From: Pertti Lounesto Newsgroups: sci.math Subject: Re: Life in 4 Dimensions Date: 21 Nov 1997 06:53:45 +0200 Hans Engler writes: > I am planning to give a presentation to eigth-graders about > four-dimensional geometry (mainly the 4-D cube) and want to > include some things about how weird life in 4 dimensions would be. I > can think of some simple things (6 compass directions, a corner room in > your house would have windows in 3 directions etc.) and know about > some fancier phenomena (Huyghens' Principle doesn't hold, so all sounds > would have fading echos). There are also some funny consequences of > scaling laws that I can think of, but that's about all. You might also mention that there are 6 regular polytopes in 4D (in 3D there are only 5 regular polyhedra = Platonic solids). Of these 6 polytopes only 3 fill the 4D without cavities in tiling. 4D is exceptional: in higher dimensions there are only 3 regular polytopes, and only 1 of them fills the space. More information about regular polytopes can be found in the Chapter "The Fourth Dimension" of my book "Clifford Algebras and Spinors" with URL http://www.cup.org/Titles/59/0521599164.html. Another topic might be rotations in 4D. They do not have an axis (of dimension 1, although some rotations have an axis of dimension 2). A rotating ball in 4D is such that it has two perpendicular planes, both rotating at arbitrary angular velocities. And a strange thing happens, if the two angular velocities are the same: then all points are rotating at the same angular velocity, along some great circles, which are pairwise linked. You can find more information about the rotating ball in 4D in the Section "Rotating ball in R^4" of the Chapter "The Fourth Dimension" of my book. > Any other contributions? Does e.g. the inverse-square law from > gravitation have to be replaced with an inverse-cube law, so that > gravitational potentials still solve a partial differential equation? > What implications would this have? Inverse-cube law (for the gradient of the Coulomb field) in connection with electron (around an atom) results in circular orbits (elliptic orbits are not possible closed orbits in 4D). The same is true in n dimensions (with inverse-(n-1) law), but 4D is again exceptional: in 4D all the electrons, on all circles, have the same angular momentum and energy (that is, only one energy state is possible). -- Pertti Lounesto http://www.hit.fi/~lounesto