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Extra Dimensions of Space
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/content/aapt/journal/tpt/51/6/10.1119/1.4818367
1.
1.E. Abbott, Flatland: An Edition with Notes and Commentary (Cambridge University Press, 2009).
2.
2.L. Randall, Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions (Harper Perennial, 2006).
3.
3.B. Greene, Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory (Vintage Books, 2000).
4.
4.D. Lincoln, Understanding the Universe: From Quarks to the Cosmos (World Scientific, 2012, revised).
5.
5.LHC Safety Assessment Group, lsag.web.cern.ch/lsag/LSAG-Report.pdf.
http://aip.metastore.ingenta.com/content/aapt/journal/tpt/51/6/10.1119/1.4818367
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Fig. 1.
Image of Fig. 2.

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Fig. 2.

A tightrope walker is restricted to motion in one dimension along the rope's axis. In contrast, the smaller ant has access to two dimensions, including one much smaller one. (Figure courtesy Dan Claes.)

Image of Fig. 3.

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Fig. 3.

In a one-dimensional universe, the flux from the source can travel along the axis and the area it encounters doesn't change at all. (Recall that in a one-dimensional world the flux can only travel along the dimension's direction. The transverse dimension shown here is to enhance visualization.) Indeed, to be mathematically correct, that “area” consists of only two points on opposite sides of the source. In a two-dimensional world, the size of the surface that the flux encounters changes with radius. The surface is the circumference of the circle, which scales linearly with the distance between the source and the surface.

Image of Fig. 4.

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Fig. 4.

In a universe with one large and one small and cyclical dimension, the situation looks quite different when one changes the size being probed. At small scales, the shape of the force field looks like a familiar two-dimensional one, while at large scales the force field looks one dimensional. The field lines fill up the smaller dimension.

Image of Fig. 5.

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Fig. 5.

It is possible for quantum mechanical wave functions to wrap around the small dimensions. Two example wavelengths are shown here. The mass of these hypothetical particles increases as the number of wavelengths increases.

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/content/aapt/journal/tpt/51/6/10.1119/1.4818367
2013-09-01
2014-04-25

Abstract

They say that there is no such thing as a stupid question. In a pedagogically pure sense, that's probably true. But some questions do seem to flirt dangerously close to being really quite ridiculous. One such question might well be, “How many dimensions of space are there?” I mean, it's pretty obvious that there are three: left/right, up/down, and forward/backward. No matter how you express an object's coordinate, be it Cartesian, spherical, cylindrical, or something exotic, it's eminently clear that we live in a three-dimensional universe.

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Scitation: Extra Dimensions of Space
http://aip.metastore.ingenta.com/content/aapt/journal/tpt/51/6/10.1119/1.4818367
10.1119/1.4818367
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