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You don't have to use gyroelongation
(lemmy.ml)
image description
Standard "they don't know" meme format, featuring line art of "That Feel Guy" wearing a party hat standing in a corner while other people are dancing. An image of an icosahedron formed by three mutually perpendicular golden ratio rectangles sits in front of That Feel Guy. The caption text says "They don't know that three mutually perpendicular golden ratio rectangles, with edges connecting their corners, form a regular icosahedron."
https://en.wikipedia.org/w/index.php?title=Regular_icosahedron&oldid=1219666251#Construction
Is it possible to create a golden ratio rectangle in CAD using "ruler and compass" techniques, or is numerical approximation necessary?
I don't do CAD, but can it be done with "ruler and compass"?
Yes, as the golden ratio is (1+sqrt(5))/2 and sqrt(5) can be found in a right angled triangle with side length 1,2 and sqrt(5).
With some effort the root of any natural number can be constructed with ruler and compass by creating a spiral of right angled triangles.
The lengths you can construct are exactly the numbers which are composed of 1, addition, subtraction, multiplication, division, and square roots