Friday, October 2, 2015

Dimension

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Dimension
Multiple Responses:
1.
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.

In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to describe string theory, and the state-space of quantum mechanics is an infinite-dimensional function space.

The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.

2.
As you've probably noticed, we live in a world defined by three spatial dimensions and one dimension of time. In other words, it only takes three numbers to pinpoint your physical location at any given moment. On Earth, these coordinates break down to longitude, latitude and altitude representing the dimensions of length, width and height (or depth). Slap a time stamp on those coordinates, and you're pinpointed in time as well.

To strip that down even more, a one-dimensional world would be like a single bead on a measured thread. You can slide the bead forward and you can slide the bead backward, but you only need one number to figure out its exact location on the string: length. Where's the bead? It's at the 6-inch (15-centimeter) mark.

Now let's upgrade to a two-dimensional world. This is essentially a flat map, like the playing field in games such as Battleship or chess. You just need length and width to determine location. In Battleship, all you have to do is say "E5," and you know the location is a convergence of the horizontal "E" line and the vertical "5" line.

Now let's add one more dimension. Our world factors height (depth) into the equation .While locating a submarine's exact location in Battleship only requires two numbers, a real-life submarine would demand a third coordinate of depth. Sure, it might be charging along on the surface, but it might also be hiding 800 feet (244 meters) beneath the waves. Which will it be?

Could there be a fourth spatial dimension? Well, that's a tricky question because we currently can't perceive or measure anything beyond the dimensions of length, width and height. Just as three numbers are required to pinpoint a location in a three-dimensional world, a four-dimensional world would require four.

At this very moment, you're likely positioned at a particular longitude, latitude and altitude. Walk a little to your left, and you'll alter your longitude or latitude or both. Stand on a chair in the exact same spot, and you'll alter your altitude. Here's where it gets hard: Can you move from your current location without altering your longitude, latitude or altitude? You can't, because there's not a fourth spatial dimension for us to move through.

But the fact that we can't move through a fourth spatial dimension or perceive one doesn't necessarily rule out its existence. In 1919, mathematician Theodor Kaluza theorized that a fourth spatial dimension might link general relativity and electromagnetic theory. But where would it go? Theoretical physicist Oskar Klein later revised the theory, proposing that the fourth dimension was merely curled up, while the other three spatial dimensions are extended. In other words, the fourth dimension is there, only it's rolled up and unseen, a little like a fully retracted tape measure. Furthermore, it would mean that every point in our three-dimensional world would have an additional fourth spatial dimension rolled away inside it.

String theorists, however, need a slightly more complicated vision to empower their superstring theories about the cosmos. In fact, it's quite easy to assume they're showing off a bit in proposing 10 or 11 dimensions including time.

Wait, don't let that blow your mind just yet. One way of envisioning this is to imagine that each point of our 3-D world contains not a retracted tape measure, but a curled-up, six-dimensional geometric shape. One such example is a Calabi-Yau shape, which looks a bit like a cross between a mollusk, an M.C. Escher drawing and a "Star Trek" holiday ornament [source: Bryant].

Think of it this way: A concrete wall looks solid and firm from a distance. Move in closer, however, and you'll see the dimples and holes that mark its surface. Move in even closer, and you'd see that it's made up of molecules and atoms. Or consider a cable: From a distance it appears to be a single, thick strand. Get right next to it, and you'll find that it's woven from countless strands. There's always greater complexity than meets the eye, and this hidden complexity may well conceal all those tiny, rolled-up dimensions.

Yet, we can only remain certain of our three spatial dimensions and one of time. If other dimensions await us, they're beyond our limited perception -- for now.

3.
A Whole New Dimension to Space
In everyday life the number of dimensions refers to the minimum number of measurements required to specify the position of an object, such as latitude, longitude and altitude. Implicit in this definition is that space is smooth and obeys the laws of classical physics.

But what if space is not so well behaved? What if its shape is determined by quantum processes in which everyday notions cannot be taken for granted? For these cases, physicists and mathematicians must develop more sophisticated notions of dimensionality. The number of dimensions need not even be an integer, as in the case of fractals—patterns that look the same on all scales.

Cantor Set : Take a line, chop out the middle third and repeat ad infinitum. The resulting fractal is larger than a solitary point but smaller than a continuous line. Its Hausdorff dimension is 0.6309.

Sierpinski Gasket: A triangle from which ever smaller subtriangles have been cut, this figure is intermediate between a one-dimensional line and a 2-D surface. Its Hausdorff dimension is 1.5850.

Menger Sponge: A cube from which subcubes have been cut, this fractal is a surface that partially spans a volume. Its Hausdorff dimension is 2.7268, similar to that of the human brain.

Generalized Definitions Of Dimensions
Hausdorff Dimension
Formulated by the early 20th-century German mathematician Felix Hausdorff, this definition is based on how the volume, V, of a region depends on its linear size, r. For ordinary three-dimensional space, V is proportional to r3. The exponent gives the number of dimensions. “Volume” can also refer to other measures of total size, such as area. For the Sierpi´nski gasket, V is proportional to r1.5850, reflecting the fact that this figure does not even fully cover an area.

Spectral Dimension
This definition describes how things spread through a medium over time, be it an ink drop in a tank of water or a disease in a population. Each molecule of water or individual in the population has a certain number of closest neighbors, which determines the rate at which the ink or disease diffuses. In a three-dimensional medium, a cloud of ink grows in size as time to the 3/2 power. In the Sierpi´nski gasket, ink must ooze through a twisty shape, so it spreads more slowly—as time to the 0.6826 power, corresponding to a spectral dimension of 1.3652.

Applying the Definitions
In general, different ways to calculate the number of dimensions give different numbers, because they probe different aspects of the geometry. For some geometric figures, the number of dimensions is not fixed. For instance, diffusion may be a more complicated function than time to a certain power.

Quantum-gravity simulations focus on the spectral dimension. They imagine dropping a tiny being into one building block in the quantum spacetime. From there the being walks around at random. The total number of space­time building blocks it touches over a given period reveals the spectral dimension.

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