Hyperspheres and Tesseracts

Sally steps closer to the board. If a hypersphere with a seventeen-foot radius came into our world, what points would be on it She points her finger in the air. Assume that its center is located at my fingertip.  

For several seconds, you stare at Sally s elegantly manicured nails before taking a string from your drawer and measure off seventeen feet. Sally, may I borrow one of your earrings  

Sally nods. Using your formula, this means that no matter in what direction you move eight feet away from my finger, the additional fifteen-foot move upsilon gives a point exactly seventeen feet away from my fingertip.  

Sally thinks for a few seconds. Now I can understand why a hypersphere consists of a series of spheres—spheres that grow smaller as one moves upsilon or delta away from my fingertip at the sphere s center. Also, the less I move away from the hypersphere s center in our space, the more I c an move upsilon or delta to be on the hypersphere s surface.  

You go over to a handsome walnut cabinet in a dark corner of your office. The cabinet door is locked, so you fumble in your pocket for a key. Nearby, a stand holds sheet music and a guitar. On your coffee table is an odd assortment of magazines, from Wired to Sushi News, Chess Life, and Scientific American. 

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Sally, we must make the time. Christopher Columbus didn t start exploring without first understanding basic principles of navigation. You ve returned to your FBI office. I want to talk more about hyperspheres and tesseracts, the 4-D counterparts to spheres and cubes. You draw a circle on the board with a dot at its center. A circle is the collection of points (on a plane) all at the same distance r from a point. A sphere is the collection of points (in space) all at the same distance r from a point. Similarly, a hypersphere is the collection of points (in hyperspace) all at the same distance r from a point. ... 

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