Last year I finished my 2nd year of studies and up until that point I had never had an experience with coding or any form of LaTeX programs. That was until I was successful in applying for AMSI Vacation Research Scholarship. With the guidance of Grant Cairns and Yuri Nikolayevsky I was opened to a whole new area of learning. Not only was I going to be studying a brand-new maths idea I had never heard of I was also going to be learning a new way to present my findings as well.

AMSI has given me the opportunity to further develop my mathematical skills as well as given me an insight of what maths research is like. And so, I began looking into Escher’s Configurations on Triangles. I started drawing triangles and with the help of Klamkin and Liu’s Theorem I was able to find Escher solutions to specific triangles. What I mean by that is finding ways I was able to draw 3 lines through a triangle and all lines to pass through the same point.

It was difficult at first. I wasn’t understanding the formulas and the values I got weren’t working for me. Again, with the assistance of my supervisors, I was able to break the triangles down to 2 specific cases. Using a right-angle triangle and cutting the 2 shorter sides in half and changing the hypotenuse, I was able to solve for Escher solutions. We found that there are no Escher solutions when p was prime and greater than 3.

This (p,2,2) case gave me the confidence to further the logic to (p,3,2) cases where I found 7 different possible configurations. These 7 different configurations led to 7 different general equations to solve for an Escher solution, all of which, had a possible solution. However, again, if p was prime and greater than 5, there didn’t exist a possible solution.

Overall I have had such a great experience from the AMSI Vacation Research Scholarship and would encourage anyone thinking of applying to do so. I got to get a taste of what PhD life might be like and was able to learn plenty of new things along the way. I would really like to thanks AMSI for the opportunity and again thank my supervisors, Grant Cairns and Yuri Nikolayevsky, for their time and patience with me.

Aidan Brohm
La Trobe University