tlhIngan-Hol Archive: Wed Dec 01 06:56:36 1999

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Re: cardinal directions

I hate my Email package.

<Shift>-S, with a slip of the right pinky becomes <Ctrl>-S and 
sends the message with no confirmation option. Bheah.

So, we have all these words we commonly use for directions. 
North, South, East, West, Northeast, North Northeast, etc.

All this has to do with MODIFIED compass directions. Don't 
forget magnetic declination (though we got along quite a few 
years before figuring that out).

Now, add the 360 degrees of the compass. While these systems 
overlay onto one another, they have no real relationship to each 
other, since the numbers are all measured from one point marked 
zero, while the directional words have four peer words for 
direction. The number system is more precise, but in 
conversation, people are more likely to use the words than the 

Now, look at maps. Suddenly, we have longitude and lattitude. 
This has nothing to do with our direction words OR our 360 
degree compass from any particular point, except in nearly 
perverse ways in terms of measurement of idealized circles 
(flat, when the Earth isn't and round, when the Earth isn't).

There is no reason that Klingons could not similarly have 
unrelated systems going between directional words, compass 
numbering systems and maps. Klingons may very well map planets 
in equilateral triangles, arbitrarily numbering the triangles 
with no reference to compass directions.

Once you look on a map which might include one or more of these 
triangles, or which might include some triangularly subdivided 
section of such a map, you could plot any point on the map in 
terms of distance and angle from any corner. Or you could just 
name everything on the map. Or you could have an infinitely 
expandable coordinate system that would continue to divide 
things down to finer grids of triangles.

My point here is that map systems don't have to have anything to 
do with the words people use to discuss directions. They may use 
those words to relate two different points on the map for the 
map's most common functions: "You are here." "Your destination 
is here."

Okrand tells us that there are three different systems for 
dealing with this. One is the direction words he describes. 
Another is a more precise numbering system for directions. A 
third is the three dimensional coordinates used in space. He 
only describes one of these systems. Don't assume that its use 
needs to extend farther than Okrand tells us it does.


On Tue, 30 Nov 1999 22:03:08 -0800 Ben Gibson 
<> 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
> 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 On
> the page 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'


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