The following text is a contribution from Cliff Pickover.Physicists can give no good reason why space has three dimensions. Perhaps the dimensionality of space in our universe was "accidentally" determined during the Big Bang, billions of years ago. It does seem that life would be more challenging in other dimensions. As we discussed, it would be difficult for digestive tracts to run through a creature in two dimensions because the tract would cut the creature into two pieces. Richard Morris in _Cosmic Questions_ suggests that if the dimensionality of space were four or greater, then stable planetary orbits would not be possible. Morris suggests that if a planet did manage to form, it would follow a path that caused it to spiral into the sun. This line of thinking is extended in Max Tegmark's wonderful recent article "On the Dimensionality of Spacetime" appearing in the journal _Classical and Quantum Gravity_.

Consider a universe with _m_ spatial dimensions and _n_ time dimensions. These universes are classified as (_n_+_m_)-universes. For example, our universe could be a (3+1)-universe with three spatial dimensions and one dimension of time. Max Tegmark of the Institute for Advanced Study in Princeton, New Jersey, suggests that all universes -- except for a (3+1)-dimensional universe -- may be "dead universes" in the sense they are devoid of observers. He believes that higher dimensional spaces cannot contain traditional atoms nor perhaps any stable structures. In a space with less than three dimensions, there may be no gravitational force. In universes with more or less than one time dimension, living creatures could not make predictions. These ideas are so fascinating that I would like to explain them just a bit further.

Some kinds of universes are more likely to contain observers than others. Here is some background. As far back as 1917, P. Ehrenfest suggested that neither classical atoms nor planetary orbits can be stable in a space with _n_ > 3. In the 1960s, F. Tangherlini further suggested that traditional quantum atoms cannot be stable in higher dimensional universes (see "For Further Reading"). For physicist readers, these properties are related to the fact that the fundamental Green's functions of the Poisson equation .eq del sup 2 phi = rho -- which gives the electrostatic/gravitational potential of a point particle -- is .eq r sup <2-n> for _n_ > 2. As Tegmark points out, this means the inverse-square law of electrostatics and gravity become an inverse-cube law if _n_=4, etc. When _n_ > 3, the two-body problem no longer has any stable orbits as solutions (see I. Freeman's 1969 paper).

In simple English, this implies that if you were in a four-dimensional universe and launched planets toward a sun, the planets would either fly away to infinity or they would spiral into the sun. (This is in contrast to a (3+1)-universe which can, for example, have stable orbits of moons around planets.) A similar problem occurs in quantum mechanics, where a study of the Schr&oe.dinger equation shows that the hydrogen atom has no bound states for _n_ > 3. This seems to suggest that it is difficult for higher universes to be stable over time and contain creatures that can make observations about the universe.

Lower dimensional worlds (such as 1- and 2-dimensional worlds) may not be able to have gravitational forces, as discussed in _Gravitation_ by John Wheeler and colleagues, and in a paper by S. Deser.

So far we have been talking about spatial dimensions, but we may also postulate the existence of different time dimensions. Tegmark believes that a universe will only be able to have observers if there is just one time dimension (i.e., _m_=1). What would it be like to live in a universe with more than one timelike dimension? Would we have difficulty going through our daily routines of life, job, and the search for an ideal mate? Even with two or more time dimensions, you might _perceive_ time as being one-dimensional, thereby having a pattern of thoughts in a linear succession that characterizes perception of reality. You may travel along an essentially one-dimensional (timelike) world line through the (_m_+_n_)-universe. Your wrist watch would work. However, the world would be odd. If two people moving in different time directions happen to meet on the street, they would inevitably drift apart in separate time directions again, unable to stay together! Also, as discussed by J. Dorling, particles like protons, electrons and photons are unstable and may decay if there is more than one dimension of time.

All sorts of causal paradoxes can arise with more than one dimension of time. However, I do not think this precludes life, even if the behavior or the universe would be quite disturbing to us. Also, electrons, protons, and photons could still be stable if their energies were sufficiently low -- creatures could still exit in _cold_ regions of universes with greater than one time dimension. However, without well-defined cause and effect in these universes, it might be difficult for brains (or even computers) to evolve and function.

None of these arguments rule-out the possibility of life in the fourth spatial dimension (i.e. a (4+1)-universe). For example, stable structures may be possible if they are based on short distance quantum corrections to the .eq 1/r sup 2 potential or on string-like rather than point-like particles.

:h2.Seven-Dimensional Ice

Recently, scientists and mathematicians have researched the theoretical melting properties of ice in higher dimensions. In particular, mathematicians Nassif Ghoussoub and Changfeng Gui, from the University of British Columbia, have developed mathematical models for how ice turns from solid into liquid in the seventh dimension and have proven that if such ice exists, it likely exhibits a different melting behavior than ice in lower dimensions. This dependence on dimension, although not very intuitive, often arises in the field of partial differential equations and minimal surfaces -- recent results suggest that geometry depends on the underlying dimension in ways that were not suspected in the past. Other research suggests that there is something about eight-dimensional spaces that makes physical phase transitions inherently different from seven-dimensional spaces. If you want to read more about what happens when you lick a seven-dimensional popsicle, see: Ekeland, I. (1998) How to melt if you must. _Nature_. April 16, 1998, 392(6677): 654-655.

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