Hacker News Reader

Comment by pgreenwood | Hacker News Reader

That is neat, I did not know this method of constructing a gold ratio. Once you have a golden ration it's easy to construct a pentagon (with straight-edge and compass).

View on HN

I'm not familiar with this pentagon trick. Care to elaborate?

The ratio of the length of the diagonal of a pentagon to one of its sides is the golden ratio -- easiest visualization is with similar triangles. Draw a regular pentagon (sides of length 1 for simplicity) and pick a side, make an isosceles triangle with that side as the base and two diagonals meeting at the opposite point. Go one side length down from the opposite point and mark that (F below). Convince yourself that triangle DCF is similar to CAD (symmetry gets you there).

Now we wish to find the length of, say, CA. From similarity CD/CA = FC/DF, and CD = DF = 1, and CA - FC = 1, so the ratio simplifies to... CA^2 - CA - 1 = 0 which yields the golden ratio.

            A
           .'.
         .' | `.
       .'  | |  `.
    B.'    | |    `.E
     \   F|   |    /
      \   |   |   /
       \ |     | /
        \|_____|/
        C       D

Reply on news.ycombinator.com