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The Teglon is a complex problem of geometry and tiling which was originally developed at the Temple of Orithena on Ecba. The problem is more than a theoretical exercise, an open space known as a Decagon is used to hold the physical tiles of the Teglon and immortalize a correct solution.
Seven varieties of Teglon tiles are used, each of which contains a groove or line on their top face. The objective is to create a pattern of tiles which provide an unbroken, continuous line over a large surface. The tiles are such that a solution is aperiodic and mathematically non-trivial.
To find a correct solution to the Teglon is extremely difficult. The aperiodic nature of the tiles mean that the solution requires a holistic awareness of the entire tiled surface. At Elkhazag Fraa Jad is able to find a correct solution and lay down the tiles in a single night, an unprecedented feat and only the fourth solution found at Elkhazag. It is later implied that Fraa Jad is able to do this because of the unique abilities of Thousanders to sense different cosmi.
The Teglon is a problem legendary for its complexity and a number of legends have formed around it. Metekoranes is said to have been contemplating the Decagon in the Temple of Orithena when the volcano on Ecba erupted. Rabemekes was working on the problem when he was killed by a Bazian soldier, a tale which echos the real-life story of the death of Archimedes at the hands of a Roman soldier. Another Arbre legend is that of Suur Charla who sketched a solution in the dust of a road only to have an army march over it.
Real World BackgroundEdit
The Teglon is similar in some respects to Penrose Tiling. The groove or line that is present in all tiles is likely inspired by Conway loops and Ammann bars that are used to decorate aperiodic tilings in the real world.
Neal Stephenson has said:
A mathematician could probably prove that my version of it is not merely fictional but ridiculous; but the same mathematician would, however, readily agree that aperiodic tilings are an important and fascinating branch of mathematics.