Poincaré Conjecture: Controversy and Eccentricity

(Image by crdotx)

(Image by crdotx)

Evidently there’s a newish biography out about Grigori Perelman, the man primarily responsible for solving the Poincaré Conjecture. Masha Gessen, a Russian journalist and author, has released “Perfect Rigor: A Genius + the Mathematical Breakthrough of the Century”, a work about the life of the curious mathematician who has vanished from the professional math community. She explains her work in an interview with failuremag.com, in which she describes the rationale behind the work, and gives some insights into both the Poincaré Conjecture and the life of Perelman.

There’s no doubt that Perelman’s response to solving one of math’s longest standing problems is part of what is so intriguing to the lay reader. He turned down a Field’s medal and withdrew from professional mathematics. Pieces on him tend to accentuate his eccentricities. Part of what makes the story so very interesting is Perelman’s response, as well as the other cast of characters involved in the solving of the problem. Though it came out in mid-2006, the New Yorker has a fantastic article about the solving of the Poincaré Conjecture and the controversy surrounding it. If you’d like to understand to a greater extent Perelman’s response to his solution, read this article. It’s long, but extremely informative and complete.

Crossword Puzzle Copies?

(Image by chipgriffin)

(Image by chipgriffin)

Matt Gaffney, a 15 year veteran of professional crossword puzzle writing, wrote an article in late November for Slate about the likeliness of two crossword puzzle creators replicating the same, or approximately the same, puzzle. Crosswords exhibit 180-degree rotational symmetry, meaning that “if you turn the grid upside down, the pattern of black squares will look the same as it does right-side up.” Couple that with a specifically themed puzzle (e.g. Halloween and Edgar Allen Poe), a certain number of long word entries, and general crossword rules (e.g. no two letter words are allowed), the probability may be higher than you think.

In a nutshell, Gaffney found that he had inadvertently used many similar aspects of a puzzle that had been released earlier in the year, and thus traces several reasons why this may have been the case. One interesting twist to the story is what happens when Gaffney asks a third crossword creator to write a similarly themed puzzle. Do you think that this third puzzle turned out to have similar entries to the first two? Read to find out!

Self-Interacting Machines

(Image by Solarbotics)

(Image by Solarbotics)

As I look through articles that I’ve bookmarked over the last many months, I realized that I had saved two that concerned machines that were built to interact with themselves in novel ways. I find this self-interaction quite entertaining for a reason I can’t quite pinpoint. Perhaps it’s because I generally think of a machine as a black box that takes an input and produces an output. I tend to segregate the input and the output entirely from the black box. But these machines are built only with the purpose of performing an action on themselves. They are input, output, and black box. And I find that extremely entertaining.

First, take this iteration of Claude Shannon’s “Ultimate Machine”, whose sole purpose when turned on is to immediately turn itself back off again:

As the video explains, you can find plans for building your own machine of the same type at Instructables. It should be said that I was introduced to this machine via Boing Boing much earlier when they directed me here. These folks call the machine the “Leave Me Alone Box”, and there are several videos on the site devoted to showing different fan-made machines in action. It looks like this site will also eventually sell a kit which allows you to build your own box. Kevin Kelly also has an article that gives some history about this “Ultimate Machine”.

Secondly, take Michael Kontopoulos’ machines that are built only with the task of nearly knocking themselves over:

The artist explains his intentions on his website as trying “to capture and sustain the exact moment of impending catastrophe and endlessly repeat it.” I love it! So close to falling, and yet so far away! There are lots of other interesting projects to find on Michael’s website as well.

Alice, Wonderland, and Math

(Image by pepsiline)

(Image by pepsiline)

Having just completed both of Lewis Carol’s books concerning Alice and her adventures in Wonderland, the recent Boing Boing post about Melanie Bayley and her research into the idea that scenes were added into the narrative after the initial draft in order to mock new math of the day, namely symbolic algebra. As one example, Bayley likens the Mad Hatter tea party scene to the concept of the quaternion introduced by William Rowan Hamilton. Without giving away the punchline, Bayley paints an interesting picture of why the three guests at the tea party are stuck at their table, constantly swapping seats. Read the full article at New Scientist here, which gives many more examples of how Carol lampooned the so-called “new math”. Who likes imaginary numbers, anyway?

Babbage’s Difference Engine

(Image by dannyman)

(Image by dannyman)

My mother-in-law made me privy to a story that aired on NPR about a group of people who built a copy of Charles Babbage’s Difference Engine using only materials available from the Victorian age, which is a feat which alluded the mathematician during his lifetime. This machine is the second of two that has been built, and is on display at the Computer History Museum in Mountain View, Calif. through the end of 2010. Concerning the physical dimensions of the machine:

“The Difference Engine fills half a gallery and stands taller than most men. It’s 5 tons of cast iron, steel and bronze woven together from 8,000 distinct parts. Though it looks like it could be a sculpture, the machine is essentially a giant calculator.”

In other words, it’s gigantic. And it works. It was the best computer that money could buy in 1840, which is probably why it was never actually built. Way too complex and way too much money. It’s worth checking out the NPR story just to see the photos of this monstrous machine. There’s also a video of the machine in action on the Computer History Museum webpage for the Babbage Engine exhibit here.

Book Review: Logicomix

(Image by Eusebius@Commons)

(Image by Eusebius@Commons)

Ever since I heard about Logicomix, a graphic novel about the 20th century search for the foundations of mathematics, I was extremely excited to read it. And I’m happy to say that now I’ve finished it, it most certainly met, and exceeded, my rather high expectations.

First, I think it’s appropriate to explain what Logicomix is, and what it is not. First, it is a work of fiction. The authors are clear on this point, and explicitly spell it out in the epilogue. This graphic novel is meant to be a piece of art, not a pristine account of history. But its account of history will definitely give the reader an accurate outline of what happened in mathematics in the 20th century, even if the details aren’t entirely accurate, and even if certain liberties are taken in order to present a story which engages the reader in its narrative format.

Logicomix tells the story of 20th century mathematics by using Bertrand Russell as a narrator. The story is framed as a talk which Russell is giving to an audience directly before Britain’s entrance into World War II. He traces his life’s story from childhood through his work as an activist, and along the way we meet the familiar set of characters from that day, including Frege, Whitehead, Gödel, Wittgenstein, Cantor, etc.

The story is compelling. The authors, Apostolos Doxiadis and Christos Papadimitriou, do an amazing job of following Russell’s career, and of showing the personality and human struggle behind a quest centered around the abstract notion of truth. While most of the characters are obsessed with matters of the brain, the story also focuses on the heart as well. For instance, Logicomix does a good job explaining Russell’s obsession with madness, and his fear associated with it. There is also an aspect of meta-story in Logicomix, whereby the authors themselves appear throughout to drive home points and debate various tidbits of the story. This is an interesting aspect that mirrors to some extent Gödel’s meta-logical statements. I thought this was a good touch.

The art, done by Alecos Papadatos and Annie Di Donna, is stunning. As a material object, this graphic novel is beautiful. I’m a huge proponent of quality, and everything about the novel breathes quality. From the art to the color to the fonts to even the dimensions of the page and panels, Logicomix is extremely well engineered. Kudos to them both!

As a work of historical fiction, this story should appeal both to lovers of logic and mathematics as well as those who enjoy a fantastic story. The authors are using history to drive home what I believe to be an accurate and well-stated point, which I will not divulge here. If there are any flaws, I would point out two. First, Gödel is treated somewhat like a deus ex machina. Rather than providing more exposition into his life, the authors are content to have him come and go quite quickly, though I must admit that I LOVED the first panel in which he appears. It’s probably my favorite panel in the entire work. Secondly, there’s nothing about Turing, other than to mention him in the meta-story toward the end. One cannot truly understand logic without knowing something about computability. Granted, it seems that a sequel may be a possibility which begins with Turing and ends in the 21st century, but only time will tell. Until then, Logicomix comes as highly suggested.