Parametric Parrot

A few weeks ago, Jonni Walker retweeted the following, which showed birds creating through use of equations discovered by Hamid Naderi Yeganeh:

Parametric birds. Formulas discovered by @naderi_yeganeh #rstats pic.twitter.com/JdAPkUQ1US

— Marcus Volz (@marcus_volz) March 25, 2018

I replied saying that I would definitely be trying to recreate one of these in Tableau. This week, I finally had a chance to sit down and give it a try. After fighting with the equations a bit, I was finally able to create this:

We tend to think of math as something that is black and white, boring, and always follows the rules. While this can sometimes be the case, math can also be breathtakingly beautiful as these equations demonstrate.

Creating Your Own

If you’d like to create one of these beautiful images, then let me briefly explain how. First, please read my blog, Beyond Show Me Part 3: Parametric Equations. This will walk you through the process of using parametric equations to plot a circle. While a circle is relatively simple, the same basic technique applies to more complex curves. Once you have this foundation, go find an image that’s been plotted using parametric equations and convert the equation to Tableau code. The hardest part of this, by far, is that translation process. For example, the parrot visualized above has the formula:

(3k/20000)+(cos(37πk/10000))^⁶sin((k/10000)^⁷(3π/5))+(9/7)(cos(37πk/10000))^¹⁶(cos(πk/20000))^¹²sin(πk/10000)

y
(-5/4)(cos(37πk/10000))^⁶cos((k/10000)^⁷(3π/5))(1+3(cos(πk/20000)cos(3πk/20000))^⁸)+(2/3)(cos(3πk/200000)cos(9πk/200000)cos(9πk/100000))^12

In Tableau, these translate to the following:

(3*[t]/20000)+POWER((cos(37*[PI]*[t]/10000)),6)*sin(POWER(([t]/10000),7)*(3*[PI]/5))+(9/7)*POWER((cos(37*[PI]*[t]/10000)),16)*POWER((cos([PI]*[t]/20000)),12)*sin([PI]*[t]/10000)

(-5/4)*POWER((cos(37*[PI]*[t]/10000)),6)*cos(POWER(([t]/10000),7)*(3*[PI]/5))*(1+3*POWER((cos([PI]*[t]/20000)*cos(3*[PI]*[t]/20000)),8))+(2/3)*POWER((cos(3*[PI]*[t]/200000)*cos(9*[PI]*[t]/200000)*cos(9*[PI]*[t]/100000)),12)

These certainly look ugly and it can be challenging to perform these conversions, especially when the source equations could be displayed in varying formats, but once you get the hang of it, you can get through it fairly quickly.

If you’d like to give this a try, then here are a couple of places where you can find some beautiful parametric equations:

Follow Hamid Naderi Yeganeh on Twitter.
The Huffington Post has featured a few of Mr. Yeganeh’s equations here: Birds in Flight and Butterflies.
A number of other equations by Mr. Yeganeh can be found on the American Mathematical Society website.
Life Through a Mathematician’s Eyes has some nice equations.
Or just search or “Beautiful Parametric Equations”

As always, if you build something using these techniques, I’d love to see them!!

And, because I enjoy creating these so much, here are a few more created from Mr. Yeganeh's equations.