Graph Florets of a Flower
TI-BASIC, 34 bytes
For the TI-83+/84+ series of calculators.
Input N
2πe^(-2sinh⁻¹(.5→θstep
AnsN→θmax
&quot;√(θ→r₁
Polar                      ;Displays polar mode graphs to graph screen
Dot                        ;Prevents lines from connecting points
DispGraph                  ;Displays graph screen

This considers the dot at the origin to be the 0th dot.
Thanks to the one-byte sinh⁻¹( token, 2πe^(-2sinh⁻¹(.5 is a short way of getting the golden angle in radians. This is derived from the fact that e^(sinh⁻¹(.5 is the golden ratio.
Here are screenshots for N=50.
(Yes, that's a 96x64 monochrome display on a TI-84+. The newer color calculators have a resolution upgrade, but still have only 3.7% the pixels of an iPhone.)

Press TRACE to step through each point.