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