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Richard Clark

Mathematics as an Art Form—Visualizing Equations

January 8, 2013 —
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When I was younger, I held the naive and incorrect view that mathematics was divorced from the arts. As I’ve gotten older, I’ve become more aware of not only how mathematics is the foundation for any of the hard sciences, but also how it is intrinsically linked to essentially any form of creativity. Certainly users of our Wolfram Music Theory Course Assistant could have told me that, but I’m not just referring to music. In truth, I’m not even trying to make some highbrow appeal to abstract art, either, although I happen to rather like that sort of thing. What I’m trying to say is that mathematical equations can make pretty pictures.

butterfly curve

How awesome is that? And you know, as much as I adore butterflies—and I love them with all of my heart—I like other things that fly, too. Many people were enthralled when we showed off the bat insignia, but while I appreciate bats and the men who follow them, in truth I’ve always found a deeper beauty in men who can fly, and without the aid of machinery. They’re super.

Superman insignia

That said, to go back to the introduction, music and mathematics do go hand in hand with each other. It’s argued that mathematics is the basis of sound, in fact. That could be true, but what I know for certain is that I see a red door and I sincerely wish it were painted black.

Rolling Stones curve

A thing I learned from a reliable source several years ago is that if you leave something that has a battery—a laptop or, say, an electric razor—plugged in too long, the battery begins to lose its zero point. Its total capacity will begin to drop. I believe an expert on electricity can elucidate further on this matter.

Pikachu curve

Electricity gets me charged up, so let’s talk about  something polarizing, like ethics. Wolfram|Alpha defines ethics as either “motivation based on ideas of right and wrong” or “the philosophical study of moral values and rules.” Ethics is one of my favorite topics of discussion, and the next expert adviser we’ll be bringing in certainly has some passionate viewpoints related to what he suggests are the light-side and dark-side binary.

Darth Vader curve

Stay tuned to future posts showing off more curves and laminae, as well as a discussion on how we make them. Until then, you can try to find some of our curves and laminae yourself. We have famous people, fictional characters, animals, logos, and more. Be sure to share your favorites with us on our Facebook page and Twitter account. Happy exploring!


Astonishing way for drawing the outlines of such beings, astonishing approach that calls attention and admiration.

Is it possible to tell the people who did you do to start that “play” with some simple equations to get that complex formulas that correspond to lovely creatures like butterflies?

Posted by Mohamed Fahmy Husssein January 8, 2013 at 1:05 pm

Is there a course or a tutorial on how to make art with equations? I want to join in on the fun!

Posted by Adam Dreaver January 8, 2013 at 1:49 pm

Wow amazing post… I’m acutely dyslexic, while failing school and missing out on basically every fundimental in the book i played chess and lots of it, which lead me to playing high stakes online poker. At school i was regarded as a moron, now im regarded many many levels above my peers with my success. But still couldnt write a basic formula to save my life. I just visualize everything I can barely explain it. But i would watch numbers change useing a live database program while playing, which lead me to just see how to manipulate statics, and now have a strangely high level of understanding of maths in the most none classical of ways.

Great post.

Posted by Jesse maguire January 15, 2013 at 4:02 pm

[…] shouldn’t be that surprising, especially after looking at the people and fictional characters we’ve turned into mathematical curves on Wolfram|Alpha. Our curves of internet memes, cartoon and video game characters, celebrities, and mathematical […]

Posted by Set the Curve with Wolfram|Alpha!—Wolfram|Alpha Blog August 12, 2013 at 7:34 am

In logarithmic functions there is sometimes a term that may have a “subscript” verses a superscript. For example, 2logsubscript7 x+25=19. How would you input the 7 as a subscript.

Posted by Terence February 5, 2014 at 11:07 am

    Terence, if you wanted to do something like 2log(sub. 7)x + 25 = 19

    You need to take into account that log(sub. x)y = ln y/ln x, therefore log(sub. 2)16 = ln 16/ln 2 = 4.,.. which is what log(sub 2) 16 is…,

    So What you would need to do is the following…

    2 * ((ln x)/(ln 7)) + 25 = 19

    Check out to see the result of the graph…

    Posted by Eliseo May 2, 2014 at 7:47 pm

cum se calculeaza eroarea unei valori logaritmice?

Posted by mihaela June 27, 2014 at 2:35 pm

My butterfly’s more geometric. -ish 😉

PolarPlot[(Abs[Cos[10 Pi/12]] + Abs[Cos[14 Pi/12]] + Abs[Cos[18 Pi/12]]) / (Abs[Cos[x + 10 Pi/12]] + Abs[Cos[x + 14 Pi/12]] + Abs[Cos[x + 18 Pi/12]]) – (Abs[Cos[12 Pi/12]] + Abs[Cos[18 Pi/12]]) / ((Abs[Cos[x + 12 Pi/12]] + Abs[Cos[x + 18 Pi/12]])), {x, 0, 2 Pi}]

Came out of another thing I’m working on. Thought it was pretty cool…


Posted by Michael Gmirkin July 27, 2014 at 4:26 pm

I’m interested in this “Visualizing Equations”, any idea on what topics of Mathematics are these associated to? I just can’t seem to find it in online resources

Posted by Neil December 11, 2014 at 4:45 am