The tesseract entered popular culture through Madeleine L’Engle’s “A Wrinkle in Time”, though L’Engle caused some of her readers confusion when one of the characters in “A Wrinkle in Time”, the prodigy Charles Wallace Murray, declared “Well, the fifth dimension’s a tesseract.” L’Engle wasn’t sure how to reconcile Hinton’s ideas about the fourth dimension with Einstein’s, so she put Hinton’s fourth dimension after Einstein’s, demoting it from fourth place to fifth.

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The 5-cell is an analog of the tetrahedron.

Tesseract is a four-dimensional hypercube – an analog of a cube.

The 16-cell is an analog of the octahedron.

The 24-cell is one of the regular polytope.

A hypersphere is a hypersurface in an n-dimensional Euclidean space formed by points equidistant from a given point, called the center of the sphere.

Yes—fixed!

]]>The documents under Four Dimensions in the examples menu are functions of a complex variable graphed as (u+iv)=f(x+iy) and projected to 3D. The controls on the right rotate the surface through rotations in the xy-, xu-, yu-, xv-, yv- and uv-planes. The white grid shows the xy plane at u=v=0.

]]>Understood, John, and look forward to the continuation of this series–it is utterly fascinating and helpful, indeed! :)

]]>Yes, thanks for catching this: I meant to write “2d plane”, not “3d plane”. My finger must have slipped.

I’m resisting saying a lot *more* about rotations in different dimensions, because this is a huge subject in its own right….

This is a fun series. :-)

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