3D Mandelbrot Sets (Sort Of)

Imagine the beauty and complexity of the Mandelbrot set in three dimensions. Fractal enthusiasts are indeed trying to tackle this, as evidenced by this page at skytopia.com.

Okay, technically these aren't true Mandelbrot sets, or even true fractals. Due to a quirk of quaternion mathematics, true fractals generated by quaternions can only exist in even-numbered dimensions of four or higher (quaternions being, by definition, four dimensional), hence these structures are referred to as pseudofractals. When attempting to measure the fractal dimension of pseudofractals, the calculation diverges. (At least this is what I've heard. I've not attempted the calculations myself, but should give it a stab.) Hmmm, space-time is 4D. Possibilities. A time-varying volumetric Mandelbrot set, anyone? Note the expanses of smooth sections. True fractals should be more...foamy.

For something closer to the real McCoy, there are always 3D cross-sections of 4D quaternion Julia sets, discussed here and here.

A basic overview of the mathematics of higher-dimensional fractals can be found here. For a little more depth, read this and this.

I Can't Believe I Totally Missed RH Day!

Yesterday was RH Day, celebrating the 150th anniversary of the publication of the Riemann Hypothesis with a series of lectures world-wide. In conjunction with this, the American Institute of Mathematics (AIM) has publish the AIM Problem Lists, listings of some of the most important outstanding problems in mathematics, which of course includes proving the Riemann Hypothesis.

Proving or disproving the Riemann Hypothesis is widely regarded as the Holy Grail of mathematics. In 1900, the legendary mathematician David Hilbert listed it as being among the most important unsolved problems in modern mathematics. Last year, DARPA included it in its list of 23 Mathematical Challenges. Settling the issue of the Riemann Hypothesis is also among the Clay Mathematics Institute's Millennium Problems.

Stupidity on Parade VIII

To paraphrase a line from a recent episode of "The Guild," there is so much FAIL in that video that I don't even know where to begin. This is a clip of "Dr." Charlene Warner, talking about the "physics of homeopathy," as if homeopathy has anything to do with reality. This is the sort of garbled nonsense that results when pseudoscientists try to use real science to support their woo. This person has absolutely NO clue what she is babbling about. If the stupidity density in her vicinity gets any higher, she runs a risk of collapsing into a singularity, but it will at least be interesting to have a chance to observe the production of Hawking radiation up close...

The Latest Song Stuck in My Head

"Voodoo Child" performed by the Rogue Traders, written by Elvis Costello (well, he wrote a riff sampled in the song), James Ash, and Steve Davis, not to be confused with the song "Voodoo Child (Slight Return)" by the Jimi Hendrix Experience.

Doctor Who fans will remember this as the song playing when the Master unleashed the Toclafane on the Earth in the Episode "The Sound of Drums." (Yes, I'm a nerd. What's your point?)

It is interesting to note that lead singer Natalie Bassingthwaighte (who has since left the band) also appeared in the Australian soap opera "Neighbors." Her appearance was disguised in this video so that the song would be judged on its own merits rather than from a celebrity halo effect. Other notable actors to have appeared in "Neighbors" include Russell Crow, Kylie Minogue, and Dichen Lachman (who is currently appearing in Joss Whedon's "Dollhouse" as Sierra).

Making Lilypond work with TeXShop

I have a confession to make. I have a secret desire to someday write a book on music theory, if only to have the opportunity to start the chapter on key signatures with a quote from Harry Chapin:

Those of you who are musically inclined will note that we have switched to a minor key, which of course means that the plot is about to thicken.

But, hey, how to pull this off? Typesetting a book that would contain equations describing the properties of sound is tough. The best tool for that would be LaTeX (with which I am fond of using the TeXShop front-end). And music engraving is even tougher. Quite frankly, most computer-generated music notation is of very poor quality. Fortunately there is Lilypond, which ostensibly can be made to work with TeXShop.