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Basic Information

A Hypersphere is a fourth-dimensional sphere.

A normal sphere is circular, it can be visualized as a projection of a circle into three dimensions. If you take a single "layer" out of a sphere, that layer would look like a circle. If you cut a sphere in half down the middle, the two halves would each have a flat circular side, and unless you moved it those two sides would be facing or touching each other. If a 3-D sphere passes through a 2-D plane, it would look like a circle to a 2-D observer within that plane.

Now apply those notions to fourth-dimensional space. The hypersphere is a projection of a sphere into four dimensions. It's made up of spheres. A single 3-D layer of it looks like a sphere. If you cut it down the middle, the two halves would have a spherical facet, each facing the other. If a 4-D hypersphere passed through our 3-D plane of existence, it would look like a Sphere to us.

But a hypersphere has even more cool properties than that. It is also made up of circles, each with exactly one point of overlap with each and every other circle in the hypersphere. This provides a type of "flow" to the hypersphere, with every point leading into the other points in a precise direction. If you had a hypersphere for a head, you wouldn't have to part your hair, you could come the whole lot of it in the same direction. A line could be drawn on the surface of the hypersphere would pass over every point of the surface and wrap back to it's starting point.

Extrapolating from that, if reality were a hypersphere, and you had really good eyesight (or a powerful telescope, or if reality were small enough) you'd be able to see all the way around it and see yourself in the distance, facing away from you. Light would bend around the surface of the hypersphere. The furthest point away in space would be the back of your own head. In any direction you looked, you'd see yourself, at a significant distance and facing away. If your eyes were offset from the horizon line, you'd be able to see an infinite chain of yourselves, each staring forward to the next you in the line. It sort of like the effect of standing between two mirrors that are facing each other, except without the mirrors effect of every other image being reversed.

If your eyes aren't good enough to see the back of yourself, the hypersphere would look flat to an observer on it's surface. This is related to the optical illusion that allowed man to view the earth as flat for thousands of years. If you kept walking on the hypersphere in a straight line, you'd pass through every point and then end up back where you started.


1. Book: Hyperspace by Michio Kaku. He also touches on the subject in Beyond Einstein: The Cosmic Quest for the Theory of the Universe, a book co-written with Jennifer Trainer Thompson. Online Text
3. Wikipedia on N-Sphere A hypersphere is an n-sphere where n=4.
4. TV: Land of the Lost The classic Sid & Marty Kroft show had an episode where Marshal, Rick and Holly climbed the highest point in the Land of the Lost and used their binoculars to see the backs of themselves atop the highest mountain. As a kid, I thought this was cool but silly. Now, I understand that they were implying the pocket dimension they were in was a Hypersphere.

Game and Story Use

  • A City In A Bottle, Pocket Reality, Another Dimension, Alien Geometries, Mobile Maze or other nightmare place might be (or be contained within) a hypersphere.
    • I pity the player who tries to map the party's progress on such a surface. You'd pass by the same landmarks again and again from multiple angles. Even when you're walking in a straight line, you're also walking in circles.
    • If you don't yet know the size of the hypersphere, be very careful with Ranged Weapons such as Guns.
      • Never fire a Laser while in a hypersphere reality! If you missed your target, you'd be shooting yourself in the back.
  • A Fourth Dimensional Lifeform, whose head is a hypersphere, would have hairdo options that we puny 3-D mortals would lack. They could comb it partless, wrap a braid around every point on their head, or other strange patterns. Perhaps this explains the charisma and beauty of the gods.
    • Did the Dwarves braid fourth-dimensional gold into hair for Sif?
    • The hair of the gods may look like fractal renderings of 4-D objects in 3-D space. See HyperSphere, from an Artistic point of View for appropriate imagery.
    • And here I thought Star Wars had some odd hairdos. Little did I know…
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