tlhIngan-Hol Archive: Tue Nov 30 22:03:32 1999

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Re: cardinal directions

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'

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