Henri's Butterfly

Obligatory Introduction

This document is so aptly named Henri's Butterfly because it is about Henri's butterfly.

To start off, here's a picture of Henri's Butterfly:

Henri's Butterfly

And its equation, of course: r = (sin(4t))^2 + cos(3t).

Yes, the lack of information of this particular plot did occur to me, in fact, my previous knowledge of this plot was little more than its equation. I originally encountered it in the book Mathematical Sorcery by Calvin C. Clawson.

Origins & History

A description about its origins can be found below by Mr.Henri Berger himself. He says it best himself:

Date: Wed, 16 Apr 2003 14:05:57 -0700

Subject: Thanks for publicizing "Henri's Buttefly"

From: "Henri S. J. Berger"

To:



Hi Yi Wen,

I happen to be the one who originally came up with the equation for "Henri's Butterfly" while I was taking an undergraduate Calculus math course at UCLA in 1988 with David Cohen who was the lecturer (whom, unfortunately, I just discovered had passed away last year). I wrote a Mac program in BASIC to perform the polar plotting and just started plugging in different equations. I showed a few of my plots to Mr. Cohen who liked one of the figures so much that he decided to publish it in two separate pre-calculus books that he published. Mr. Cohen is the one who came up with the name, "Henri's Butterfly."

If you would like to add this description to your web page, you have my permission. I would prefer that you not refer to my email address but you may use my name. (I believe the book mentions my name in the margin)

After attending a book tour lecture with Stephen Wolfram (yesterday) and seeing some of his mathematical plots, I decided to do a search on "Henri's Butterfly" to see if it was ever mentioned on the Internet. I was very surprised to see your page! I agree with you it is a really cool plot.

-henri berger

Here's the original description from my old page for uh... historical reasons:

What can possibly be uncool about Henri's Butterfly?

It looks great, the equation is simple, it comes in a variety of flavours... It's cool enough to be my mascot. And not just anything can be my mascot.

In fact, IMHO, I think it looks cooler than the dark, dark Heart of Mandelbrot (but its more than meets the eye), which reminds me too much of lacy valentine hearts for my taste. What's more impressive is that the mandelbrot heart is fractals, while Henri's Butterfly is a mere function on polar coordinates. Cool, huh?

The equation for Henri's Butterfly is none other than: r = (sin(4t))^2 + cos(3t).

And now we know.
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