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*From*: "William H. Martin" <whm2m@virginia.edu>*Subject*: Re: cardinal directions*Date*: Wed, 1 Dec 1999 09:38:25 -0500 (Eastern Standard Time)*In-Reply-To*: <3844BA1C.9A89BD7E@vip.best.com>*Priority*: NORMAL

You have so many presumptions. Realize that here on Earth, we have the following inconsistencies: 1. We speak of directions in terms of North, South, East, West, Northeast, Southeast, Southwest, Northwest, North Northeast, East Northeast, East On Tue, 30 Nov 1999 22:03:08 -0800 Ben Gibson <drakon@vip.best.com> wrote: > One of my other interests is mathematics. And I admit a > possibly unhealthy concern in how Klingons do math, how > their maps are laid out, etc. The various triangular > overlays we see on Klingon ship displays have a fascination. > Mathematics is one of the basis of science and without > science a culture does not advance. It becomes “kuve” to a > bigger, badder, smarter species. As the Klingons have > survived into our 24th century, and have star flight, they > must have some form of advanced mathematics. The > identification of the cardinal points, and specifically that > there are three, I find intriguing and possibly useful for > extrapolation. > > You bring up several good points. You are correct that a > choice of co-ordinate systems, as well as units of measure > is arbitrary. However each coordinate system has its > strengths and drawbacks. So the question is whether an > “isonormal” coordinate system has any advantages over a > orthonormal one such as what we use. (I think ‘isonormal’ is > the proper term. Orthonormal is an coordinate system in > which the axis are at a right angle to one another, such as > Cartesian coordinates. Isonormal are coordinates in which > the axis are at 60 degrees. In this discussion, I will use > this term to describe a possible Klingon coordinate system. > I realize that we are given cardinal directions that are 120 > degrees apart, however, just as south can be viewed as > “anti-north” or “negative-north”, I don’t see this as too > much of a problem.) > > My initial objection to such an isonormal coordinate system, > which I think is strongly inferred by the arrangement of the > cardinal points, is that when translating from a polar > coordinate system to such an isonormal system, to uniquely > identify a point requires one extra coordinate. You are > correct that in polar notation, one can say “‘ev chan ‘ev, > chuq cha’maH qelI'qam”. But to identify the map coordinates > of the same point, you would have to give a set of 3 vectors > instead of two. The intersection of two rows in such a map > gives not one triangle, but two. So the third coordinate is > required to tell the map reader which triangle the location > is in. This appeared inefficient, and inefficiency in a > warrior society does not appear to be healthy. That is one > of my primary assumptions. That as a warrior society, > efficiency would be very important. While the battle may not > always go to the strong, or the swift, the inefficient will > eventually get their heads handed to them. The proverb > “Ha’DIbaH DaSop ‘e’ DaHechbe’chugh yIHoHQo’” appears to > partially confirm the assessment that Klingons are concerned > with efficiency. From a military standpoint, wasting > resources through inefficiency is the same as doing the > enemies job. > > Now you may argue that such an objection can be overcome by > more precisely defining the length of the basis vectors. And > Klingons are an exact culture. That would be a good argument > for your side. > > An even better argument for your side would be to note a > flaw in using Cartesian coordinates on a globe, such as the > Earth. Earlier I said that all coordinate systems have their > strengths and weaknesses. A weakness of the system used on > this planet is evident from looking at the north and south > poles. All those neat little squares along the equator get > compressed into triangles. In mathematical terms they form a > ‘coordinate singularity’ at the poles. You can give the > exact latitude of the north pole, but its longitude is > undefined and meaningless. From a mathematical perspective, > this is a nasty thing. > > Now guess what happens when you go to an isonormal > coordinate system such as the Klingons apparently use. These > nasty coordinate singularies disappear! They simply do not > exist. It becomes easier to map a globe using such isosceles > triangles than using Cartesian squares. You get far less > geometric distortion in mapping a globe to a flat surface > using the isonormal system, than in using the system we > presently use. (For an example, take a look at > http://www.bfi.org/map.htm and compare it to a Mercator > projection. > > It might be interesting to see what happens to various > trigonometric relationships using a 60 or 120 degree angle > instead of 90. The reason for our use of the 90 degree right > angle has partially to do with the apparent uniqueness of > Pythagorean theorem. I do not know that much work has been > done in using this unusual basis, except by Buckminster > Fuller, and some high energy physicists. But it would be > intriguing. > > You make a comment about how it appears “that Klingons do > not have any special sense of orientation to the poles.” > That I think would be detrimental to the culture, and > unrealistic. (Yes, I know we are talking about a fictional > culture. I am not that loony. However, I am the type of guy > who can’t help but nit-pick such things. It is a sickness.) > An army that does not have food cannot fight. Farmers grow > that food, therefore their efforts are important to the > empire. The time of planting and harvesting is directly > dependant or determined by the climate, which in turn is > governed by the axis of the planet. (Or to be more precise, > the relationship of the farm to that axis and the > orientation of that axis to the orbital plane of Qo’noS) > Since the length of the jaj as well as the growing season is > determined by these relationships, ignoring them appears > dangerous to the culture. > > In short, the relationship to the axis of the planet is of > vital importance to the empire. To ignore such a factor is > problematic for me. > > Two other factors also are involved, first the direction the > sun comes up in the east changes from day to day throughout > the year, yet secondly, every cloudless night, one can watch > the fixed stars wheel around a common point in the sky. > While the sun wanders in its dawn position, that fixed axial > point does not. > > Also there appear to be certain physical restraints to the > internal magnetic dynamo of a planet. The laws of physics > appear to indicate that the axis of the dynamo spin is > roughly the same as the rotational axis of the planet > surrounding it. These dynamos generate the magnetic field > which in turn is what makes a compass point the way it does. > While the sun can be obscured behind clouds, a compass will > still work. > > (Counterargument: While a compass needle will always point > north, there is nothing that says what that needle is > mounted on. A compass card with the appropriate labeling > need not suffer that problem.) > > (One may bring up precession of the axis at this point, but > note that the frequency of axial precession compared to the > frequency of the sun’s dawn position. It is much longer and > is not noticeable in the span of a single lifetime. However > the solar drift is quite noticeable every year. In 3000 BC > Thuban (Alpha Draco) was the north star. Today, 5000 years > later, it is Polaris) > > For information, including means of building your own Klin > Zha set, goto http://www.fyi.net/~kordite/klinzha.htm. On > the page http://www.fyi.net/~kordite/takzh/variant.htm it > discusses the means by which one identifies a specific > position on the triangular board. (I found it a rather fun > game, until my son got to the point he could quickly beat me > every time.:) ) > > > > Ben (DraQoS) > > > > charghwI' Will Martin UVA ITC Computer Support Services

**Follow-Ups**:**Re: cardinal directions***From:*Ben Gibson <drakon@vip.best.com>

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