I’ve thanked them elsewhere, but I feel I need to thank them again and again.

Special thanks go to my wife, Kristen Schaefer, and daughters for supporting me and putting up with my constant daydreaming.

Perry Stoll has devoted countless hours helping me perfect to explore this interesting mathematical structure.  Truly a talented and insightful programmer whose software contributions helped me to have better understanding of the hexagon.

Asbjørn Kvalheim, who on his own initiative, with no more help me other than the underlying algorithm, built a lovely Javascript driven site to visualize plots in the hexagon.  How remarkable to have this useful model – and his always encouraging words – early on.

I also tip my hat to the eminent mathematician David Mumford who kindly read my draft paper, gave me advice, but mostly he simply took me seriously despite my amateur status and unusual work.

Tim Lee, a new friend from Facebook’s Math page, suggested I look at the values of numbers near pi to show significance, and I’m certainly glad he did!

There were many others who helped to varying degrees – usually though enthusiastic encouragement, others through thoughtful discussion; my brother, Darrell Gallion, and my parents, Bob and Carol Gallion, my father-in-law, Bill Schaefer, David Clark, John Buterbaugh, Eric Adams, Marc Anderson, Niel Bernas, Paul Jacobson, Ted Wong (who encouraged me to learn Python and to make the video), Eric Steadman, Conor Thompson (who independently found the modulo 6 activity in the hexagon, something I’d been calling clock math (and thought I’d invented) which relied on base 7 counting, digital roots, and re-baselining the numbers), Ed Ruthazer, Mike and Luke Raffanti, and many others.

And special thanks to the persons I’ve forgotten to list and won’t hold against me.

I salute a number of resources found around the web, that allow us to learn and think in new ways:, Wikipedia, Numberphile, 3Blue1Brown,,, and others that I will add as I remember or discover them.