If you have trouble visualizing dimensions beyond the three that make up our spatial world, you're not alone; three-dimensional beings in general lack the capability to perceive in four or more dimensions. Nonetheless, it's possible to imagine higher dimensions using extrapolation exercises. Even with visualization aids, however, you may find imagining these dimensions difficult. If so, be thankful you aren't one of the mathematicians or physicists who try to imagine 10 or more of them.
The Three-Dimensional Coordinate System
You need only a piece of paper to help you understand the three familiar dimensions of space, although there are online interactive tutorials that make it even easier. Start with two points, neither of which have dimensionality, and connect them with a line -- that's the first dimension. Draw a perpendicular line through the center of that one, and you've created a two dimensional drawing. To depict the third dimension, imagine drawing a third line through the paper; for convenience, draw it perpendicularly through the intersection point of the other two lines. This gives you a set of axes that you can use to describe the position of any point in three-dimensional space.
Time as the Fourth Dimension
Mathematicians call the axes of the three-dimensional coordinate system that you just created the x, y and z axes, and they use the coordinates to specify position. Objects in the real world can change position, however, and mathematicians are able to describe that movement by adding a fourth axis -- the t axis -- to denote time. Albert Einstein introduced the concept of time as the fourth dimension in the Special Theory of Relativity. It is the dimension of the space-time continuum in which objects move and change, but some physicists consider movements perceived in time as indicators of other spatial dimensions.
A Fourth Spatial Dimension
To understand the perception of higher-dimensional shapes, it helps to go back the sheet of paper. If an invisible three-dimension shape, such as a sphere, moved through the paper and became visible only when it touched the paper, three-dimensional observers would see a series of circles that grew larger and then smaller. An imaginary two-dimensional observer on the paper would have less indication of the true nature of the sphere, being able to see only a line changing length. In the same way, four-dimensional figures moving through three-dimensional space produce a series of three-dimensional shapes that change size, position and orientation, and fifth-dimensional shapes have even more cryptic signatures in three-dimensional reality.
10 Dimensions and Counting
It isn't possible to draw four-dimensional shapes, but it is possible to perform thought experiments to visualize them. In four dimensions, a three-dimensional cube becomes a hyper-cube, a kind of curled-up version of the three-dimensional shape. Mathematicians Theodor Kaluza and Oskar Klein were the first to propose a fourth spatial dimension based on this geometry. The idea was developed by mathematicians Eugenio Calabi and Shing-Tung Yau, who described six-dimensional shapes, called Calabi-Yau shapes, which are created as microscopic space curls in on itself. These shapes are essential in superstring theory, which is a leading candidate for a Theory of Everything; together with the three dimensions of space and the one of time, they provide the 10 dimensions needed for the mathematics to work.
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