Here’s a video my dad and I shot in 2013, when I was 13, on the subject of paradoxes, which I find pretty interesting. We discuss and show examples of paradoxes from the worlds of math, philosophy, and pop culture:

On Paradoxes

Below is a critique of a Pi visualization mural planned by the Green School in Brooklyn, funded by the Kickstarter community. In this 12 minute video, I explore the difference between the Fibonacci and Golden spirals, and explain why neither one of them is helpful in visualizing Pi (as planned by the Green School). I’d like to thank my calculus tutor, Jono Schrode of Mill Valley Math, and my media studies tutor, Craig Sarachene.

Critique of Green School Pi Visualization Mural

Pi Day (March 14th, i.e. 3-14) is the greatest holiday of the year, in this or any other culture. Math ninjas, please enjoy my past Pi Day videos!

Pi Day 2012 – “Taking Heron’s Fortress – A Trigonometric Assault”
Pi Day 2010 – “Binomial Theorem Delirium”

There’s also one of me deriving the quadratic formula when I was 8. If I can stand my squeaky voice I’ll post that one too.


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  • Skye,
    I suggest that you write a follow up note to the Green school, suggesting that they include at least one prominently located image of a circle in their mural that clearly shows pi as the origin of the inspiration for the rest of the mural.
    Tom Singer

  • On Apr 2, 2013, at 6:02 PM, Kickstarter wrote:

    Ellie Balk says:
    Hi Lou and Skye!

    Thank you so much for your thoughtful feedback and intrigue in our project. I have to tell you….I am an artist, not a mathematician. When I was your age Skye, math did not come very easy to me. In order to understand it, I had to build a visual language. I still do this now as a professional artist. The math teacher I have been collaborating with over the past five years has become a kind of tutor. When we plan a project, he shows me what he is teaching his students and I pull inspiration from that. When we met to talk about Pi, the conversation of the PHI came up. I was excited to put the two ideas together to show PI within PHI. I toyed around with circular patterns, but I loved the shape of the Golden Spiral and how it showed infinite space. I also liked how the Golden Spiral was a visualization of the Fibonacci sequence. I do understand that the lines should get thinner as they get smaller, but taking artistic license…I thought it looked kind of like a s kyline moving towards the inside and I really liked that idea for a community mural. The whole business about seeing the negative space of Pi is me just getting excited about the image and seeing Pi in a whole new way!

    I showed you video to the math teacher Nathan Affield and here is his response:

    We’re trying to get our students to act like mathematicians. One important way that mathematicians differ from math students is that they make conjectures to be proven or disproven by themselves or others. In one of my favorite moments from your video – which I did really enjoy – you state that there is no known connection between pi and phi. That’s why we chose it! As the oldest of the sciences, it’s sometimes hard to generate original explorations accessible to students without advanced math backgrounds, but in this experiment we use only pre-calculus concepts to engage students in something in math that – to the best of our knowledge and apparently yours – has never been tried. The fact that this exploration was interesting enough for you to try it for yourself is an accomplishment for us. And, like mathematicians throughout history, we’ve created the discourse in the proving and disproving of conjectures which is the foundation upon which the body o f knowledge we call mathematics was built.

    Thank you again…You are a very smart kid and have really great ideas! You should totally come to Brooklyn and paint with us!! One question….how did you make those computer generated visualizations you showed of Pi and the random numbers..that was really cool!

    We would be happy to continue the conversation and stay in touch!

    Thanks again,
    Ellie Balk

    • On Apr 5, 2013, at 4:41 PM, Lou Hawthorne wrote:

      Hi Ellie –

      SKYE: If you call what you’re doing an “artistic interpretation,” then the flaws in the math become less critical. I really like the urban “skyline” idea; that is really appropriate given where the mural’s going to be – and again it’s a huge improvement on the prior wall. But since you say your intent is to “visualize Pi”, then those words carry weight and the math really does have to support the stated goal. As pretty as what you’re doing is, it doesn’t clarify what Pi actually is.

      LOU: I very much appreciate your intellectual honesty in revealing both your math knowledge as well as the role of artistic license in your process; our concern is that the result may be a “Wall of Confusion”; people who think they understand Pi (and actually do) will pass your wall and then not be so sure. Use of the Golden Spiral clearly implies a pattern within the digits of Pi, which does not exist. This is simply not open to subjective interpretation any more than the fact that the Earth is round.

      SKYE: It’s cool that Mr. Affield wants to teach his students to think beyond what’s been proven. However, when doing so, you can’t ignore what HAS been proven over the centuries – such as the irrationality of Pi. The definition of “conjecture” is “An opinion or conclusion formed on the basis of incomplete information.” I don’t think Mr. Affield means “conjecture”; I think he means “hypothesis”. However, a hypothesis must be testable, and this mural does not support testing of any math hypotheses, because the representations are not mathematically rigorous.

      LOU: In that case, maybe “conjecture” really is the right term. There’s nothing wrong with opinion as long as it isn’t presented as something other than that. The problem arises from statements regarding the visualization of patterns within the digits of Pi as if this is a mathematical possibility, instead of a conjecture as formally defined.

      SKYE: But even if he did mean hypothesis, I don’t agree that that’s what math is about. Math is not a guessing game; math is a sequence of proofs based on prior proofs, which in turn are based on axiomatic discoveries. This mural basically involves speculating about connections instead of proving those connections.

      LOU: The proof you seek may be occurring in the Green School’s process around the mural-making process itself. Mr. Affield suggests as much by referencing “the discourse in the proving and disproving of conjectures”. This is similar to how conceptual artists view the role of the art object itself, i.e. as almost incidental to the dialogue around the art. In that sense, this mural is really a collaboration between two artists – or perhaps an artist and a philosopher?

      SKYE: So if the goal is to produce a work of conceptual art, they shouldn’t change a thing, and neither should we in our critique. It will be a dialectic; the truth in the art will be the debate around the art, which includes our critique, and the critique of others.

      With warmth and respect from both of us,
      Lou and Skye