Comments on: Phi and Fibonacci in Kepler and Golden Triangles

By: Just Cause

Just Cause — Wed, 29 Oct 2025 11:22:55 +0000

I have a bit of an issue with the ‘analogy’ of a Kepler triangle being “in contrast” in a geometric progression; just ‘like’ 3²,4²,5² is arithmatically ? Actually 1,2,3 is, as a arithmatic square sum of their roots. Yes, the roots of Phi are also powers of root phi, but that is sort of missing the point of the special (Fibonacci) case of a √2/3 : √1 : √5/3 rationally adding up as consecutive ‘powers’ , of phi. Their roots do not. Just like 9,16,25 is nowhere near an arithmatic progression. In my opinion you are comparing 2 (totally) different things, even thought 5² – 3² = 2×8 and therefore Phibonacci-alike. Fibonacci & Pythagoras are not the like. Be asured, Mathologer makes (exactly) the same mistake, as his bend(over) method accounts just as well for any (a+b) × (b-a). Not any (root phi) triple in particular. 🤣
https://youtu.be/94mV7Fmbx88?si=bZ8pYAeRkzP0bXtP

By: Mr. M

Mr. M — Fri, 09 Dec 2022 14:47:55 +0000

I have found a different approach
(a+b)²-(b-a)²=4(ab)
(a²+b²)²-(b²-a²)²=4(ab)²
(1²+2²)²-(2²-1²)²=4(1×2)²
5²=4²+3²
13²=12²+5²
34²=30²+16²
(3²+5²)²=4(3×5)²-(5²-3²)²

https://math-journal.blogspot.com/2012/02/fibonacci-meets-pythagoras.html?m=1

By: Mr M

Mr M — Tue, 08 Nov 2022 11:23:20 +0000

144 is the only Fib.square nr. that can be related to 3x 2,3,5,8 (6,9,15,24) or (15-9)(15+9) and a 9,12,15 Pythagoras triangle (9²+12²=15²) but also to a Pythagoras 5²+12²=13² which gives a nice equation:
15²-9²=13²-5²
225-81=169-25
250=160+90

By: Mr. M

Mr. M — Tue, 30 Aug 2022 11:56:40 +0000

In reply to Gary B Meisner.

Somewhat less mathemagical
(b-a)(a+b) = b²- a²
(5-3)(5+3) = 5²-3²=4²
5² = 4² + 3²

By: Gary B Meisner

Gary B Meisner — Tue, 30 Aug 2022 02:49:28 +0000

In reply to Mr. M. There are likely many combinations that will work. The article is just showing how Fibonacci sequence numbers consistently follow this pattern, which is not meant to imply that only those numbers work.

By: Mr. M

Mr. M — Sat, 27 Aug 2022 14:44:27 +0000

ps I notice(d) that it is not typically Fibonacci:
as 7, 8, 15, 23 works too, or 12² + 11 x 35 = 23².

By: Mr. M

Mr. M — Tue, 26 Jul 2022 09:25:09 +0000

I am sort of missing the link between the square sum of the 3 consecutive natural numbers i.e. 9 + 16 = 25 and the sum of the 3 consecutive Fibonacci numbers: 3 + 5 = 8 as 3 times those numbers gives: 9 + 15 = 24 which reveales the similarity a little bit more. as there is only 1 less on either side. Maybe it is all to obvious for math enthousiasts; it really was not to me.

By: Mr. M

Mr. M — Fri, 22 Jul 2022 14:17:53 +0000

I should have used the roots of Fibonacci-products: (2×13),(3×13),(5×13) for all sides, to make it an even better one.

By: Mr. M

Mr. M — Tue, 12 Jul 2022 07:41:27 +0000

In reply to Gary B Meisner.

A right triangle in the Euclidean plane is a Kepler triangle if and only if it is similar to a triangle with side lengths: 1 , sqrtPhi, Phi.
1 Aside, Fibonacci-numbers are (of course) not similar to those dimensions, nor are their roots. In that sense single Lucas nr. can be use to generate Phythagorian Kepler-triangles with similar dimensions The higher the number the closer.
123^0 : 123^1/20 : 123^1/10

By: Gary B Meisner

Gary B Meisner — Sat, 09 Jul 2022 09:41:23 +0000

In reply to Mr. M. Yes, but the sqrt3, sqrt5 and sqrt8 are NOT Fibonacci numbers. They are square roots of Fibonacci numbers, which is not the same thing. Good analysis though!