Angel Isle

APPENDIX

Fodaro’s Equations

According to Big Bang theory, in the very early stages of the formation of our universe there were more than the four dimensions we are familiar with. String theory, for instance, involves tiny strings of energy vibrating in ten dimensions of space time. Thirteen also seems to be a popular number. Anyway, once the first few microseconds of the explosion were over, the unwanted dimensions were “folded up and tucked away,” whatever that may mean.

These conclusions were arrived at by applying complex mathematical processes to the observed behavior of sub-atomic particles—quarks and so on—in gigantic particle colliders, and of light arriving from the most distant reaches of the universe.

Fodaro came to his rather different conclusions by very similar means. He was able to observe such things as the formation of stars and the outermost limits of the universe through a very powerful telescope—Benayu’s pool—and the behavior of sub-atomic particles in the form of magical impulses. He was also a mathematical genius. He wasn’t, incidentally, an untaught genius. Before he abandoned his career and took up shepherding he was Imperial Professor of Applied Mathematics at the University of Balin-Balan—the youngest ever to hold that prestigious post.^*1 He continued his work in his spare time after his retirement and made his breakthrough when his observations of certain magical anomalies led him to the discovery of a place where our four-dimensional universe comes almost, but not quite, into contact with Jex’s seven-dimensional one. He called such places “touching points.”

It is impossible for our four-dimensional minds to imagine what such a universe might be like. We can to some extent describe it with mathematical equations, but we can’t envisage it, can’t form any kind of mind-picture of it.^**2

The dimensions of any universe must compose a perfect whole. This means a five-dimensional universe, say, cannot simply have our four plus an extra one, because our four are already a whole and the extra one has nowhere to fit in. This makes it impossible for even one atom of four-dimensional matter to be transferred to a five-dimensional universe, and vice versa. The consequent destruction of matter would involve the creation of energy, as in an atomic explosion.

Sub-atomic particles are another matter. We are already familiar with these in our own universe, in the form, for instance, of the aurora borealis, which is caused by streams of charged particles arriving from the sun and being trapped by the earth’s magnetic field in the ionosphere. Rather more mysterious are the bursts of gamma rays that appear to emanate from the central bulge of our galaxy, and arrive in our atmosphere carrying the extraordinary charge of 511 kilo-electron volts.

Particles of antimatter are created in the big particle colliders, but survive only an instant before they collide with particles of matter, wiping each other out and releasing a burst of energy in the process. Fodaro knew nothing of particle colliders, but was able to use the touching point he found in a very similar fashion, observing the steady leakage of particles from Jex’s universe into ours and measuring the bursts of energy released when they collided with their four-dimensional counterparts and were mutually destroyed. Owing to the inherent weirdness, to our minds, of a seven-dimensional universe,^*3 this energy took the form of what was known to pre-Fodaran thaumatology as “wild magic.”^**4 Without understanding its source and nature, over the centuries the magicians of the Empire learned by trial and error to exploit and shape it into a powerful set of tools, usually described as “made magic.”

Pre-Fodaran theory evolved a complex system of “levels” to account for the observed phenomena. Like pre-Copernican astronomy, this worked well enough for most practical purposes, but at the cost of ignoring an increasing number of anomalies and unexplained phenomena, such as the acknowledged difficulty experienced by almost all magicians in progressing from third-level to fourth-level magic. In Fodaran thaumatology this is accounted for as a result of the different degrees of complexity between universes with numbers of dimensions other than our own. First-level magic, “hedge” magic, uses only the impulses from the stray particles inherent in our own universe; second-and third-level magic uses impulses from the simpler two-and three-dimensional universes; fourth-and fifth-level, confusingly, use impulses from universes with five and six dimensions, and sixth-level uses impulses from any universes beyond that.

Certain people are born with innate magical power. This is something like a magnetic field, in that it can capture the magical impulses from other universes. The stronger the field, the greater the number and variety of the impulses it can capture. But a magician cannot use it until he or she has learned to control and shape the flow of impulses within the field. What in pre-Fodaran thaumaturgy was called wild magic was the result of random and transient combinations of impulses; made magic was produced by magicians using their craft; while natural magic arose from certain combinations in wild magic that turned out to have survival value and evolved of their own accord into more and more complex forms, such as dragons and unicorns.

Demons are another matter. Most of them arise from attempts by ambitious magicians to shape and control some phenomenon of natural magic for their own purposes. This then proves to be beyond their powers, and they themselves are absorbed into the result. Since their original motive for making the attempt was almost invariably bad, the resulting monster is more or less powerfully malign. Angels arise from well-intentioned but failed attempts to do the same thing. Hence their rarity.

Further information can be found on the Web site of the Thaumatological Department at the University of Balin-Balan.