# Re: cardinal directions

• From: "William H. Martin" <whm2m@virginia.edu>
• Subject: Re: cardinal directions
• Date: Wed, 1 Dec 1999 09:38:25 -0500 (Eastern Standard Time)
• 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
>
> 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

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