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.:)
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.
If you meant cube root, most of us are aware that there exist cube roots. Regarding extracting cube roots, I think of prime factorization as the only possible method. Logarithms help, but they are an approximation. For higher roots also, prime fasctorization is the ideal method.
PS:- The forum you host is quite interesting.
It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Yes, we know about the cube root. How about the tetrahedron root, octahedron root, etc;? The extracting of 3D figurate numbers is what has me stumped. I'm glad you like my forum.
Roots aren't given in shapes with different number of sides. They are all cubes of different dimensions. A 2-dimensional cube is called a square, and so we say a number is squared. A 3-dimensional cube is simply called a cube, so we say the number is cubed.
Why? Because if you take some length x, x*x is the area of a square and x*x*x is the area of a cube.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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.
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!
Perhaps you can define things such as "triangle" roots, and the like, because I (nor Google) haven't the slightest idea what you mean by such.
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.
Last edited by R3hall (2007-07-04 15:02:22)