From the BBC Four program The Joy of Stats*, an excellent, Tuftesque demonstration of the power of visual design combined with the presentation of statistics.

I as much as anybody (more than most, probably) am often distressed and cynical about a world that feels like it’s spinning out of control. It’s nice to see a clear statistical demonstration that things are, in fact, getting vastly better for many millions of people.

*Unfortunately, apparently not (easily) available in the US.

Were you the bored kid in math class who doodled rather than paying attention? (I was — I remember discovering the “crazy checkerboards” and star patterns myself, and still draw Sierpinski gaskets when I need to think-doodle.) Vi Hart’s Mathematical Doodling series is an impressive display of clear and simple explanations of topology, graph theory, etc. that arise from sketching and simple rule-following.

Her website is full of tons of other cool mathematical distractions.

From BoingBoing.

Speaking of the recent 3D visualization of the Mandelbrot set, here’s a really cool zoom-in on the plain, boring ol’ 2D version (maybe you can put on your Avatar glasses and see the 3D version). At the least, you should break out your bong, Doritos, Pink Floyd, and set up the laser light show, because this thing is seriously trippy.

As a reminder, the Mandelbrot set is an infinitely detailed curve surrounding the region defined by those points which, when iterated through a simple equation (z_n+1 = z_n² + c) remain bounded (the black region); and where the colors in the outside region are determined by the speed at which those points escape to infinity. And when they say that the boundary of this sucker is infinitely long and infinitely complex, they ain’t kidding – this video zooms into an almost indescribably small region. I don’t know that this is the deepest penetration into the Mandelbrot set (probably not), but the scale is so mind-bogglingly huge that it’s impossible to mentally visualize: This quick voyage claims to cover 214 orders of magnitude. Now the ratio of the diameter of a proton to the diameter of the universe is 10⁴². Thus, if you blew a proton up to the size of the universe, and then blew a proton in that structure up to the size of the universe again, you’d have to do this five times to observe a level of detail similar to what is reached in this video. (I don’t know how they programmed this thing to hold numbers of this precision, but I’m impressed.) And at this level of depth, we can see that it is clearly just as complex as it was way back at the beginning.

People get hung up on fractals: How can a 2D curve be infinite in length, yet contained within a finite area? I think this video gives something of a glimpse of how that can be.

I can begin to see why gazing so long into infinity can be so addictive; why people calculate pi out to trillions of digits (as in the last chapter of Contact): by around eight or nine minutes into the video, the radial eightfold symmetry of the journey takes on a particularly mesmerizing quality . . . or maybe I just need to wash out my bong.

It’s the birthday of Newton, inventor of physics! He wrote daring equations; confounded his critics!

Actually, the date on the calendar when Sir Isaac was born was Christmas Day, 1642 – hence “Newtonmas.” However, England at the time was still on the Julian calendar, while most of the rest of Europe had converted to the Gregorian. Thus, it was ten days behind in its reckoning of the date at that point, and what was Jan. 4th, 1643 for Rome et al. was Dec. 25, 1642 for the Brits. (Also, it gives me an excuse for procrastinating and writing this post ten days late.)

I recently wrote about the Royal Society of London’s 350th anniversary, for which they’ve made available 60 seminal scientific articles from that span to celebrate the occasion. Over the holidays, I killed some time on and in various airplanes and airport lounges reading some of these, and I thought I’d write a little bit about one in particular: Namely, an article that was, so far as I can tell, the first one that Newton ever published, the beginning of his glorious scientific career, and one of the most impressive papers I’ve ever read.

In this 13-page article, the 29-year-old Newton solves several of the most baffling and long-standing mysteries about the nature of light at the time. He did so through a very simple and straightforward application of the scientific method; through clear and simple reasoning, he was able to come up with an impressively large number of explanations. There is nothing complicated in the paper – no partial differential equations or Ricci tensors; it involves shining lights through pieces of glass and some basic algebra and geometry. This is research that a smart 7th grader should be able to conduct for a science fair today.

But part of its genius was Newton’s ability to observe the simple and deduce the profound where no one ever had before; to step back, disregard common wisdom, and tackle things simply and one step at a time. All with no equipment other than two triangular glass prisms and a window shade. From so simple a beginning, he was able to deduce that:

1.) Prisms diffract light, because circular incident beams emerge creating oblong spots.

2.) That a second prism will reverse the refraction of the first and recreate the spot of light.

3.) That these light rays traveled in perfectly straight lines.

4.) That intervening objects, reflections, or transmissions do not affect the color of this light.

5.) That single colors of light from the prism can’t be individually refracted or combined back into white light;

6.) but that the entire range of colors can. Thus, white light is composed of individual rays of different colors, and these colors are differently refractable.

7.) And the inherent color of all objects is due to their selectively reflecting one or more of the individual colors incident on them – i.e., color is not something an object adds to the light, but an inherent property of the light itself.

8.) Also, this is why rainbows exist. Explained here for the first time.

You’d think that’d be enough for one paper for most people, but

9.) he also takes a two-page sidestep to reinvent the frakking telescope! Yeah, this 29-year-old pauses to say, “Oh, by the way, this is why all existing telescopes produce colored rings around objects, and always will. But if you design them this way instead, the problem will go away.” And all large telescopes today still use this design.

10.) Oh, and another thing: There was no known way to make mirrors for this new kind of telescope of sufficient quality to make them better than the lenses, even with the color problems. That’s OK – he just takes some time off from Cambridge, because of the plague, and invents a new way to polish perfect mirrors . . . at the same time he was inventing calculus!

I’m 31, and I can barely understand the calculus this guy invented on a school break (of course, he wasn’t constantly distracted by the latest iPhone app or YouTube video).

To be fair, I should also point out that Isaac Newton was also a giant anti-skeptic: he was a religious fanatic, producing in his lifetime more writings on theology than science; a huge crank, devoting half a century to alchemical and occult pursuits; and a cantankerous old bastard that no one liked, who prossibly died a virgin. (However, to be fair to my being fair, alchemy was largely inseparable from chemistry at the time, theological pursuits were nearly universal among natural philosophers, and being a completely unlikable jerk has no bearing on whether or not you are correct.)

Anyway, happy 367th birthday, Izzy!

Maybe.

But probably not.

The Voynich manuscript, for those who don’t know, is a fascinatingly enigmatic document that has been puzzling modern puzzlers for almost a century. It is a purportedly 500-year-old manuscript written entirely in an unknown script that has resisted all efforts at decryption over the last five centuries. Its provenance is unclear at best; it has (apparently) been owned by royalty, scholars, and clergy and been lost, found, captured, sold, lost, and found again. It contains copious illustrations of the astronomical, botanical, and anatomical, the latter largely of the female variety. And of the nude variety. Its code has been attacked by myriad hobbyists, by NSA and WWII British cryptographers, and by modern computers. It presumably contains, guarded by encryption in a script of fiendish complexity, information of inestimable value. Or it may be meaningless gibberish. It may have been authored by DaVinci or Roger Bacon; or it may be a hoax.

But someone named Edith Sherwood claims to have at last at least partially broken this enigmatic code. She claims to have discovered that the glyphs are simple substitutions for Latin characters as used in medieval Italian, and that the words are anagrams of medieval Italian words using this substitution cypher.

I don’t know anything about this Dr. Sherwood, but her discussion doesn’t sound like that of an outright crank – her jargon and technique sound on the up-and-up; there is no ranting or claims of conspiracy, or suggestions that the document encodes the secrets of Atlantis. There are, however, a couple of at-least-orange flags which make this story seem doubtful, though:

1. She apparently doesn’t speak medieval Italian (hey, not that I hold that against her – I don’t, either), so her familiarity with the linguistics she’s claiming to translate is presumably minimal. Also, she’s using Wikipedia and online dictionaries and anagram finders as primary translation/deciphering/historical reference tools, which hardly screams “professional” (not that this type of decoding would be something an amateur couldn’t do – only orange flags, as I said).
2. The postulated substitution+anagram encoding is relatively simple; it seems very likely that it would have been detected before by cryptographers, and almost certainly by computer letter-frequency analysis. She doesn’t seem to address this at all, which seems odd.
3. The offered set of decrypted words is both relatively small, and (seemingly – I don’t speak medieval Italian, either) comprised of remarkably few distinct letters. Apparently in Italian you can make almost any plant name with Os, Is, Ls, and Cs – which means that an acronym composed of only those letters can likely be decoded into several or many words.

Still, it’s interesting, and bears keeping an eye on – if she comes up with decipherments of large chunks of the body text of the book, it would be quite an accomplishment!

The Mandelbrot set is a famous and beautiful fractal that has been known for more than 30 years – anyone who went to college, can identify a Deadhead by smell, or ever owned a black light has probably seen a picture of one, or had a poster on their freshman dorm wall.

But apparently the 3D version has been elusive for decades.

The Mandelbrot set is really just a very crinkly (technical word) boundary on the complex plane between points which satisfy a set of equations (the interior) and points which don’t (the exterior, stretching to infinity). Being a fractal, this boundary is infinitely long (though bounding a finite area), infinitely complex, and infinitely self-similar (though never exactly repeating). The psychedelic coloration usually associated with its dorm posters are generally derived from the speed at which the values of each point diverge from the stable solution required to be within the set.

This structure is, however, ultimately 2D – not in the fractal sense, but in the sense of existing in a plane and not in a space with depth and volume and texture and shading. Apparently, it has been very difficult to find a 3D analog of the infinite complexity of the Mandelbrot set. (3D fractals certainly exist – the Menger sponge, Romanesco broccoli, etc. – but these are repetitively, generically self-similar; not infinitely diverse and non-repeating like the Mandelbrot set.)

This is particularly surprising given the simplicity of the generating function for the Mandelbrot set: Any complex number c, where iteratively finding z_n+1=z_n²+c (starting from z_n=0) doesn’t explode to infinity, is in the set. (Part of the problem may stem from the fact that there is no 3D analog to the complex numbers – only 4D analogs and higher. Still, it intuitively seems that the extra space of 3D should allow something similar to the Mandelbrot set – after all, knots can only be formed in “roomy” spaces of 3D or higher.)

Well, it appears that the 3D equivalent to a Mandelbrot set (or something damned close) has finally been found. And it is amazing and gorgeous.

It’s definitely worth a click through to see all of the incredible images of this thing. (The formula for the beast certainly does appear to be appreciably more hairy, though.) And it looks to have infinite levels of unique but self-similar complexity stretching to the smallest levels:

I, for one, can’t wait to get my black light out.