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u/Shadowmaster_70

▲ 18 r/desmos

The Updated Ultimate Guide to The Golden Ratio

This was a silly Desmos project I made in my free time.

I was messing around with equations and I rediscovered The Golden Ratio.

It starts with the equation x/y = (x+y)/x , I then put 1 as y and it gave me the equation x=phi.

I then got the y intersection with the original equation and made that into another equation y=1 then calculated the x intersection with it and repeated this process 14 times.

I also created some borders on top to show each square inside the open shape then got their areas.

I then placed a couple circles fit and cut just right so they fit in the squares aka The Fibonacci Spiral (Approximation of The Golden Spiral).

I noticed how there were lots of Euclidean Triangles embedded in the open shape, I calculated the "diagonals" and the areas of the triangles, and because they are Euclidean Triangles, I compared the similarities in side length and area of the couple triangles I defined.

User u/Circumpunctilious pointed out that The (approximated) Golden Spiral could be expressed with parametric equations, and created an approximation for the spiral.

I then modified it so it's closer to the original spiral.

I wanted to try polar equations, so I started copy pasting a bunch of equations and tinkered with them till I got something very close to the spiral.

In the process, I found that no matter how hard I try, I couldn't get them to fit exactly.

This is because The Fibonacci Spiral is an approximation of the actual Golden Spiral (which I didn't know at the time).

- I'm open to any modifications with explanations.

- I'd love to know more about this topic or tangent topics since I'm still learning (so if you got any tips or info, feel free to share them!)

Updated (5/10/2026): Added the Phyllotaxis / Sunflower spiral shape

Hope y'all enjoy it!

The Golden Ratio

u/Shadowmaster_70 — 4 days ago