**Physicist**: There are a couple of different contexts in which the word “dimension” comes up. In the case of fractals the number of dimensions has to do (this is a little hand-wavy) with the way points in the fractal are distributed. For example, if you have points distributed at random in space you’d say you have a three-dimensional set of points, and if they’re all arranged on a flat sheet you’d say you have two-dimensional set of points. Way back in the day mathematicians figured that a good way to determine the “dimensionality” of a set is to pick a point in the set, and then put progressively larger and larger spheres of radius R around it. If the number of points contained in the sphere is proportional to R^{d}, then the set is d-dimensional.

(Left) as the sphere grows, the number of points from the line that it contains increases like R, so the line is one-dimensional. (Right) as the sphere increases in size the number of points from the rectangle that it contains increases like R^{2}, so the square is two-dimensional.

However, there’s a problem with this technique. You can have a set that’s really d-dimensional, but on a large scale it appears to be a different dimension. For example, a piece of paper is *basically* 2-D, but if you crumple it up into a ball it seems 3-D on a large enough scale. A hairball or bundle of cables seems 3-D (by the “sphere test”), but they’re really 1-D (*Ideally* at least. Every physical object is always 3-D).

A “crumpled up” set seems like it has a higher dimension than it really does. You can get around this by using smaller and smaller spheres. Eventually you’ll get the correct dimension.

This whole “look at the number of points inside of tiny spheres and see how that number scales with size” thing works great for every half-way reasonable sets. However, fractal sets can be “infinitely crumpled”, so no matter how small a sphere you use, you still get a dimension larger than you might expect.

The edge of the Mandelbrot set “should” be one-dimensional since it’s just a line. However, it’s infinitely twisty, and no matter how much you zoom in it stays just as messed up.

When the “sphere trick” is applied to tangled messes it doesn’t necessarily have to give you integer numbers until the spheres are small enough. With fractals there is no “small enough” (that should totally be a terrible movie tag line), and you find that they have a dimension that’s often a fraction. The dimension of the Mandelbrot’s boundary (picture above) is 2, which is the highest it can be, but there are more interesting (but less pretty) fractals out there with genuinely fractional dimensions, like the “Koch snowflake” which has a dimension of approximately 1.262.

The Koch snowflake, which is so messed up that it has a dimension greater than 1, but less than 2.

That all said, when *somebody* (looking at you, all mathematicians) talks about fractional dimensions, they’re really talking about a weird, abstract, and not at all physical notion of dimension. There’s no such thing as “2.5 dimensional universe”. When we talk about the “dimension of space” we’re talking about the number of completely different directions that are available, not the whole “sphere thing”. The dimensions of space are either there or not, so while you could have 4 dimensions, you couldn’t have 3.5.

According to Falconer, one of the essential features of a fractal is that its Hausdorff dimension strictly exceeds its topological dimension.^{[1]} Presented here is a list of fractals ordered by increasing Hausdorff dimension, with the purpose of visualizing what it means for a fractal to have a low or a high dimension.

As musical old-timers repeatedly sing the sad song of the supposed demise of the full-length album, a funny thing has happened. Lovers of games have taken up a growing passion for game music, and in particular the indie score for indie games. Independent game publishing and independent music composition – from truly unsigned, unknown artists – go hand in hand. Indeed, the download and purchase charts on Bandcamp are often dominated by game scores. Fueled by word-of-mouth, these go viral in enthusiast communities largely ignored by either music or game reportage.

Far from the big-budget blockbuster war game, these scores – like the games for which they’re composed – are quirky and eccentric. They reject the usual expectations of what game music might be, sometimes tending to the cinematic, sometimes to the retro, sometimes unapologetically embracing magical, sentimental, childlike worlds.

And now, defying music’s typical business models as well as its genre expectations, you can get a whole big bundle of games for almost no money. Pay what you want, and get hours of music. Pay more than $10, and get loads more. You just have to do it before the deal ends (five days from this posting), at which point the bundle is gone forever. In a sign of just how much love listeners of these records feel, there’s a competition to get into the top 20, top 10, and top-paying spots, which with days left in the contest is already pushing well into the hundreds of dollars. But for that rate or just the few-dollar rate, these are the true fans. You’ve heard about them in theory in trendy music business blogs and conferences, in theory. But here, someone’s doing something about it, and it’s not a fluke or a one-time novelty: it’s a real formula.

**http://www.gamemusicbundle.com/**

Game music itself is, of course, a funny thing. Game play itself tends to repetition, meaning you hear this music a lot. So it says something really extraordinary about the affection for these scores that gamers want to hear the music again and again. This gets the musical content well beyond the level of annoying wallpaper into something that, even more than a film score you hear just once or a few times, you want to make part of your life. That endless play gets us back to what inspired ownership in the first place, to buying stacks of records rather than just waiting for them on the radio. And in that sense, perhaps what motivates owning music versus treating it like a utility or water faucet hasn’t changed in the digital age at all. Maybe it’s gotten even stronger.

We’ve already sung the praises of Sword and Sworcery on this site; it’s notably in the bundle. But I want to highlight in particular one other score, the inventive and dream-like *Machinarium*. Impeccably recorded, boldly original, the work of Prague-based Tomáš Dvořák, Machinarium mirrors the whimsical constructed machines of the games. There’s a careful attention to timbre, and music that moves from film-like moments to song to beautiful washes of ambience, glitch set against warm rushes of landscape. For his part, Dvořák is a clarinetist, and his musical senstitivity never ceases to translate into the score. It’s just good music, even if you never play the game, and easily worth the price of admission for the bundle if you never listened to anything else (though you would truly be missing out). It’s simply one of the best game music scores in recent years.

And another look at Jim Guthrie’s score to Sword & Sworcery:

Game Meets Album: Behind the Music and Design of the iPad Indie Blockbuster Swords & Sworcery[Create Digital Music]

Game Meets Album: Behind the Music and Design of the iPad Indie Blockbuster Swords & Sworcery [Create Digital Motion]

Also in this collection: Aquaria, To the Moon, Jamestown, and a mash-up, plus a whole bunch of bonus games when you spend a bit more that feel heavily influenced by Japanese game music and chip music.

And some of the best gems are in the repeat of the last bundle, which you can (and should) add on for US$5 more:

Minecraft: Volume Alpha, Super Meat Boy: Digital Soundtrack, PPPPPP (soundtrack to VVVVVV), Impostor Nostalgia, Cobalt, Ravenmark: Scourge of Estellion, A.R.E.S. Extinction Agenda, Return All Robots!, Mighty Milky, Way / Mighty Flip Champs, Tree of Knowledge

I’ve sat at game conferences as composers working for so-called AAA titles lamented the limitations of the game music production pipeline. Quietly, indie game developers have shown that anything is possible, that the quality of a game score is limited only by a composer’s imagination.

More music to hear (and some behind-the-scenes footage), including a really promising Kickstarter-funded iPad music project from regular CDM reader Wiley Wiggins:

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Tim Hawkinson's Möbius Ship sculptures are nautical, single-surfaced and have fractional dimensionality. Yo-ho-ho and a bottle of rum!

Echoing the working methods of ship-in-a-bottle hobbyists, Hawkinson created a painstakingly detailed model ship that twists in upon itself, presenting the viewer with a thought-provoking visual conundrum. The title is a witty play on Herman Melville's novel Moby Dick, which famously relates the tale of a ship captain's all-consuming obsession with an elusive white whale. The ambitious and imaginative structure of Hawkinson's sculpture offers an uncanny visual metaphor for Melville's epic tale, which is often considered the ultimate American novel.

(*via Kottke*)