CTK Insights

Until very recently my interest in tessellations was rather utilitarian. I could not help but admire the proofs of the Pythagorean and Napoleon's theorems that used tessellations. I have also illustrated in Java how a certain kind of a hexagon that M. C. Escher mentioned in his notebooks tessellates the plane and the same property of any simple quadrilateral. And that was it, except of course, that Escher's Metamorphosis has been adorning my little boy's bedroom from the time he settled in it for the first time.

Then in the middle of December, 2010, I received a letter from Dr. Paul Brown with an inquiry about hinged tessellations mentioned in one of D. Wells' books. Paul - a very successful math teacher and a frequent presenter at math teacher conferences - himself an author of a remarkable book on proofs.

Paul pointed to what he called the "Dancing Squares Tessellation" and asked whether it is possible to implement hinging in other semi-regular tessellations. At the time, I was about to depart for a winter break trip South to Savannah, GA and decided to take my laptop and this project on the road.

The result is several Java applets that illustrate various hinged tessellations and ways of inserting hinges into an existing tessellation:

• Dancing Squares or a Hinged Plane Tessellation
• A Hinged Realization of a Plane Tessellation
• Semi-regular Tessellation on Hinges A
• Semi-regular Tessellation on Hinges B
• Semi-regular Tessellation on Hinges C
  

Back from the trip, there was a backlog of letters to respond, projects to complete and the necessity of returning to our life routine - the winter break did not last forever. So the matter of hinges and tessellations took, for a while, a back seat. Sometime later I received a message with a link to a hinged tessellation of plus signs. No sooner I received the link as I misplaced it and could not locate it all my efforts notwithstanding. For some reason I was sure that it was from John Smart. As it happened I was mistaken, but John proved to be the right address. He almost immediately responded to my request with the misplaced link.

The plus sign - if blown up - comes to resemble the Greek Cross, and, indeed, this is how the corresponding tessellation is most commonly referred to. So, now I also have a Hinged Greek Cross Tessellation.

Finally, about a hundred years ago it was observed that some materials when stretched become thicker - not the commonly expected behavior. "Fool's gold" is one example. Very recently the scientist found an explanation in a tessellation arrangement of rectangular units.

References

1. J. Sharp, Surfaces: Explorations With Sliceforms, QED Books (May 31, 2004)
2. D. Wells, Hidden Connections, Double Meanings: A Mathematical Exploration, Cambridge University Press (April 29, 1988)

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