Pictures with equations

Check out this awesome blog post by Richard Clark on the Alpha Blog.

 

Follow the link to see lots of great pictures made with equations. These pictures are so complicated it makes you wonder, is there any picture we can’t make with equations? My first answer is NO.

Think about vector-based graphics. Vector graphics, for those who’ve never heard the term, are pictures/graphics that are stored as a set of instructions for redrawing the picture rather than as a large array of pixels. You’ve used vector graphics if you have ever used clip-art or used the drawing tools in Microsoft Office, or if you’ve ever used Adobe Illustrator, or Inkscape. The advantages of vector graphics include very small files and infinite loss-less resizeability. How can vector graphics achieve this? Well, like I said, vector graphics are stored as rules not pixels. And by rules, we could just as easily say equations.

So the answer is certainly YES we can make any picture using equations. I think the harder question is can we make any picture using ONE equation? Or one set of parametric equations? Or one implicit equation?

What constraints do we want to impose? Do fractals/iterative/recursive rules count?

I am curious to find out how the creators of these picture-equations came up with them. It seems infeasible to do this by trial and error, given the massive size of these equations.

Oh, and if you haven’t yet seen the Batman Curve, you better go check that out too.

 

Food Triad

Have you heard of the “The Incompatible Food Triad” problem? This was first introduced formally here by George Hart (father of the now famous Vi Hart). Here’s the statement, taken straight from his website:

Can you find three foods such that all three do not go together (by any reasonable definition of foods “going together”) but every pair of them does go together?

Hart’s page is full of various suggested solutions, most of which are shot down in one way or another. It’s obviously not a rigorous mathematical question, since much of the success of a solution depends on culturally-defined taste. But it still doesn’t stop us from trying solutions.

pictured above: chocolate and strawberries on a waffle, one suggested pairing. just don’t you dare also add peanut butter!!

Here’s mine. My wife and I sat down to have waffles one weekend and we think we discovered an Incompatible Food Triad: chocolate, strawberries, and peanut butter.

  • Chocolate & strawberries obviously taste good together on a waffle.
  • Chocolate & peanut butter also obviously taste good together on a waffle.
  • Strawberries & peanut butter is okay if you think of it like a peanut butter & jelly sandwich (mash the strawberries!)

But we found that all three did not taste good together on a waffle. A quick web search reveals that people do sometimes eat these three things together, but not on a waffle.Let me make a slightly more rigorous statement of my suggestion then: I claim (chocolate AND waffle), (strawberries AND waffles), and (peanut butter AND waffles) are an incompatible food triad.

Can you see how this is not a very mathematical/scientific question? :-)

Would you agree with our incompatible food triad? Do you have any other suggested solutions?