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Mathematics Illuminated

Other Dimensions

Is there such a thing as a higher dimension, a parallel universe where otherworldly things can happen? Over the years, artists, writers and filmmakers have tried to answer that question, creating some dazzling works of science fiction in the process. But are the higher dimensions we see in sci-fi really fiction?

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Hypercube with Rulers

Hypercube of Rulers

The conventional notion of dimension consists of three degrees of freedom: length, width, and height, each of which is a quantity that can be measured independently of the others. Many mathematical objects, however, require more—potentially many more—than just three numbers to describe them. This unit explores different aspects of the concept of dimension, what it means to have higher dimensions, and how fractional or “fractal” dimensions may be better for measuring real-world objects such as ferns, mountains, and coastlines.

Unit Goals

  • Dimension is how mathematicians express the idea of degrees of freedom.
  • Distance and angle are measurements that exist in many types of spaces.
  • Lower-dimensional analogies extend qualitative understanding to spaces of four dimensions and higher.
  • The techniques of projection and slicing help us to understand high-dimensional objects.
  • High-dimensional space is one way to compare two people mathematically.
  • Hausdorff dimension is a re-envisioning of our normal thinking of dimension due to behavior of objects under scaling.
  • Fractal dimensions describe many real-world objects that exhibit statistical self-similarity.

Additional Unit Resources: Bibliography




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