahedron and tetrahedron sharing a common face.

A close-up of the fine work Lynne did.  The interior blades were not sharp but intended to provide crimping of folds.

The only problem with that kind of cutout is that it is relatively difficult to tie new pieces into the framework. 
I wanted something that would provide integral flanges. The solution I came up with, in 1993, was to draw a regular hexagon and inscribe an equilateral triangle by scoring the edges. Alternating the flanges allowed new paper units to be added to a construction, with enhanced strength due to multiple flanges glued around each joint.

The three polyhedra to the left are the octet cell, the octahedron, and a curious space filler whose name eludes me just now.  The paper cutouts on the right are the basic building block, a regular hexagon with an inscribed equilateral triangle forming triangular flanges. These particular two specimens were cut from an edge of sheet stock and so each have one truncated flange.

The next two are a cub-octahedron and octetruss, respectively.   
 a

Nesting regular tetrahedra end to end gives tetrahelices, left. On the right is an octetruss building unit with flanges extended outward.
  

An icosahedron. This model used the same paper hexagons but like the space filler above it was sprayed with rubberized truck body liner to provide increased stiffness. 
 
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