Updates to Turnabout 2, "Bracing Regular Polygons",
in Hinged Dissections: Swinging & Twisting, by Greg N. Frederickson

Gavin Theobald's improved pentagon bracing

In 2004, Gavin Theobald improved on Thomas H. O'Beirne's pentagon bracing scheme, reducing the number of additional rods from 64 to 56, as shown below. (Thanks to Gavin for allowing me to reproduce his excellent colored GIF figure!)

Since there are several new neat ideas in Gavin's design, and since it's not so easy to see that the regular pentagon is indeed braced, I'll supplement the figure with an explanatory discussion. Gavin adapts two key ideas of O'Beirne and also introduces some new ones: With Gavin's kind permission, I included a discussion of this bracing in an article:
"Bracing regular polygons as we race into the future,"
College Mathematics Journal, Vol. 43, No. 1, pp. 43-49, 2012.
The article was also selected for inclusion in:
Martin Gardner in the Twenty-First Century, ed. Michael Henle and Brian Hopkins, Mathematical Association of America, Washington, DC (2012), pp. 11-18.
(Note that my middle initial is erroneously identified as "W" in that book.)

Andrei Khodulyov's bracings

In an update to Math Magic, Erich Friedman reported better results for bracing regular polygons when the rods are allowed to cross each other. Erich learned from Serhiy Grabarchuk that Andrei Khodulyov had found the following: bracing a square with 15 additional rods, a pentagon with 26 rods, a heptagon with 72 rods, an octagon with 23 rods, an enneagon with 42 rods, a decagon with 45 rods, a hendecagon with 144 rods, a dodecagon with 37 rods.

Gerard 't Hooft's Meccano Math

A related, yet interesting, topic is how to take strips containing holes at fixed integral distances apart and bolt them together to make planar figures such as regular polygons. Physicist (and Nobel Prize winner) Gerard 't Hooft has written a paper, "Meccano Math", about such constructions. Meccano is the tradename for sets of such strips and connectors that have been produced for children for some number of years. (In the United States, such sets were produced under the name of Erector Sets.) 't Hooft develops a theory that allows him form rational numbers and takes square roots, form straight lines, and bisect angles. He gives several constructions that form regular pentagons, one with as few as seven strips, a construction that forms a regular heptagon using only 15 strips, and a general construction that creates a regular n-gon out of 10n-27 strips, whenever n is odd.

Copyright 2001-2012, Greg N. Frederickson.
Permission is granted to any purchaser of Hinged Dissections: Swinging & Twisting to print out a copy of this page for his or her own personal use.

Last updated January 19, 2015.