When I was growing up in Canada, one of my favorite playthings was my Meccano set—a wondrous collection of metal strips, plates, wheels, gears, and other parts that could be assembled into all sorts of mechanical contraptions.
Over the years, ingenious mechanics have assembled available parts into cars, trucks, cranes, construction equipment, and much more. Enthusiast Tim Robinson has even used standard Meccano parts to create scale models of mechanical computers (two difference engines and a differential analyzer) invented by Charles Babbage (1791–1871) in the 19th century.
Theoretical physicist Gerard 't Hooft of the University of Utrecht also has fond memories of playing with a Meccano set as a child. He has recently turned this long-ago interest into an intriguing and entertaining mathematical pursuit. 't Hooft shared the 1999 Nobel Prize in Physics for his part in elucidating the theory of the electroweak interaction in particle physics.
As described in a paper available at his Web page, 't Hooft focuses only on the metal strips, each one having a certain number of holes a fixed, integral distance apart. You can fasten these strips together in various ways using nuts and bolts.
The Meccano strips shown above have 7 holes (blue, length 6) and 4 holes (yellow, length 3). They can be fastened together at any of the holes, using bolts. All images courtesy of Gerard 't Hooft.
What sorts of planar geometric figures can you make from these strips?
"One can basically construct all of the fundamental figures that can be generated using the axioms of Euclidean geometry," 't Hooft contends.
"Usually, Euclidean geometry refers to compass and straightedge as the only legitimate tools," he continues. "Here, I replace compass and straightedge by Meccano strips."
In doing so, 't Hooft says that he finds, perhaps surprisingly, that "Meccano strips are more versatile that compass and straightedge."
In his constructions, 't Hooft assumes that the strips are strictly two-dimensional and that as many of them as desired may be completely or partly overlapping. The bolts take up no space.
't Hooft shows how it's possible to combine strips to construct new units that behave as strips in which the holes can be chosen at any rational position (p/q) on the strip. It's also possible to construct strips of length a + √(b), where a and b are any integers.
This construction produces a generalized strip that effectively has holes at the positions A, A + 1, . . . B, and C, where distance BC is the rational number qr/d. The resulting construction is equivalent to the strip shown to the right, which isn't available as a standard part.
His constructions also include examples in which strips can be combined so that one of the end points can move only along a perfect straight line segment. This feature can then be deployed to not only bisect but also trisect an angle.
't Hooft goes on to describe ways to construct a rigid regular pentagon. "We start with five strips of the same length, but to fix them rigidly . . . more strips are needed," he notes. His most straightforward construction of such a pentagon requires 11 strips (below).
One way to construct a rigid regular pentagon (brown).
Other polygons are also accessible. A bit of serendipity, 't Hooft says, led to a regular heptagon of only 15 pieces.
There's a lot more to explore in this Euclidean realm of integral Meccano strips. 't Hooft has a number of suggestions for further investigations.
And, luckily, Meccano sets are still available, though the brand name and assets, originally British, have been transferred to a Japanese company (Nikko Toys). Kids in the United States may be more familiar with the Erector Set brand, a similar toy construction kit. Erector sets are now also made by Meccano factories in France and China.
Peterson, I. 2006. Constructing difference engines. MAA Online (May 1).
't Hooft, G. Preprint. Meccano math.