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Review of dimensions

Visualizing higher dimensions. We can think about objects with a higher dimension by looking at cross-sectionsusing the terminal points/sides" argumentReview from Flatland (pp. 60-61)Introduce vertices, edges, faces, and cubes. Motivating Question. What is the dimension of a circle? (Note: a c

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Review of dimensions

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    1. Review of dimensions We can find the dimension of a space by counting the number of “sufficiently different” directions of travel doubling an object in all dimensions and seeing what exponent is used to describe the change in size

    2. Visualizing higher dimensions We can think about objects with a higher dimension by looking at cross-sections using the “terminal points/sides” argument Review from Flatland (pp. 60-61) Introduce vertices, edges, faces, and cubes

    3. Motivating Question What is the dimension of a circle? (Note: a circle, not a disk.) First Strategy: Exponentiation: what happens to the circumference of a circle when its measurements are doubled?

    4. Motivating Question Second Strategy: Direction counting: if you lived in the circle, how many directions could you move?

    5. Motivating Question Third Strategy: Covering dimension: if you covered the circle completely with pennies with no overlap but with the least overlap possible, what is the thickest overlapping of pennies you would have? Note: Avoid the very small gaps left when two pennies are placed edge to edge.

    6. Exercises Repeat this process with A hand-drawn curve A finite collection of dots A rectangle

    7. Covering dimension The covering dimension of an object in the plane is the one fewer than the least number of overlaps needed to completely cover the object with disks. The covering dimension is an intrinsic property: it remains the same regardless of how the shape is continuous deformed. Continuous deformations are those where an object is changed in a slow and reversible manner (and we pretend the object is infinitely stretchable). You cannot cut or join objects.

    8. Covering dimension in space The definition of covering dimension stays the same in three dimensions except balls are used instead of disks. Find the covering dimension of a sheet of paper a hollow cylinder How do these computations agree with direction counting?

    9. Motivating Question So what’s the difference between a straight line segment and a circle (in terms of where they reside)? Review the standard spaces. The embedding dimension of a object is the dimension of the smallest dimension standard space it can live in. Examples with the segment and circle.

    10. Art Example John Robinson’s Immortality What is the covering and embedding dimensions of this object?

    11. Art Example

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