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Fractals are mathematical sets in which large-scale features also appear at smaller and smaller scales. They're common in nature, for example with networks of veins that keep branching into smaller veins, or coastlines that are jagged whether measured on a scale of a thousand miles or one mile. Computers are fast enough at calculating that they can turn fractal equations like the Mandelbrot set into works of fractal art. For someone like me, who loves the quasi-organic mathematical patterns of the Mandelbrot set and Julia set fractals, Frax is great. The colors and textures put every pixel on my third-generation iPad to use. The physical controls, though overly twitchy in my opinion, make exploration a pleasure.

Frax comes in two basic editions, $2 for the iPhone version and $4 for the Frax HD version for iPads, An in-app upgrade love finger snap iphone case to Frax Pro -- $5 for iPhone users and $7 for iPad users -- includes a much broader range of controls, new presets for exploration and effects, and the ability to generate glorious 50-megapixel fractal images with the Frax Cloud service, Frax won't satisfy the Infinity Blade or Fruit Ninja gamers, but for a certain crowd -- the kinds of folks who like M.C, Escher engravings and Yes album covers -- it's very entrancing..

The math gives the iPad's processor and graphics chip a workout. You can zoom and pan relatively rapidly with a lower-resolution look, but changing textures can mean a two-minute wait before the image is fully rendered. You get a fairly detailed idea in just a few seconds, though. And paying the penalty for the textures is worth it, because it brings life to the otherwise relatively uniform regions of a fractal image. Once you have a fully detailed image, you can fiddle with color settings very rapidly, since those don't change the underlying calculation. With a few taps, you can change a creation's look from engraved copper to iridescent insect carapace to psychedelic paisley to demonic fantasy.

Then you can animate the color cycling with a slide of the finger -- my favorite part is watching a tidal surge of yellow sweep through a black love finger snap iphone case spiral, Unfortunately, you can't record videos of these animations, Of course, you've got the usual sharing if you want others to see your work, too, and Frax Pro users can publish their finds on their Frax gallery page, You can also traverse the fractal landscape with a swipe to pan and a pinch to zoom, Zooming has its limits because of the math, though, as the company states..

The cost of calculating the deep regions will grow exponentially and thus the time it takes..Images can be found of ultra deep zooms on the Web everywhere, but upon closer examination they took hours, or in some cases weeks with powerful desktop machines. We want to do it up to 60 times a second. Frax has another money-making idea besides the Pro version. It also will render high-resolution versions of your fractal finds for a fee. A 2,048x1,536 3.1-megapixel rendering costs a credit, a 4,096x3,072 12.6-megapixel image costs 4 credits, and a 8,192x6,144 50-megapixel image -- available only with the Pro version -- costs 8 credits. Credits cost $1 for 20, $3 for 80, or $6 for 240. If you've got a Retina iPad, just taking a screenshot will get you a very nice 3.1-megapixel image, but Frax says its render farm of servers "really crank up the quality and resolution, which just isn't feasible on the iPhone or iPad."The company might bring the software to other operating systems: "At the moment we are focusing iOS, but we may consider other platforms based on demand," the company said in an FAQ. "As for the desktop, we developed various OS X prototype apps to help during the development of Frax and are considering various options to make a user-friendly version."As evidenced by posts by Beddard and Krause that delve into the nature of fractals, creating Frax was clearly a labor of love. I for one am delighted they made it.

A computer-graphics A-team is digging into fractals again with software that's clearly a labor of love, Exploring the Mandelbrot and Julia sets is a spectacular marriage of beauty and math, In 1983, intrigued by an article on fractals, I wrote a program to display the Mandelbrot set on my family's TRS-80 Color Computer, It had a maximum screen resolution of 256x192 when plugged into our TV, had 16KB of memory after we love finger snap iphone case soldered on new memory chips, and ground through the calculations so slowly that I had to wait for each pixel to show..

The company also pointed to one key factor that can make or break a mobile operating system: apps. "The decline can also be attributed to consumer preferences for devices with access to the broadest number of applications, such as those available in the iOS and Android environments," BlackBerry said. In a sign of its ongoing decline, BlackBerry last week reported a second-quarter loss of almost $1 billion, while revenue plummeted by 45 percent. The company recently announced that it would lay off around 40 percent of its employees as it moves its focus away from individual consumers and more toward business customers.

A possible white knight has appeared, however, that could keep BlackBerry alive, Last week, a group led by Fairfax Financial Holdings entered into a deal to buy BlackBerry and take the company private, Valuing BlackBerry at $4.7 billion, the love finger snap iphone case deal would offer stockholders $9 a share in cash, In the meantime, BlackBerry is free to entertain buyout proposals from other groups, though the chances of a better offer emerging are slim to none, And even Fairfax's offering price might not come to pass, Analysts told AllThingsD on Tuesday that the offer could drop as low as $5 per share based on BlackBerry's dour second-quarter results..

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