What's 1/2 Doing in Φ? - the Golden Ratio

[@imaginaryangle]

Note for colorblind people: I noticed that the choice of border and fill color for the isosceles triangle is not optimal 😣, but I think you should still be able to follow this part from the verbal explanation. The triangle is defined by the points (-1/Φ, 0), (1/2, 0) and (0, 1). It becomes clearer when the coordinate system gets turned around to show the matching angles. I'm still learning how to make my videos more accessible to my colorblind viewers and future videos should improve significantly in this regard. Thank you for your patience! 🙏

[@mag-icus]

Always a treat with these videos of yours! I'm eagerly awaiting the next one.

[@imaginaryangle]

Thank you!

[@soninhodev7851]

i know this video just released... but i am already eager for the next, phi is soo wonderful when in relation to its integer powers spoilers, i guess... Φ^n+1 = Φ^n + Φ^n-1 is a formula i played around a bunch, you will notice however, that it is very similar to the fibonnacci formula, in fact so similar that if you do the limit as N aproaches infinity of the Nth fibonnacci number divided by the N+1st fibonnacci number... you get Φ, our beloved, and that is but 1 of the relations between phi and the fibonnacci sequence!

[@imaginaryangle]

This "power series" equality you mention is one of my favorite properties too! I'm looking forward to your comments on the next video in the Φ series! I'll do my best to put it out soon, but it will have to wait for me to finish another topic I'm working on now.

[@chrisg3030]

Φ^(1+1) = (Φ^1)+1

[@chrisg3030]

Here's another bracket shift equation: μ^(2+1) = (μ^2)+1. The character "μ" designates what I call the Bovine ratio, pronounced "moo", since it's the common ratio of the so-called Narayana's Cows sequence (OEIS A000930), just as the Golden Ratio or Φ is that of the Fibonacci sequence Yet another is 2^(0+1) = (2^0)+1. Putting them in order together with their corresponding sequential recursions we get 2^(0+1) = (2^0)+1 A(n) = A(n-1) + A(n-1) Φ^(1+1) = (Φ^1)+1 A(n) = A(n-1) + A(n-2) μ^(2+1) = (μ^2)+1 A(n) = A(n-1) + A(n-3)

[@Kram1032]

you spent a lot of time constructing these triangles, then went "that's not special for this choice of angles" (i.e. it's just true in general for any choice of angle 2alpha + beta = 90°) and then in a single side-remark mentioned almost off-hand why that particular choice is interesting anyways. I feel like the general construction here is more interesting than its connection to phi. "We get the sweetspot where we get the 2:1 relationship in both the domain of side lengths and the domain of angles, and all of this happens around the unit square." - this sentence, I feel, would have needed more explanation. I don't really see, based on what you did here, how the unit square is important to that.

[@imaginaryangle]

The double-angle phenomenon is indeed not specific to phi (and yes, there's a rabbit hole that starts there that I didn't go into). But the special case of this phenomenon for phi ties it together with a half and a whole in both 1 and 2 dimensions, and has a 1:2 ratio both in angles and the sides of one of the triangles. The unit square represents a 1:1 ratio, so you get 1:1, 1:2 and phi related to one another.

[@MichaelBarry-gz9xl]

The Phiangle!

[@imaginaryangle]

I like it 👌

[@PermanentExile]

Fascinating video. Thank you. You can see the similar angles of alpha by noticing that there is a radius from x=1/2 toward x=phi, and x=1/2 toward y=1, which makes the green triangle isosceles.

[@imaginaryangle]

Thank you! Exactly, the matching radii give it away.

[@ojas3464]

👍