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Example research essay topic: Dimensional Space Straight Line - 1,540 words

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... stability is most intriguing: that the Tree of Reality is a multidimensional fractal. Fractals (to make a long story short) arise from unbounded convoluted lower-dimensional curves filling a higher-dimensional space. The fractals we are familiar with are formed using lines (sometimes these are contour lines defining the properties of certain regions when a function is applied to them), which are 1 D objects, but partially fill a sheet of paper, which is a 2 D space.

The 3 D volumes of the Many Planes may be so convolutedly embedded in a higher-dimensional space that they partially fill it. Dimensions like dri and grl would then reach out into the space in which the fractal is embedded. Many fractals are governed by equations; if this equation varied over time, it would change the fractal, which could correspond to the Tree itself changing over time. In the examples below, we will use the Sierpinski triangle, a fractal with a simple, geometrical construction method; although more complex fractals are more likely to be a good model of the Tree, the Sierpinski triangle is easy to envision. Fractals have two properties that seem to fit well with the concept of the Tree. The first property is that of self-similarity, where if one magnifies an area of a fractal that contains certain structures, similar structures will be found in the magnification, no matter how far one zooms in!

This could possibly account for the branches and sub-branches of the Tree at any one moment in Time. Each feature of a fractal, such as a triangle (which is a 1 D object if we only consider its boundary) in the Sierpinski triangle, could be an entire branch of reality, complete with its own Tkumel and Many Planes. Sub-branches, or twigs, of a branch would be connected to or contained within the branch they spring from. For example, a branch of reality in which a peasant travels into Jakalla would contain branches in which the peasant seeks clan aid there, is slain in an intrigue, consults with Temple administrators, and so on.

Each sub feature in a fractal has all features of any larger feature, and thus contains its own Tkumel and Many Planes. The larger a feature is, the more significant or probable a branch of the Tree it constitutes. Note that the larger a feature is, the less likely it is to disappear completely or merge into its parent feature as the fractal morphs over time with changes to its governing equations; these attributes fit well with Part description of lesser branches dissolving or returning to more probable branches. Further, the shape and size of the features determines its multidimensional spatial curvature (about which, see below); below a certain size limit, features may become so curved that only small areas are accessible to its 3 D inhabitants, which fits well with the fact that improbable Shadow Planes are often of limited size. As we know well, history unfolds differently as one follows different branches of the Tree.

The shape of a feature does not determine the state of the multiverse it contains. Different branches aren't the same to their inhabitants, except on the grossest spatial scale; events can and do unfold completely differently, and that unfolding may be determined by (or may determine! ) its location in multistage (i. e. , the coordinates of a point in the fractal Tree with respect to the higher-dimensional space it is embedded in). Perhaps, if history changes course on a meaningful scale, a branch shifts position or alignment? Could this be why the dread minions of the Goddess seek to manipulate events on Tkumel they seek to shift its branch to a region or direction of multi-space where the Goddess holds sway? Another puzzling attribute of the Tree is that seemingly nearby branches, where events have unfolded in very nearly the same way as on our own branch, are very difficult to reach, while distant branches where history is much different are much easier to travel to.

If each feature of a fractal is a branch, and travel through the undifferentiated spaces between features is easier than travel along or through the features of the fractal (perhaps crossing gradients of the fractals equation is difficult, or perhaps navigating the ever-increasing complexity of the smaller features is unmanageable? ), then this makes sense. In the image, compare traveling along the drawn line between two distant branches to traveling back down the branches from either point. A traveler either follows the straight line in the diagram to a distant structure, which because it's distant has a more divergent history, or tries to get to a similar Tkumel (which must be nearby), which entails either crossing color zones in a straight line (bad) or following the infinitely convoluted contours of the features between them and a nearby feature / multiverse (worse -- maybe impossible). A complementary, if less convincing, possibility is that one or two spatial dimensions in a representation of the fractal represent time: in this case, moving back in time might correspond to moving toward the center of a cluster of features. This would explain why one can only go back so far in time: there is an upper limit to the size of features in many fractals, indicating a limit to how many steps one could return, while one could trace forward through smaller and smaller features to the End of Time. Some of the Many Planes even sound like temporally shifted versions of other Planes: does not the Land of Que, for example, sound somewhat like ancient Tkumel?

We have not yet exhausted speculation regarding the nature of the Many Planes. Not only the flora, fauna, and magic of distant worlds can be drastically different from those on Tkumel. It is also possible that certain planes have a locally perceptible spatial curvature, unlike the (at least locally) zero-curvature world we live in. Some planes may have positive spatial curvature, for instance, giving an elliptic geometry where the angles of a triangle sum to more than 180 degrees and where nearby objects would appear distant to viewers from a flat universe. Others may have a negative spatial curvature, giving a hyperbolic geometry where the angles of a triangle sum to less than 180 degrees and where distant objects appear close. Such vistas would prove maddening indeed for planar explorers.

Lovecraft knew whereof he spoke when he described the horrors of non-Euclidean geometry in connection with his Old Ones! VII. Adventure Seeds Remarkable as it may sound, mathematics itself could provide a basis for adventure on Tkumel. Some possibilities: I) A Tinaliya discovers what we call Russell's Paradox (the inherent contradiction of a set of all sets that are not members of themselves; is this set a member of itself, or not? ) and "holy war" erupts among the "gnomes." A Tsolyni would be unable to distinguish between this and theological dispute, and it might well produce a comparable level of violent conflict among reason-loving races, especially if they were committed to the view that every collection could be associated with some number. Other mathematical dilemmas, such as Zeno's paradox (how can we move from one point to another when, having moved half the distance to our goal, we must then move half of that distance, and half of the remaining distance, and so on, requiring an infinite sum of movements? ), the irrationality of the square root of 2, or the transcendence of e or might be enough to terrify them as well. II) If tally sticks are employed to record substantial quantities when each party wishes a record of the amount, they could be a fruitful source of intrigue.

Should a debtor lose his or her tally stick for a transaction, what is to prevent the lender from doubling or trebling the amount, particularly if the debtor is untrustworthy? Or, if a lender is thought untrustworthy, the debtor could make it look as if he or she had lost their stick through foul play, then claim that the lender had falsified the record; this would be even more effective if they could steal the lenders stick as well. III) The characters are contacted by scholars or magi who wish to test some of the hypotheses presented here regarding the arrangements of the Many Planes. The characters might be hired to carry complex surveying equipment to measure the angles of a large triangle on a plane. Aside from the practical and magical difficulties of the project, this may also may draw the attention of denizens of the plane - who find surveyors delicious!

Another option is an assignment to travel through many planes, some quite dangerous, in a specific sequence. In both these cases there is a danger that if the party finds evidence that contradicts a powerful and deranged mages pet theory, he or she may prevent their return! Bibliography web web Edwin Abbott, Flatland, (Please put these in Mail bad at that sort of thing &# 61516; ) M. A. R. Barker, Flamesong, Grammar of Sun, Numerology, and Swords & Glory Vol. 1 M.

A. R. Barker & Joe Saul, Interview, Seal of the Imperium 1 Georges Iraq, The Universal History of Numbers, Wiley, 2000.


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Research essay sample on Dimensional Space Straight Line

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