Projections of [phyllotaxis] patterns

The literature of phyllotaxis is full of baroque descriptions of the features of projections of patterns with golden ratio angles. In the pictures below, the n^{th} point has position (Sqrt[n]{Sin[#], Cos[#]} &)[2π n GoldenRatio], and in such pictures regular spirals or parastichies emanating from the center are seen whenever points whose numbers differ by Fibonacci[m] are joined. Note that the tips of many growing stems seem to be approximately paraboloidal, making the n^{th} point a distance Sqrt[n] from the center.