One of the simplest shapes to be created by math is the so-called “rose curve”, well documented in this Wikipedia page. It’s created by a simple set of equations:
(hate the stupid WordPress scaling, blurs my images) but anyway here’s what the first few look like (k = 1…9):
Pretty boring, huh, so let’s see if we can kick it up a notch.
First, let’s experiment with non-integer k.
Not only does this create more “petals” but it also overlaps them in the interior in interesting ways.
So let’s try making the r have multiple harmonics:
4/5 sin(4t) + 1/5 sin(7t): A bit more interesting.
4/5 sin(3t) + 1/4 sin(3.5t) + 1/10 sin(6t), or perhaps
4/5 sin(4t) + 1/5 sin (3-1/3t) + 1/10 sin(3-2/3t)
And we could go on but it might take a while to find something very different, hence the need for RSE (Rapid Shape Experimentation) as mentioned in previous post.
Or what about taking r to some power?
But let’s also try superimposing some.
And with different petal lengths
or more symmetrically changing petal lengths
or multiple symmetries combined
Can you tell which values of k I used?
Or multiple symmetries with different petal lengths?
Now this should give a flavor of how a rather simple parametric equation can be permuted into a variety of shapes. None of them, however, to my eye even begin to look like a “rose” so I’m not quite sure how this function got named. Even more experimentation than I’ve shown in this post can be done and at some point some interesting shapes are going to be found, just a matter of how patient you are and creative at tweaking tieflume code to generate lots of variations.
So I’ll leave you with this one, you figure out what generated it (and how to make it better).
So stay tuned