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Dimension

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timothy

coondoggie writes "At the Design Automation Conference (DAC) here this week, John Kubiatowicz, professor in the UC Berkeley computer science division, offered a preview of Tessellation, describing it as an operating system for the future where surfaces with sensors, such as walls and tables in rooms, for example, could be utilized via touch or audio command to summon up multimedia and other applications. The UC Berkeley Tessellation website says Tessellation is targeted at existing and future so-called 'manycore' based systems that have large numbers of processors, or cores on a single chip. Currently, the operating system runs on Intel multicore hardware as well as the Research Accelerator for Multiple Processors (RAMP) multicore emulation platform."

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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 Rd, 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 R2, 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.

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In the past, debutante balls were opportunities for introducing noble daughters to high society. Photographer Olivia Harris discovers what London’s Queen Charlotte’s Ball means to the young girls of today.

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In a world made small and accessible by technology, it is easy to forget the magnitude of nature’s infinite complexity. But sometimes technology reminds us, such as when trawling planet Earth on Google’s Satellite View, zooming across landscapes partitioned by natural and unnatural boundaries.

While searching Google Earth, Paul Bourke, a research associate professor at the University of Western Australia, discovered an amazing sight—the patterns of the Earth seemed to form a delicate geometric pattern when viewed from the sky. Not only delicate, but almost perfect. Bourke was captivated by the geography—lacy tracks of rivers and mountain ranges stretching across the Earth in unison as if digitally cloned.

Fractals are recognized as patterns of self-similarity over varying degrees of scale. There are both mathematical fractals as well as natural fractals—the former are idealized and found across a range of scales, while the latter generally only exist across a smaller scale range.

Bourke explains that fractals are found in all parts of life, from the brain sciences and astrophysics to geographic formations and riverbeds. “Fractal and chaotic processes are the norm, not the exception.”

“I always knew these amazing natural patterns would be there,” he said. “They are literally everywhere—it’s just a matter of finding them.”

And find them he did. Bourke, an authority on fractals and visualizations, showcases more than 40 different fractals he’s uncovered while zooming through the satellite views of 25 countries. Through his website, he encourages users to submit examples they’ve found in their own browsing, and provides KMZ coordinate files for each image, allowing users to visit the exact views of the fractal features. Bourke’s collection realizes the power enabled by the open-ended tools of modern technology and applies them to a practical and popular aesthetic end.

To see more natural fractal patterns, visit Bourke’s website.

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Kristiina-lahde-list

Kristiina Lahde’s work is preoccupied with measurement, regulation, and the possibilities for boundary-breaking that such systems present. Using material that traditionally provides and facilitates order, like measuring tapes, phone books, and rulers, the Canadian artist produces graphic works and sculptures that reconsider their original function and refresh them as something new and extraordinary. Her Beyond Measure series of works examines the role of measurement in our day-to-day lives; in the Metric system pieces, she takes one mode of structuring and regulating physical reality (inches and centimetres) and reconfigures them as abstracted cubes that nevertheless retain a sense of uniformity and consistency. Her 2009 bookwork, Compilation, meanwhile appears to play on the hive of information currently available to us, and perhaps ties the intentional graphic consistency of something as humdrum as a telephone book with the startlingly beautiful consistency of natural geometric forms. Wow.

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