Comment by yababa_y

Phi is conceptually defined like so:

    Suppose you have a rectangle whose side length ratio is ϕ. You draw a line across the rectangle which divides it into a square and another rectangle.

    Then the side length ratio of the new, smaller rectangle is also ϕ.

The diagram is straightforward to set up:

       a        b
    +-----+--------+
    |     |        |
    |  ϕa-|        |
    |     |        |-b
    |     |        |
    +-----+--------+
     \            /
      -----  -----
           \/
           ϕb

This gives us a system of two equations:

    ϕa = b
    ϕb = a + b

If you substitute b = φa into the other one, you get

    ϕ(ϕa) = a + ϕa

And since a is just an arbitrary scaling factor, we have no problem dividing it out:

    ϕ² = 1 + ϕ

Since we defined φ by reference to the length of a line, we know that it is the positive solution to this equation and not the negative solution.

(Side note: there are two styles of lowercase phi, fancy φ and plain ϕ. They have their own Unicode points.

HN's text input panel displays ϕ as fancy and φ as plain. This is reversed in ordinary text display (a published comment, as opposed to a comment you are currently composing). And it's reversed again in the monospace formatting. (Which matches the input display.)

The ordinary text display appears to be incorrect, going by the third usage note at https://en.wiktionary.org/wiki/%CF%95 )