So... does the Teglon have to do with sending a message/signal/call from Arbre to Urnud? The arrival of the Geometers at Arbre was the Fourth Advent. Jad made the fourth solution to the Teglon at Elkhazg
Intriguing. Who knows what Jad was up to. To solve the Teglon, he had to envision the completed solution. When he placed a tile in the middle, he had to also know what effect that would have on the edge many tiles away. Was he also envisioning the "butterfly effect" all the way to the Geometers' Ship?
No. The message was an analemma (the figure eight on the cover). Orolo sent the analemma by tracing it out in the sky with the guide lasers for the directional telescope. The analemma is the pattern traced out on the floor of the temple of Orithena by the sun's light going through the oculus in the ceiling.
The Teglon is a of the class (NP-complete) such that it is easy to verify a solution, but cannot be reduced into smaller pieces which may be solved independently. These are the sorts of problems that quantum computers can outperform silicon, since quantum computer memory exists in the superposition of all possible solutions or in the polycosmos/many-worlds interpretation it exists in many universes at once. Thus the point here is that Fraa Jad is able to use his brain as a quantum computer to solve the Teglon.
To create a teglon, you'd need a computer that could generate arbitrary penrose tile sets, then trace a single unbroken line at random across the tiles. Then scramble the tiles. The question then becomes, are there any solutions to putting the puzzle back together other than the one originally 'discovered' by the computer?