Explore the spiral structures of daisies and sunflowers.
Run the code to draw a spiral. Try angles other than 5 degrees:
Note: to try other angles, change both occurrences of the angle.
This gives the coordinates of eight points with 45° spacing around a circle (type deg to get the degree symbol (°)):
This draws the points:
Make the angle smaller and the number of points larger to get a tighter circle of points (a lot of the points overlap):
Multiply by the square root of t to make the points spiral out from the center:
Make a spiral with the golden angle. Try numbers of points other than 400:
Plant structures like the daisy flower above are often arranged in spirals whose angle is the “golden angle”:
When you draw spirals with the golden angle, the points pack together evenly throughout:
Make the points larger and red. Try other sizes and colors (like Blue or Orange):
Make the points larger with PointSize (put multiple graphics elements in a list, indicated by curly braces):
Make the points red:
Explore different angles. Drag the slider to change the angle:
Hint: press (or ) for fine control while dragging the slider; press + (or +) for even finer control.
The arrangement you get with the golden angle is very sensitive to changes in the angle. Changing the angle by a tiny amount from the golden angle—just one tenth of a degree here—destroys the nice packing:
Make an interactive interface to explore angles using Manipulate.
Wrap the Graphics expression with Manipulate, replace the fixed angle ga with the variable a, and specify that a ranges from 0 to 360, with an initial value of ga. The option Appearance“Labeled” puts the angle value next to the slider, so you can see how it changes as you drag the slider:
Add controls for the size and number of points:
Replace the fixed point size .03 with the variable s and the fixed number of points 400 with n: