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... based on weather, obtain accurate demographic data using small samples, compute probabilities, use tests of significance to determine if a lot of weapons with a given failure rate had indeed come from an armorer clan whose weapons have a historically known failure rate, and so on. Unsophisticated devotees of Hr or Was would presumably consider such calculations heretical, but it seems very Tkumelani that the Inner Doctrines of these Temples would include a great body of work in statistical technique, including formulae for means and variances, significance tests, and painstakingly, experimentally compiled tables of normal probabilities. The confirmation or correction of even a portion of such a table would qualify as a Labor of Reverence!
More advanced statistics could well also be included in the Inner Doctrines. Other Temples, mainly those of Hill and This, could also be interested in these techniques for both ideological and practical reasons. VI. Topology and Higher-Dimensional Geometry There is, however, one area in which the mathematics of the Five Empires has advanced substantially beyond the relatively primitive, disparate and un-generalized disciplines described above. The highest adepts of Belkhnu, the Inner Circle of Karl, and a few great scholars have mastered both the practical and intuitive aspects of the discipline of topology to a degree that few, if any, Earth mathematicians have ever attained.
They understand multidimensionality, circular and linear dimensions, spheres, toroid's, holes, and indeed have identified many other topological concepts of practical value at least for plane-travelers -- which we have scarcely formulated due to our differing focus on foundational studies. None of this could properly be called 'differential' or even 'algebraic' topology in our sense; their texts are full of strange diagrams comprehensible only to the initiate, filled with curves and arrows and the glyphs of a hundred different tongues. The motivation for such studies is obvious: the mastery of the Many Planes. To the adept of inter planar travel, the three dimensions of ordinary visible space are supplemented by dri, tree, another one or two known only to such plenipotent luminaries as the sorcerer Waba, and perhaps still others beyond these, hinted at only in the most ancient of manuscripts and in the few recorded conversations on these topics with the Mihalli. Many sorcerers have attained a certain practical competence in negotiating the Many Planes, of course. But the few true researchers among them have tried to develop both general theories and practical concepts which go beyond the functional competence of the middling adept, and from this research was born the splendidly baroque and endlessly multifaceted mathematical science of Tkumelani topology.
Lacking as we do reproductions of Waba's Almanac or the Periplus of the Planes one of the authors has been attended by strange nightly visions since requesting the former through interlibrary loan we can only speculate about the nature of this discipline, using analogies and terminology from our mathematics. Nonetheless, we believe the subject of sufficient interest both in its own right and in relation to Tkumel to be worth discussing in more detail. It is difficult for the denizens of a three-dimensional world to conceive of four or more dimensions, even using special techniques such as projection and lower-dimensional cross-sections. It is useful to imagine two-dimensional beings that live on a plane or a sphere (the latter being a closed universe, about which more later), and to consider what their interactions with three-dimensional beings or objects would be like; Edwin Abbotts classic book Flatland is excellent for this. (Note that these hypothetical two-dimensional beings would not perceive the three-dimensionality of the universe beyond their plane or sphere, just as most inhabitants of Tkumel do not perceive the additional dimensions noted by Waba and others. ) There are many possible temporo-spatial configurations of the Many Planes. One possibility is that the Planes form the three-dimensional faces of a vast polytope (the higher-dimensional analogue of a polygon in two dimensions or a polyhedron in three), or perhaps a collection of polytope's. For comparison, consider the 2 D faces of a gaming die that together form a 3 D object.
Each plane would be a closed universe: if you travel far enough in a straight line, you would eventually return to your point of departure, as if a two-dimensional being living on a sphere traveled all the way around it. The distance that would need to be traveled could be so large that it would be impossible to verify that each Plane is in fact closed. (Contemplating such a titanic feat of spatial engineering begs the question of what purpose such a vast construct could serve. Those versed in Tkumelani lore may suspect that they are intended to keep something in - or out! of their higher-dimensional interior! ) This theory lends itself most readily to empirical investigation. Certain planes would be adjacent in the polytope, and thus perhaps more accessible, while other planes would not be accessible save by hopping through a chain of adjacent planes, or by jumping through the higher dimensions. When Alpha the Mihalli travels through several planes in sequence in Flamesong to arrive at some particular destination, for instance, he explains that the seemingly circuitous route he takes is in fact the shortest path.
A point in favor of this model is that nexus points, where two planes meet, are two-dimensional, just like the 2 D edges between 3 D faces of a 4 D polytope. However, Alujas voyaging also points up a potential problem with this model. Normally, we think of a polytope as a static construct, so that each branch of reality would contain its own polytope (s); the structure of the multiverse would not change over time. Everything we know about planar travel from Flamesong and elsewhere, however, clearly indicates that different boundaries become accessible at different times, and that the best (and sometimes the only possible) paths between different planes change over time.
This does not necessarily refute the polytope theory, but it does seem to necessitate that we regard the structure of the polytope as temporally evolving rather than static. If so, it seems quite likely that the secrets of this shifting are known only to the Mihalli, if indeed to anyone at all. The nature of time in Tkumel's universe, which is intertwined with Pavr concept of the Tree of Reality, is a difficult subject. The Grammar of Sun gives us some guidance for understanding it: The universe, says the Engsvanyli philosopher Pavr of Ga, may be likened to a great tree extending from roots at the beginning of time to its highest fronds at the end. The Tree extends along the time-line from Creation to the Final Leaf at the Terminus.
There is nothing before or after the Tree, and no possibility of there being other, similar "trees.".. The trunk of the Tree of Time contains those Planes of Existence which are primary and most probable. The larger branches are those major "bundles" of Planes which have split off due to event (s) occurring at one or another "decision node. " Lesser branches are minor time-lines, and these in turn are diversified into still smaller Planes as they spread farther and farther away from the main trunk. The leaves that grow from each branch are pocket universes, each with its own reality.
Some are tiny "bubble-verses, " while others are almost as extensive as those of the main trunk itself. Each has its own specific contents: suns, planets, creatures, and all the rest. Some are empty void; others are crammed full of matter, like the crowded stars of our own Milky Way. Between them lies the amorphous void of the Space Between, filled with formless, roiling power. At this point Pavr's metaphor falters: the branches of the Tree of Reality, he says, have a tendency to turn and grow back into the trunk or branch from which they split. They thus coalesce into the larger time-line again.
Pavr likens this phenomenon to rivulets that wander away from a stream only to rejoin it later. Those leaves and branches that fail to re-enter the Tree become Shadow Worlds: Planes of Unreality that eventually become tenuous and dissipate into formlessness. It is this structure which the intuitive topological theorists of Tkumel strive endlessly to understand. For example, the structure of the Tree rules out the superficially plausible idea that the multiverse is simply an inchoate multidimensional blob, which has only certain 3 D cross-sections that are comprehensible to (or habitable by! ) 3 D beings. For comparison, consider the many possible 2 D cross-sections of a cube: squares, triangles, even hexagons can be obtained by varying the angles of the cross-section.
While it may or may not be possible, in the context of our mathematics, to embed the structure of the Tree in some single higher-dimensional space, it is quite clear that the internal structure of Tkumel's multiverse is more articulated than the blob-metaphor would allow. A third pos...
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Research essay sample on Three Dimensional Failure Rate
