The
so-called mounds in the Cydonia region of Mars are certain formations which
were first pointed out by Richard C. Hoagland. Among the many triangles that
Hoagland indicates in his diagram of the Geometry
of Cydonia, several of them have their vertices contained
within these mounds. Hoagland points out that one of these triangles
seems to be equilateral. (In his diagram, it's the small triangle to the
very left, indicated by three A's for its angles. Recall that A=60^{o}.)
Several others seem to be right triangles and Hoagland proposes that at least
one of them has the angle E = 34.7^{o} as its smallest angle. This is an example of what Hoagland means by redundancy because this angle is the same as one of the
angles proposed by Torun for the D&M
Pyramid. (It is half of the angle D=69.4^{o}.)
Here is another picture showing these same triangles: City
Geometry. As Hoagland points
out in __The Monuments of Mars__, the margin of error in measuring these
angles is rather large, perhaps as much as 3^{o}.

A different suggestion was put forward by Horace Crater. Using the letters in the picture "City Geometry"(not to be confused with the angles from the
D&M Pyramid), Crater proposed that the triangles ABD, AEG, and GAD are similar right triangles and that triangle ADE is isosceles, with the angles at A and E equal to each other. A simple geometry argument then shows that the smallest angle of each of those three right triangles must be exactly equal to 45^{o}-t/2, where t =arcsin(1/3). This angle is approximately 35.25^{o}. The two equal angles in the isosceles triangle must be exactly equal to 45^{o}+t/2. This angle is approximately 54.75^{o}.

Here is a summary of the paper ** Mound Configurations on the Martian Cydonia
Plane** by Horace Crater and Stanley McDaniel, which has been published
in the

The reader can find substantially more background in Cydonian Mound Geometry: A Closer Look The paper itself is now available on the internet: Mound Configurations on the Martian Cydonia Plain. On that same webpage, just after the paper, the reader will find objections to the methodology of the paper which were raised by the referee P.A. Sturrock. That is followed by Crater's response to those objections. Readers interested in Cydonia may want to consult the JSE volume cited above which also contains an interesting article about Cydonia by David Pieri - a scientist from JPL - and a response to that article by Mark Carlotto, Horace Crater, James Erjavec, and Stanley McDaniel. Some of those articles and other somewhat related articles can be found here: Peer Reviewed Journal Publications and Other Recent Articles.

In May, 1998, Horace Crater sent me a copy of his paper with McDaniel and asked me if I would be willing to carefully critique it. I devoted considerable time during the month of June to studying the paper and then prepared a four-page critique in July. Crater replied to the critique soon after, responding to some of the points I made, and agreeing that additional statistical analysis would be useful in dealing with other issues that I raised. He also told me that he appreciated my "roll-up-the-sleeves" type of skepticism. In the paper itself, Crater acknowledges my input. What follows is an abbreviated version of my critique. I have continued to think about the approach of the Crater-McDaniel paper, and may add more comments in the future. Preceding my critique are some illustrations which describe the two special triangles referred to above and show some
of their properties which are relevant to the discussion.

ILLUSTRATIONS

Illustration 1. This shows the Cydonia region where the mounds are found interspersed with large mountainous-looking formations. A number of the mounds are identified by letters and some of the triangles with vertices inside of these mounds are shown.

Ilustration 2. The photograph on this page shows the twelve mounds with their identifying letters.

Illustration
3. This is an illustration of the
two special triangles in the Crater-McDaniel study. The triangle to the
left is a right triangle whose angles are approximately equal to
54.75^{o}, 90^{o} and 35.25^{o}.
The triangle to the right is an isosceles triangle whose two base angles
are approximately equal to 54.75^{o} and whose apex angle
is approximately 70.5^{o}. Notice that this isosceles triangle
can be folded in half (along the vertical line through its apex) to form
the right triangle. To give the precise values for the angles in
these special triangles, let t=arcsin(1/3)=19.47122043....
Then the special right triangle has angles exactly equal to 45^{o}+t/2,
90^{o}, and 45^{o}-t/2.
The special isosceles triangle has its two base angles exactly equal to
45^{o}+t/2 and its apex
angle equal to 90^{o}-t.

Illustration 4. This illustration shows the relationship of the special triangles (in green) to the rectangle whose sides have a ratio equal to \/2. If the shortest side of that rectangle has length 1, then the longer side has length \/2. In the first picture, the hypotenuse of the right triangle coincides with a diagonal of the rectangle and its length is \/3. (One sees this by using the Pythagorean Theorem.) In the second picture, the special isosceles triangle has the longest side of the rectangle as its base. I will refer to this rectangle as the "\/2-rectangle."

Illustration 5. The \/2-rectangle just mentioned has a very simple "self-replicating" property. If one forms a larger rectangle by placing two such rectangles next to each other with their longer sides in common, then the larger rectangle has the same shape as the smaller ones. The ratio of the sides is still equal to \/2. In the illustration, the two smaller rectangles have sides of length \/2 and 1. The larger rectangle has sides of length 2 and \/2. The ratio \/2/1 equals the ratio 2/\/2. Thus, two of these rectangle produce a third rectangle of exactly the same shape when they are placed along side of each other as in this illustration.

Illustration 6. The self-replicating property of the \/2-rectangle discussed above results in a self-replicating property for the special triangles. This illustration shows an example of this self-replicating property. By placing four of the special right triangles (of the same size) as in this illustration, one uses just six points as vertices. The four triangles are indicated by colors and the six vertices are labeled A-F. But fourteen of the special triangles are actually produced using these same vertices. (Try to find them. Twelve are the special right triangle and two are the special isosceles triangle.) If one used an arbitrary right triangle, instead of the special one, and laid four of them out as in this illustration, then four more would be produced. In addition, two isosceles triangles (which give the right triangle when folded in half) would be produced. Thus, ten triangles would be produced instead of fourteen.

Illustration
7. This shows four of the mounds and three special
triangles that are formed by choosing vertices within the mounds. The two
triangles AEG and GAD are the special right triangles and the triangle
ADE is the special isosceles triangle. The self-replicating property of
the special triangles is also illustrated here. The special right triangle
AEG and the special isosceles triangle ADE, arranged to have the side AE
in common as illustrated, produce the third triangle GAD which is
in fact the special right triangle.

CRITIQUE: The repetitive appearance of these special triangles is intriguing and puzzling to me. I am not able to dismiss this kind of evidence. However, for various reasons, I find it far from convincing. Here are some of those reasons. The most serious is the second. It concerns the self-replicating property that has been illustrated above.

1. There is no doubt that
regular geometric patterns of some kind or another can be found in any
given set of points. This is analogous to finding numerical relationships
in any given collection of numbers. Such patterns might seem rather
surprising and striking. They might give very impressive results statistically.
But unless the pattern is chosen in a completely *a priori *way, it
is very difficult to evaluate the significance of finding them. This issue
is discussed very clearly by Tom Van Flandern in the following quote (which
readers of my essay on the D&M Pyramid
will have already seen):

* "...... In any truly random
data set, many regular patterns can always be found. For example, if we
have a star chart with a million stars, we might find an unusual shape
formed by stars that has less than one chance in a billion of happening
by chance. So are some mysterious super-beings moving stars around? This
is not as likely as the simpler explanation: In every random data set capable
of forming billions of random patterns, it is virtually certain that some
1-in-1 billion pattern will be found formed by chance.*
*
In general, we tend to be deceived because our minds often do not recognize
how truly vast is the number of possible coincidences that can occur. So
when a few of them do occur, as they must if the odds are right, we tend
to be amazed simply because the odds against that particular coincidence
were very great. The odds against a flipped coin coming up tails ten straight
times are 1024-to-1 against. But if we make several thousand attempts,
the odds become pretty good that it will happen one or more times.*
*
In science, an improbable event that has already happened is called
"a posteriori" (after the fact), and generally is taken to have no significance
no matter how unlikely it might appear. By contrast, if we specified a
certain specific highly improbable event in all its detail "a priori" (before
the fact), and it happened anyway, that would be significant, and we would
be obliged to pay attention."*

The repetitive appearance of the
interesting special triangles that Crater and McDaniel discuss does seem
rather striking. But there is an infinite variety of patterns that might
be exhibited by a collection of twelve (or any number) of mounds and many
such patterns might seem (subjectively) equally striking. For this
reason, it is not clear to me how unusual it really is to find such a striking
pattern that does extremely well in a suitably designed statistical test.

2. It is important to keep
in mind the sequence of events. Hoagland and others had been thoroughly
scrutinizing the locations and shapes of various objects in the Cydonia
region. Hoagland pointed out that certain triangles formed by the mounds
appeared to be right triangles or isosceles triangles. Crater then noticed
that if one chose points within four or five of the mounds as vertices,
then one could find several of the special triangles shown above in illustration
3. This by itself does not seem very remarkable, especially given
the leeway allowed by the size of the mounds (which amounts to a few degrees
of leeway for the angles). But Crater then discovered that these very same
triangles continue to appear. In the end, he found that 12 of the 220 possible
triangles formed by the twelve mounds would be in the shape of the special
right triangle and that 7
would be in the shape of the special isosceles
triangles. It is the appearance of the additional triangles beyond
the several which Crater originally noticed that seems remarkable.
It is for that reason that I cannot dismiss the pattern discovered by Crater.

However, the self-replicating property of the special triangles is clearly
a very significant contributing factor to the appearance of these additional
triangles. Illustration 7 shows an example of this self-replicating
property among the triangles initially noticed by Crater. As
another example, the three special triangles PGL, PGE, and LEA produce
three more special triangles PLA, PEL, and AEG. It is very difficult
to judge how strongly this self-replicating property affects the outcome
of the statistical analysis in the Crater-McDaniel paper.

In my opinion, this is a serious flaw. As I mentioned above, the
pattern that the authors have studied was first noticed by looking at four
or five of the mounds pointed out by Hoagland. It is clear that the
self-replicating property of the special triangles played a role at that
early stage. Crater noticed the appearance of triangles of a roughly similar
shape
formed by those initial mounds. This led him to ask if those triangles could be exactly similar. Crater determined that if this were so, then the triangles would have to be the special triangles. In essence, it was the self-replicating
property that led Crater to single out those triangles. The statistical analysis of the paper does not take this order of discovery into account and so
is not really "a priori" in the sense described by Van Flandern.

This issue
could be studied by additional statistical tests. One indication that
the mounds which were initially examined may have a very significant effect
is that if one counts the number of times each mound occurs as a vertex of
one of the 19 appearances of the special triangles, one finds that
some of those initial mounds occur quite disproportionately. Mound A, which is a vertex in each of the original set of four triangles singled out by Crater, turns out to be a vertex in 10 of the 19 appearances of the special triangles. Mound E is a vertex in 7 and mound G is a vertex in 8 of the 19 appearances. If one omits just one or two of the mounds A, E, or G from the set of 12 mounds, then the statistical anomaly is reduced dramatically. For example, if one omits E and G, then there are 6 appearances of the special triangles with vertices among the remaining 10 mounds. If one omits A and G, then there are just 3 appearances of the special triangles.

The 12 appearances of the special right triangle are as follows:
AEG, GAD, ABD, EAB, PGE, JPD, PGL, GKL, LEA, PLA, MQA, and GKO. The 7 appearances of the special isosceles triangles are: ADE, PMA, KAE, PEL, QJG, KLQ, and GMO.

In his referee's report
for JSE (as published in the same issue of JSE as the Crater-McDaniel paper), Sturrock begins as follows:

*The authors state "attention was first drawn to these objects because of the apparent regularity of arrangement... of the mounds P, G, E, A, D, and B," then they carry out analyses of all mounds including P, G, E, A, D, and B. This is not a valid procedure. One should not use the same data set to search for a pattern and to test for that pattern." *

In reply, Crater writes:

Sturrock is objecting to the fact that the statistical analysis in the Crater-McDaniel paper involves the initial data which had led to the discovery of the pattern in the first place. That is, the approach is not truly

In addition to Sturrock's objection, the point that I am making is that the statistical analysis must also be designed to take into account the fact that the appearance of the special right and isosceles triangles among the initial mounds E, A, D and G strongly influences the additional appearances of those triangles, and thus the continuation of the "predicted pattern." This is because of the self-replicating property of those special triangles. The crucial role of those initial mounds is clearly indicated by how often they occur as the vertices in the 19 observed triangles. Furthermore, each time that a new special triangle appears, it increases the chance of more such triangles appearing, again because of the self-replicating property.

3. The statistical analysis in the paper tests what the authors refer to as the "random geology hypothesis." However, geological processes are definitely not random. One could just think of mountain ranges or the locations of volcanoes to realize this. Whatever processes led to the formation of the mounds (if they are not the result of intentional design) would create some rough kinds of regularities. Perhaps, there could be some rough tendency towards collinearity or parallelism which could be a contributing factor to the repetitive pattern. This possibility is not taken into account in the paper.

4. If the twelve mounds were
indeed intentionally placed with geometric considerations in mind (e.g.,
the repetitive appearance of certain triangles), then it is very difficult
to understand why there isn't much more symmetry in their placement.
Of course, as one would expect to find in any set of points, there is
some small degree of symmetry. As an example, the three triangles EAB, AEG, and
PGE seem to be the beginning of a nice
symmetrical pattern - what one might refer to as a "tiling pattern."
But that pattern does not continue.

However,
as a partial response to this lack of symmetry, it is worth mentioning
an observation of McDaniel. He points out that there could have been
a larger collection of mounds whose centers are the points of a grid based
on the \/2-rectangle
and which included the twelve mounds studied in the Crater-McDaniel paper.
Thus, it is possible that the repetitive appearance of the special triangles
in the placement of the 12 mounds is a remnant of a much more highly symmetrical
placement of a larger number of mound-like structures.

5. Michael
Malin makes the point that the Viking photographs can only give an
extremely imprecise determination of the location of the mounds and other
formations. He therefore discounts the study of geometrical properties
which depend on the determination of those locations. Nevertheless, the
statistical anomaly in the Crater-McDaniel paper remains puzzling and Malin's
remark does not explain it away.

One objection that could be raised is the following: As I mentioned above,
only one point within each of the twelve mounds is used as a vertex of
the triangles identified by Crater. Thus, the configuration of 19 triangles
that Crater has found form what he calls a "coordinated-fit." However,
based on the location of the mounds as determined from the Viking photographs,
this common vertex within each mound is not always the apparent center
of the mound. For a number of the mounds, the point is far from the apparent
center of the mound. That means that if one considers the actual
centers of the mounds (based on the Viking photos), and forms triangles
using those centers as the vertices, then one will find 19 triangles
which are only approximations to the special triangles, and, in some cases,
rather rough approximations.

On the other hand, because Crater insists on a "coordinated-fit," the vertices
of the 19 special triangles he identifies would be 12 specific points in
the Cydonia region. It is therefore at least conceivable that the actual
mounds would have those 12 points as their precise centers.

Back to the essay on the D&M Pyramid

***