If you have any knowledge of these famous series of numbers, then you know that they all are closely related in many ways. Obviously the Lucas series is based on the Fibonacci sequence. The Golden Ratio can be found using the Fibonacci sequence rather easily. All of the above can be found in Pascal’s Triangle. My question is this: Did Pascal realize all of these things on his own?
I understand that Pascal did not actually come up with the pattern in the famous triangle, but he is credited with finding many of its properties. I also know that it holds many other patterns. Basically I was wondering if the person who first happened upon this pattern could have possibly understood all of the special properties it holds. Also, does anyone have more information on who actually first came up with the pattern in the first place? Thanks.
This doesn’t seem to be quite what you’re looking for, but it may be of interest to you. The following is from Journey Through Genius by William Dunham:
One of the earlier users of Pascal’s triangle was Omar Khayyam. Incidentally, if you’re interested in learning more about the properties of Pascal’s triangle, one of the sources you can check is my paper “Khayyamic n-simplices: A generalization of Pascal’s triangle”. It’s in the March 1975 issue of The Journal of Undergraduate Mathematics.