Using Trigonometry to Make Flowers

In this post I want to talk a bit about trigonometry and flowers. The basic point is that there are a lot of different ways to figure out where you are. In most math classes we have a couple directions, horizontal (or x) and vertical (or y). Another way to navigate is to start at the middle of the plane, pick a direction and a distance and go that far in that direction.

Occupy math has already noted that, in order to do math
effectively, it is good to select a point of view from which your task
is easier.

The x,y view and the direction-and-distance view of navigation are called coordinate systems. Some things are easier in one of these systems, others are easier in another. The x,y system is called the rectangular or Cartesian system, after Rene Descartes who popularized it. The direction-and-distance system is called the polar coordinate system. The origin or center of the plane is the “pole” of the system. The two coordinate systems are equivalent, in the sense that if you know a small amount of trigonometry and have a bit of common sense you can translate a point in one into the other. That, however, is not my goal today. If directions are taken as angles counterclockwise from due east (the x-axis) then we can get graphs like this:

Distance=Cos(3*Direction) Distance=Cos(4*Direction) Distance=Cos(5*Direction)

These are called petal curves or rose curves. You may notice that the curves with an odd parameter, like 3, have the same number of petals as the parameter. The curves with even parameters have twice as many petals as the parameter, so a parameter of 4 gives 8 petals. This shows that a little basic trigonometry gives us access to the outline shape of flowers. That’s still a long way from building the letters at the top of the page.

Making a solid flower isn’t too hard – you draw a large number of
outlines of increasing size to fill in the flower shape.

The next step is to choose six numbers that give the starting point and wave size of three sine waves. The sine waves control the red, green, and blue color channels to give us a palette of colors, once we combine the three channels into RGB colors, as shown below.

To color flowers, we pick a small segment of the palette and use it to color the outlines that make up the flower, moving along the colors as the outlines get larger. Here are ten flowers from a part of the traveling wave palette, above. Look about how the colors travel across the flowers below.

Now we add a center of variable size to the flower outlines – this amounts to replacing the cosine term with a cosine plus a fixed number. This makes a disk in the center of the flower where the petals start and give us different flower shapes.

Almost there! The last step is that we add some wobble – another sine wave – to the position within the palette of colors depending on the distance from the center of the flower. The two flowers below have no wobble and lots of wobble. The wave that makes the wobble must be much faster that the one that makes the petals.

Okay, now we have a model, using seven different sines and cosines, that lets us make at least slightly plausible – if idealized – flowers. Once the model is worked out we can get the computer to generate thousands of flowers. This is where the “TRIG” picture at the top of the page came from. I took really big, black pictures of letters and dropped flowers so that their centers were always over black. Here is what happens when we flower-blast the letter Z:

Occupy Math hopes you liked the flowers. This post is supposed to reiterate the point that even simple math, like trigonometry, can be  used to do things that can pass for artistic. Occupy Math has almost no formal training in art: we wish artists got more training (and less boring training) in trigonometry and other types of math. Occupy Math has a flower alphabet of 26 capital letters and would be happy to post it if someone tweets or comments a request for it.

I hope to see you here again,

Dan Ashlock
Department of Mathematics and Statistics
University of Guelph,
Ontario, Canada

9 thoughts on “Using Trigonometry to Make Flowers

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