305
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
The lengths you can construct are exactly the numbers which are composed of 1, addition, subtraction, multiplication, division, and square roots