Teacher resources and professional development across the curriculum

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5 Other Dimensions



Dimension is how mathematicians express the idea of degrees of freedom—aspects of an object that can be measured separately.

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Flat Land

Points in two-dimensional space require two numbers to specify them completely. The Cartesian plane is a good way to envision two-dimensional space.


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The hypercube is the four-dimensional analog of the cube, square, and line segment. A hypercube is formed by taking a 3-D cube, pushing a copy of it into the fourth dimension, and connecting it with cubes. Envisioning this object in lower dimensions requires that we distort certain aspects.


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A point in four-space, also known as 4-D space, requires four numbers to fix its position. Four-space has a fourth independent direction, described by "ana" and "kata."

In Euclidean four-space, our standard notions of Pythagorean distance and angle via the inner product extend quite nicely from three-space.


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A sphere can be thought of as a stack of circular discs of increasing, then decreasing, radii. The process of slicing is one way to visualize higher-dimensional objects via level curves and surfaces. A hypersphere can be thought of as a "stack" of spheres of increasing, then decreasing, radii.

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Line Land

A point in one dimension requires only one number to define it. The number line is a good example of a one-dimensional space.

number line

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A point in three-dimensional space requires three numbers to fix its location.

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