Triskewers

[Triskewers]

Triskewers: A collection of 144 wooden skewers glued into a symmetric, repeating pattern.

Designed and constructed in April, 2010.

[Triskewers: 3-fold symmetry axis]

Each skewer lives in a triangular "hole" created by the other skewers.

A subset of an infinite, repeating arrangement of interwoven cylinders, made from 144 wooden skewers and held together with glue. The skewers form four identical, intersecting triangular prisms, giving the figure chiral tetrahedral symmetry—or chiral cubic symmetry if the spiked and non-spiked ends are not distinguished. The high density packing and repetitive lattice pattern create mesmerizing networks of passageways through the sculpture.

George Hart's 72 Pencils was an inspiration for this sculpture, but there is an important mathematical difference between the two: they are based on different infinite cylinder arrangements! The easiest difference to spot is the fact that, in Triskewers, each skewer fits into a triangular hole formed by the other skewers (see this image above). In 72 Pencils, these holes are hexagonal. For this reason, these cylinder arrangements have been termed tristakes and hexastakes respectively by Conway et al. in The Symmetries of Things. (If the ends of the skewers or pencils were not distinguished, they would be called tristicks and hexasticks instead.)

You may ask if glue is really necessary for this (or something similar) to support itself. Maybe a sufficiently complex weaving of (unbent) skewers could create a rigid structure? For the Triskewers sculpture, any skewer can slide out while leaving the others fixed, so the structure is not rigid without glue. Interestingly, this is true for any skewer construction, and far more generally still: All configurations of nonoverlapping convex (or even star-shaped) regions are not rigid and can be completely disassembled by continuous motion without overlap. A separating motion can be constructed by, roughly, "exploding" the pieces linearly away from a chosen origin.

Exercises

  1. Exactly how dense is the tristicks/tristakes packing? In other words, if this pattern were extended infinitely in all directions, what fraction of space would be inside a cylinder?
  2. This density is obtained in the center of the physical Triskewers sculpture, where the four large triangular prisms intersect. What polyhedral shape describes this intersection?
  3. What is the density if the cylinders were replaced with infinite triangular prisms that exactly filled their holes?
  4. Repeat these computations for hexasticks instead of tristicks. Which is more dense?
Copyright © 2011–2016 by Zachary Abel. All rights reserved. Last updated on 9/29/2016.