There was a young man who was the first in his family to graduate from High School and go off to college. He came out of a rural backwards community. At Christmas break, he returned home to his family. Relatives were crammed into the house. The father, all proud and exited said, "Well son, this is your first semester in college. Say something real smart to the family." The young man replied "πr2" "What!?" exclaimed the father. "Why son, that is the dumbest thing I ever heard you say! Any dern fool knows that pie are round, cobbler are square!"
OK. Let us take a trip to the bowling alley, shall we? Now how many pins are set up? Usually 10! That's right. How many rows are there? Four. We know that 1+2+3+4=10. Ten is a triangle number. What is its triangle root? 4! Did you know that 666 is a triangle number? Yikes! Its triangle root is 36. Why? Because if you add the numbers from 1 to 36, you will get the sum of 666.
Wikipedia has an article on triangle numbers. At the very bottom, they give a formula for testing if a number is a triangle number. If it comes out as a whole integer, it is a triangle. If it comes out with numbers right of the decimal, it is NOT. The same idea would apply for a square root. If I placed a number in the square root sign, and it came out a whole integer, it's a square number. If it has numbers right of that decimal point, it ain't.
My forum http://forums.delphiforums.com/figurate/start also goes into more details.
I just posted charts that give proof that you CAN INDEED have roots of other polygonal numbers besides squares. Just look under "Triangle Root" and "Pascal Genesis" for the most recent postings and you see that the door to yet another world is opened to you. Enjoy!
Actually, they can! Soon, I will be posting a chart at my forum that shows the triangle root, pentagonal root, etc; we simply have the square and cube in their simplest form. After all, Pascal's triangle does not have squares and cubes. It has triangle and tetrahedron numbers.
Please consider visiting our forum. Please let me know if their is any way I can contribute to yours. Say, as you may well be aware, just as one can generate square numbers (n x n), one can also generate triangle numbers. Where any of you aware that just as there is a square root, there is also a triangle root? I'm not kidding.
Good afternoon. A member of my forum that I host referred us to you lovely site. We are looking for formulas that help generate and extracts roots of figurate numbers. It is an interesting but often neglected field of study. You could be the ones who help us make great discoveries.:D You are welcomed to share info with us: http://forums.delphiforums.com/figurate/start
Have a great week.:)