It is my extravagant hypothesis that the purple gem is relating to people who generally wear rings and such items that have gems on them. The ring is in such a fashion that it is circular in nature. This circle in conjunction with cycles refers to a continuous loop in the programming. With loops established as an important entity, the field must thus be created in the shaped curvatures of an enormous donut. Although, the center circle, often referred to by high ranked historians as the 'donut hole', is not in the shape of a circle. Yes that is right, FIRST is trying to through us off with a doughnut hole not in a typical design, but rather we must look further into the riddle. The pi situation is not only that of a tasty cherry pie that can be eaten through
la bouche. A pie is constructed in a pan in which you can find the circumference by applying a method known as the "angle referencing technique". You simply take two triangles, most usefully the George burns triangle and the John Neun triangle, and add up the sum of the angles you get 360 degrees. With a circle, there is 360 degrees of rotation and with the "angle referencing technique" giving us 360 degrees as well, we can approach the assumption that there will be a triangular donut hole. So now we have a large circular field perimeter and a triangle in the middle. FIRST, I am on to you.
The gem is also known as a purple type of stone that, when light is shone through it, the light is magically transformed into purple. Have you ever mixed red with blue, I bet not, but it becomes purple as well! With a blue alliance and a red alliance, mixing them would cause a purplish tint. The game must some how include some sort of working together between the alliances. What that task is exactly, well, FIRST made it quite clear. If it won’t for my vast knowledge in baseball, we would not be blessed with knowing that George Burns and John Neun are the only players to have completed a triple play. This is not to be looked to far into, overanalyzes can get you nowhere. Thus, to compete a purple task, both alliances must go to the baseball field, tangent to the circle field, and complete a triple play. Ha, FIRST, you can't get me.

If this is your reaction to my hypothesis, then I am sorry for my congruently advanced knowledge of FIRST robotics riddles. I have no choice but to leave you with this simple summary.
In summary of my life long career of studying FIRST, there will be a circular ring with some sort of triangle in the middle where you can complete a task that involves one robot from each alliance partner to achieve extra points.