A Description of the Four Canoes Sculpture
and
A Tiling Problem

Also see"The Mathematics of Helaman Ferguson's Four Canoes" by Loe and Borovsky in  Mathematics Magazine vol. 81, #3, June 2008 (journal available through UST Library).

Dimensions: 9' x 7' x 9' (standing aspect) 21' x 20'  (plaza tiIing)
Weight : 9 Tons
Materials: Red and black granite, quarried and cut by Cold Spring Granite, MN.

Helaman Ferguson sculpted Four Canoes as the keystone outdoor piece located in the quadrangle bordered by two new "wet" and "dry" science and engineering buildings at the University of St. Thomas in St. Paul, Minnesota.

Installed in September 1997, this granite sculpture includes a plaza of alternating red and black hexagonal tiles, two hexagonal prisms rising from the plaza, and two three dimensional non-orientable double cross cap toroidal forms resting on the prisms. 
Six feet in diameter, the toroidal forms weigh three tons each, and stand linked upon the prisms, as if to span the gap between. Each toroid can be thought of as two canoes sewn together and bent round. Translated past and rotated through one another, these two double canoe Klein bottle forms, solid granite toroids with double cross-caps, couple inextricably and mysteriously.

Twenty-eight serrated-edge hexagons, each three feet in diameter, tile the plaza. It is possible to cover the plane periodically with these double canoe hexagonal tiles, with a fundamental region equivalent to just 3 hexagons, but the randomness of Ferguson's placement of the tiles appears to be non-periodic

Description by Jonathan Ferguson, Helaman Ferguson
Photo by Sam Ferguson 
Copyright © 1999 Helaman Ferguson

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A Tiling Problem

In October of 1997, shortly after UST's new Frey Science and Engineering Complex opened, Helaman Ferguson's sculpture, Four Canoes, was installed on the plaza outside the buildings. This massive piece consists of two linked three-ton granite "donuts", each six feet in diameter. These are supported by granite pedestals and rest on an unusual and beautiful tiling. The sculpture itself is laden with fascinating mathematics, but the tiling is especially intriguing to a topologist. (Topology is the branch of mathematics that is often described as "rubber sheet geometry", and is concerned with properties of figures that remain invariant under various transformations).

The tiling that supports Helaman Ferguson's sculpture, "Four Canoes", is comprised of 30 identical tiles, shaped like hexagons, with small notches cut out on the edges (a).

(a)

Under identification of opposite pairs of edges in a certain way indicated by the notches, each hexagon represents a complex mathematical surface known as a Klein bottle ­ a bottle with no inside or outside that lives in four dimensions, and so it's hard to describe in our three dimensional world. (Each of the granite "donuts" also represents a Klein bottle).

The only rule for fitting the tiles together is that two tiles can be adjacent along an edge whenever an "edge arrow " is created (b), and not when a "zig-zag' is created (c).

(b)  legal

(c)  illegal

According to the sculptor, his method of fitting the tiles together on Foley plaza was somewhat haphazard: at each step of piecing the tiles together, he just rotated a tile by some multiple of sixty degrees until it fit, and looked attractive to the artist’s eye, being careful not to violate the edge arrow criterion. Consequently, there is no apparent pattern to the position or orientation of each tile in the tiling. This raises several questions:

1. Is there a way to fit the tiles together (still following Ferguson's rule) that is "periodic" rather than random? That is, will they fit together so that a small patch of the tiles can be repeated over and over (just by sliding the patch around) to create an entire tiling that doesn't violate Ferguson's rules?

2. Is there more than one periodic tiling possible? How large is the "fundamental region" that will create the entire tiling by "translation", i.e., by sliding a portion of the tiling around?
 
3. Following Ferguson's rule, is it always possible to fit these tiles together, or are there some impossible situations one could create so that no tile would fit in a certain spot?

Problem by Melissa Loe