So I took out compass and paper and constructed Golden Rhombuses. (OK, I got the first two wrong... but the third one worked.) You can build a Rhombic Triacontahedron with Golden Rhombuses.
To build the rhombus ... get a compass, a ruler, and some paper.
- With the ruler, draw line AB.
- With the compass, draw a circle centered on line AB. The center of the circle will be point a.
- Draw an identical circle centered where the first circle intersects line AB. The center of the second circle (also where the first circle crosses line AB) will be point b.
- The two circles will cross each other twice; once above line AB, once below line AB. Use the ruler to draw a second straight line over these two places. The straight line will cross line AB at point c. Point c will be the mid-point between points a and b.
- Draw another identical circle centered on point c.
- Open the compass wide (I like to double it) and center it on point b. Draw an arc above and below line AB. Place the compass point on the other side of the first circle and draw another arc above and below line AB. This will create two points directly above and below point a on line AB. Draw a third straight line through the three points (the two sets of arcs and point a). Where the new third line meets the first circle is point d.
- Center the compass on point c and open it until the pencil touches point d. Draw a circle. The forth circle should cross line AB just to the left of the center of the first circle (which isn't labeled in these instructions). Call this point point e.
- Use the compass to measure the distance from point b to point e. Using this distance, center the compass on point c and draw a fifth circle. Where this circle crosses line AB, mark points f and g. (The whole point of drawing five circles and three lines is to get the distance ad and be, which can be used to create a Golden Rectangle)
- Where the vertical line crosses the third circle (running through point c), mark points h and i.
- Draw the rhombus through points f, h, g, and i.