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=60o.)  Several others seem to be right triangles and Hoagland proposes that at least one of them has the angle E = 34.7o 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=69.4o.)   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 3o

       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 45o-t/2, where t =arcsin(1/3). This angle is approximately 35.25o. The two equal angles in the isosceles triangle must be exactly equal to 45o+t/2. This angle is approximately 54.75o.

       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 Journal of Scientific Exploration (Vol. 13, #3, Summer, 1999, pages 373-400). In that paper, twelve formations referred to as "mounds" are singled out by certain simple criteria.  Following Hoagland, Crater and McDaniel look for triangles whose vertices lie within the mounds. Having noticed that two specific types of triangles are formed by some of the mounds (a certain right triangle and a certain isosceles triangle, as briefly described above), they look for additional appearances of those same triangles when they consider all twelve mounds.  There turns out to be a surprising repetitiveness in the appearances of those two triangles.  The result is that it is possible to find a way of arranging 19 of those triangles so that their vertices are within the twelve mounds and so that only a single point within each mound is used as a vertex.  Most of the paper is devoted to performing an elaborate statistical analysis which shows that this repetitive pattern would be extremely rare if the locations and sizes of twelve mounds are chosen in a random way. The approach is experimental, based on making a comparison with a large number of configurations of twelve mounds whose sizes and placements are generated by a random number generator.  This is designed to test what the authors refer to as the "random geology hypothesis" for the location of the twelve mounds at Cydonia.  The conclusion that they draw is that the random geology hypothesis fails by a very large margin.

      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.


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  54.75o,   90o and  35.25o.  The triangle to the right is an isosceles triangle whose two base angles are approximately equal to 54.75o  and whose apex angle is approximately 70.5o.  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 45o+t/2,  90o,  and 45o-t/2. The special isosceles triangle has its two base angles exactly equal to 45o+t/2 and its apex angle equal to 90o-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:

" . . . What drew attention to the mounds initially were claims by another writer about an apparent isosceles and right triangle among just mounds EADG. When we examined that configuration we found an unexpected geometric peculiarity not noticed by the earlier writer; the existence of two similar right triangles (GEA and ADG) and one isosceles triangle (ADE) defined by just one angle. Predicting a priori upon this discovered peculiarity, it was subsequently confirmed that mounds B and P fall within the predicted pattern, and step by step as each further mound was taken into consideration, the predicted pattern was found to be evident among all 12 mounds. . . . "

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 a priori. Crater responds that the surprising thing is the continuation of the appearance of the special right and isosceles triangles. However, the analysis in the paper does not distinguish between the appearances of those triangles among the initial mounds E, A, D and G and the appearances of those triangles which were subsequently discovered. All those appearances are treated equally and so, in my opinion, Sturrock's objection is valid. The statistical analysis should have been designed to test just the continuation of the pattern.
  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.

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