C/A ~ \/2 ,           B/D ~ \/3  ,         C/F   ~  \/3  ,        A/D  ~   e/pi

To explain the notation,  if a and b are positive numbers,  then  a ~ b  will mean that a and b are approximately equal.  The error will always be expressed as a percentage,  namely  the difference between a and b expressed as a percentage of the number b.  For example,  one might write  199..~..200. The difference is 1 and so the error is .5%  because 1 is .5% of 200.   The percentage errors for the above four approximations are    .5%,   .17%,  .7%,  .08% , respectively.  The percentage errors in the numerical relationships listed by Hoagland are in the range .05% to 1.6%,  with most of them in the range  .1% to .5%.   In what follows, the numerical relations listed in Hoagland's book will be referred to as the "Torun-Hoagland Model."
 

ADDITIONAL RELATIONSHIPS:  We will give a number of additional numerical relationships exhibited by the above eight angles which have a similar level of accuracy to those listed by Hoagland.  These were either overlooked or ignored by Hoagland and Torun.  Some of these new relationships fit into nice sequences with some of those from the Torun-Hoagland Model.  The percentage errors will be given in parentheses.   For comparison, one can find a tabulation of the percentage errors for all of the numerical relationships given in The Monuments of Mars  here.
 
 

 Ratios of square roots.    Here is a set of six relationships involving ratios of the above angles and ratios of consecutive terms in the sequence of square roots of positive integers:     \/1 ,  \/2,    \/3 ,    \/4,     \/5,     \/6 ,    \/7    .

C/A~.\/2 /\/1........................( .5% )
H/G ~ \/3 /\/2........................(.11%)
D/A ~ \/4 /\/3........................(.17%)
H/F~ \/5 /\/4.........................(.27%)
F/G ~ \/6 /\/5.........................( .4% )
A/H ~ \/7/\/6.........................(.45%)
 

 The first of these approximations is one of the relationships in the Torun -Hoagland Model. The percentage errors (indicated in parentheses) are well within the range which Hoagland and Torun allow for the relationships which they found.  Also ,  it should be pointed out that these relationships are not independent.  For example, multiplying the relationships involving F/G and H/F will give the one involving H/G.
 

More square roots.  The first two numerical relationships of the Torun-Hoagland Model  are the beginning of another nice sequence of approximations.  But another angle from the D&M pyramid is needed which Hoagland denotes by the letter  I and which has the value I = 100.4^o.

C/A ~ \/2....................( .5% )
B/D ~  \/3....................(.17%)
D/E ~ \/4....................( .0% )
I/G ~..\/5....................(.45%)
C/E ~  \/6....................(.35%)
B/G~ \/7....................(.6% )

Gamma.  There are several relationships involving the number gamma  ( the Euler-Mascheroni Constant ).
 

D/B    ~    gamma ........( .2% )                        COT(A)    ~    gamma ..............(.025%)
F/C    ~    gamma..........(.75%)                        2SIN(A)     ~   1/gamma             (.025%)

These relationships should be compared to some of the relationships in theTorun-Hoagland Model ,  namely those which involve \/3. It just so happens that  \/3 ~  1/gamma  with an error of about .025%.  Therefore, any relationship involving the number \/3.can be replaced by one which involves the number 1/gamma. The percentage error will change slightly.
 
 

Logs.   Here are a couple of relations involving natural logarithms.
 

D/I   ~   ln(2)               (.3%)

F/G  ~   ln(3)               (.1%)
 
 

CRITIQUE:   I have given so many additional relationships just to make the point that with the rather generous level of accuracy that is allowed by Hoagland and Torun,  it is fairly easy to find a variety of striking numerical relationships.  This same point will be made on another page involving a "Telephone Number Experiment"  which I did in the Fall of 1996.  That experiment was an attempt on my part to convince Richard Hoagland that one should not put any significance on the existence of such relationships.  In his book, his public lectures, and his frequent appearances on the Art Bell Show,  he has often talked about the "Geometry of Cydonia," describing it as an exquisite set of mathematical relationships exhibited throughout the Cydonia region. He claims that these mathematical relationships contain some kind of message which is related to the geometry of a regular tetrahedron inscribed in a sphere and related furthermore to what he calls "hyperdimensional physics."   On the basis of this claim, he argues that this set of relationships must be part of an intentional design and therefore constitutes strong evidence that various formations, including the D&M Pyramid and the Face on Mars, are artificial.
              Undoubtedly humans will eventually visit Mars and have an opportunity to take a close look at the D&M Pyramid, the Face, and some of the other formations in that region. It then may be obvious whether or not they are artificial.  At the present time, we only have the photographs taken by the Viking Orbiter in 1976 and by the Mars Global Surveyor since 1998.  These have been extensively examined by various people.  (Here is a small collection of  links.)  Concerning the D&M Pyramid, it is just a conjecture that this object was at one time a symmetrically-shaped,  five-sided pyramid. It is not even clear that it ever had five well-defined corners or a true apex.  By themselves, the photographs of the D&M Pyramid (which date from 1976) don't seem to me to provide a good case for the artificiality of that object.  But do the mathematical relationships that were noticed by Erol Torun and Richard Hoagland strengthen the case? In my opinion, the answer is no!
             I made my objections known to Art Bell and he told me several times that he would try to arrange a debate between myself and Richard Hoagland. But it didn't happen and so I decided in May, 1998 to directly challenge Hoagland to a debate. He never responded to my letter, but on May 26th, to my surprise, Art Bell confronted Hoagland on the air with my challenge.   Hoagland refused my challenge and chose to be rather impolite and insulting in his refusal.  Some of my acquaintances who listened to him thought that he seemed terrified of the prospect of debating me.   Although I believed that my mathematical arguments were quite strong, I felt rather terrified myself because I had absolutely no experience in such a confrontational, debating situation.  Nevertheless,  I tried for several months after that to pursuade Art Bell to let me have just fifteen minutes on the air in order to briefly explain my point of view and to put more pressure on Hoagland to accept my challenge to a debate, but to no avail.  In any case, here are the main points that I would try to make in such a debate.

         1.  Hoagland  and Torun have given themselves too much freedom in finding the relationships that they present.  These relationships are just approximations and the level of accuracy that Hoagland and Torun accept is rather generous.  Perhaps they believed that allowing such a level of accuracy was justified because of the fact that the angles themselves could not be estimated very accurately.  No matter what their reason,  the existence of many such relationships is not at all surprising, as I've already pointed out above.  My telephone number experiment was intended to show that even if eight angles were chosen in a random way, one could still find many striking relationships involving those angles and with a level of accuracy similar to that in the Torun-Hoagland Model. The reader should make a close comparison of the percentage errors in my telephone number experiment  with those occurring in the relationships found by Torun and Hoagland (which I have tabulated..here).  The percentage errors vary considerably in both cases, but the overall level of accuracy is quite comparable.
           One way to view the matter is as follows.  To start with, there are the nine angles A, B, . . . , I.  One can then compute the ratios formed by these angles. This would give 72 numbers (excluding the ratios A/A, B/B which are equal to 1). In addition, one can evaluate the trigonometric functions sin, cos, and tan at those nine angles.  This gives 27 more numbers.  One can also include the radian values for those angles, which gives 9 more numbers. Altogether, one has a collection of more than one hundred numbers. They turn out to be mostly in the range between 0 and 4 and more than half of them are in the range between 0 and 1. On the other hand, there are certainly many important mathematical constants which occur in the same ranges, forming another collection of numbers. In effect, one has two large collections of numbers crowded into a fairly small interval. That some numbers from the first collection turn out to be close approximations to some numbers from the second collection is inevitable. The number of such occurrences depends on how close one requires the approximations to be.
           In The Monuments of Mars,  Richard Hoagland dismisses the possibility that the numerical relationships and alignments that he refers to as the "Geometry of Cydonia." could just be coincidental.  He seems to find those relationships too remarkable to be just a set of random coincidences.  Apparently, that is what his intuition about such things led him to believe. But I have looked at all of those relationships too and my intuition told me that they could easily be completely unintentional coincidences.  My purpose in doing the "Telephone Number Experiment" was to demonstrate in a simple and concrete way to Hoagland and others that my intuition about such things is quite well-founded.  My choice to use eight numbers formed from telephone numbers in the experiment was to make it obvious that whatever relationships involving e, pi, or other mathematical constants that exist among those numbers must be coincidental since no one could reasonably suggest that such relationships were intentional.  I discovered many such relationships, some of which fall into rather striking patterns.   And so, if one can find an abundance of numerical relationships from a set of numbers found in the telephone directory, why should one put any significance on similar kinds of relationships for a set of numbers found in some other way?   I have raised this issue a long time ago, but Richard Hoagland and his associates have failed to offer an adequate response.  Coincidences abound. One needs good, sound arguments to believe that something is not a coincidence and so the burden must be on Hoagland to provide such arguments, especially because of the quite unextraordinary nature of the relationships that he presents.
           It is usually impossible to prove that something is just a coincidence. For example, I cannot think of a way to definitively prove that the architect who designed the Great Pyramid of Giza did not really have the number pi in mind when he chose the proportions of that structure - a popular idea that was first suggested by John Taylor in 1859.  The relationship of pi to the Great Pyramid which was proposed by Taylor is impressively accurate. That accuracy is the primary reason that many people have taken the proposal seriously. But, as it turns out, that reason is rather deceptive. See my page Pi and the Great Pyramid for a discussion of this interesting issue.
          I also cannot think of a way to definitively prove that the Torah does not intentionally include a prediction of the assassination of the Israeli Prime Minister Rabin, as has been suggested by Michael Drosnin . See my page The Bible Code for a survey of this topic.  Immediately after Drosnin published his book about this, various people did "control experiments"  to show how easily one could find such "predictions" in various other texts and documents. Such experiments are in the same spirit as my experiment with telephone numbers.    
       A Martian architect could indeed design a five-sided pyramid incorporating the relationships proposed by Hoagland and Torun as rough approximations. However, one of the points that I will explain below (in 2) is that it would not be geometrically possible to incorporate all, or even most, of those relationships with a high degree of accuracy. This means that if one is willing to grant the premise that an architect who is designing a massive object such as the D&M Pyramid would strive for a high degree of exactness in his design, then one must conclude that many of Hoagland's mathematical relationships are in fact merely concidental. Also, I will point out in 5 that if one accepts a theory known as the Exploded Planet Hypothesis, then one is led to conclude that Hoagland's very accurate relationship relating the latitude of the D&M Pyramid to the mathematical constant e/pi  must indeed be purely accidental - just another nice example of the ubiquity of elegant, coincidental mathematical relationships.
 
 

          2.  An obvious question to ask is whether the percentage errors occurring in the relationships given in the Torun-Hoagland Model could turn out to be significantly better overall when (and if) more precise measurements of the angles occurring in the D&M Pyramid are made (assuming that it is even like a pyramid and has precise angles to be measured).  The answer is no! It would be possible for some of the percentage errors to be better, but then others would necessarily be worse.  Even if one just considers the relationships involving the four angles A,B,C, and D, it turns out to be impossible for all of them to be accurate.  Here are some of those relationships.

A=60^o
B=120^o
C/A= \/2
B/D=\/3
A/D=e/pi
C/D=e/\/5
C/B=\/5/pi

The source of the problem is that the four angles A, B, C, and D are not independent. Elementary geometry implies that there are two simple relationships between them.  These can be seen by looking at the diagram here where the angles are clearly indicated by colors.  The two geometric relationships are: B=2A  and  B+2C+D=360^o. When one examines the Torun-Hoagland Model with these two relationships in mind, one finds very quickly that their relationships cannot all be accurate.  To illustrate this point, suppose for example that A=60^o  and C/A=\/2 turn out to be accurate to within a percentage error of  .1%. Then B=2A would be in the range 119.88^o < B < 120.12^o and D=360^o-B-2C would be in the range  69.83^o< D < 70.76^o .  But then B/D would be a rather poor approximation to \/3=1.73205.... The value of B/D would be at most 1.72018 and the percentage error of the relationship B/D=\/3 would be almost .7%.
             This problem pervades the Torun-Hoagland Model.  To put it simply, the relationships that they propose in their model are contradictory.  It is impossible to design a structure for which all or even most of these relationships are achieved to a high degree of accuracy.   The seriousness of this problem becomes more apparent if one tries to imagine a Martian architect in the process of designing the D&M Pyramid.  Let's assume hypothetically that he wants to incorporate the two relationships A=60^o  and C/A=\/2 into his design.  He would certainly realize that the angle B=2A must be 120^o and that the angle D is then completely determined and would calculate it to be

                                      D  =  360^o - B - 2C = 360^o - 120^o - (2 x \/2 x 60^o)  =  70.29437...^o

Using this value for the angle D,  it turns out that B/D=1.70710..., which is a very poor approximation to \/3 = 1.73205....  The percentage error is almost 1.5%.  For this reason, it seems inconceivable that the Martian architect would also have the relationship B/D=\/3 in mind.   The relationship A/D=e/pi would have a percentage error of .6%. The relationship  C/D=e/\/5  would have a percentage error of more than .7%. The relationship C/B=\/5/pi  would have a percentage error of more than .6%.  And so, in this hypothetical situation,  it is difficult to imagine that the Martian architect would seriously consider any of these relationships as part of the design.
            Let's consider another hypothetical possibility which turns out to be somewhat better. The architect might want to incorporate the relationship A=60^o and B/D=\/3 into his design.  Then B must again be 120^o,  D would be 69.28203...^o  and C would be 85.35898...^o.  Using this value of C,  it turns out that C/A=1.42264... and this is a poor approximation to \/2.  The percentage error is almost .6%.   The relationship C/D=e/\/5 would have a percentage error of more than 1.3%.  But the relationship C/B=\/5/pi is fairly accurate, the percentage error being just .06%.   Since A=B/2, the ratio A/D  is exactly equal to (B/D)/2 = \/3/2 and the relationship A/D=e/pi would have a percentage error of .09%.
            There are other elementary geometrical relationships between the various angles considered by Hoagland and Torun.   In addition to the two mentioned above, one has: C+G+F=180^o,  D+2H=180^o,   G+H=I,   2E=D. Taking all of these geometric relationships into account, the architect would only need to choose values for three of the above nine angles. The remaining six angles would then be completely determined.  For example, just choosing specific values for the angles A and C would determine the angles B, D, E and H. If the architect then made some choice of the angle F, the angles G and I would be determined. The choice of those three angles could conceivably be guided by a desire to incorporate a small number of additional relationships into the design of the structure, possibly even a few of the relationships in theTorun-Hoagland Model. Other relationships would arise as direct consequences (such as tan(A)=\/3 if A is chosen to be 60^o).  And still more relationships would arise with various degrees of accuracy as merely accidental, unintended consequences of the choice of those three angles.
            I have examined a number of hypothetical possibilities for choosing three of the nine angles in a way which incorporates some of the relationships in the Torun-Hoagland Model.  Just as in the above illustrations,  it always turns out that many of the other relationships become quite inaccurate.  Torun and Hoagland were aware of the relationships coming from elementary geometry (which are highlighted above in red). This is apparent from the fact that the angles A,B,. . . ,I that they propose satisfy these relationships exactly.  But it is also obvious that Torun and Hoagland did not take these simple geometric relationships into account in proposing the set of relationships in their model.  In my opinion, this makes their proposed model untenable because there is no doubt that an architect sitting at his desk to design such a massive structure would take elementary geometry very much into account.
 
 

         3.  Richard Hoagland has described the D&M Pyramid as the "Mathematical Rosetta Stone of Cydonia." He seems to see a message encoded in all of the mathematical constants which occur in the Torun-Hoagland Model for that object - a message involving specifically the geometric properties of a regular tetrahedron inscribed in a sphere.  Furthermore, he claims that this message reveals that the designers of the D&M Pyramid and the other objects in the Cydonia region had knowledge of some kind of "hyperdimensional physics."
                I am baffled by Hoagland's interpretation of those mathematical constants. Even if it were somehow known that the D&M Pyramid had really been designed by some intelligent beings and that all of the relationships that he and Torun have found were really intentionally incorporated into its design (overlooking the discussion in 2), his interpretation would make no sense to me. The problem is that the various mathematical constants that occur in those relationships don't support the interpretation that Hoagland is imposing on them.
                I don't see any connection between the geometry of a tetrahedron and constants such as e/pi,  e/\/5, and pi/\/5 (or their reciprocals).   There are many constants which play a very important role in mathematics. Certainly, the numbers e , pi, and square roots such as \/5 are of critical importance throughout mathematics.  But ratios of such numbers are not necessarily  important. The  ratio e/pi, which is a number that Hoagland especially singles out, seems somewhat boring to me.  I cannot think of any formula or theory where that specific number occurs naturally.  Of course, some ratios are indeed important numbers, e.g. the ratio \/3/2 occurs often in geometry, as do both \/3 and 2.   I have no doubt that I could easily find some nice geometric interpretation of the ratio pi/\/5 involving the geometry of Platonic solids and spheres. I could probably find something interesting to say about the ratios e/\/5 and e/pi , but nothing which would make those numbers worthy of being incorporated in a monument.
              Some of the other constants which occur in the Torun-Hoagland Model (e.g.   pi/3,  2pi/3, and \/3 ) are connected with the geometry of an equilateral triangle.  Since the four faces of a regular tetrahedron are equilateral triangles, this would perhaps be an indirect connection with the geometry of a regular tetrahedron. But equilateral triangles occur in a countless number of ways in various designs and geometric objects, of which the regular tetrahedron is just one example.
                 Hoagland and Torun wrote an article in 1989 (the "Message of Cydonia") where they argue that the constant e/pi is the key and that if one replaces e=2.7182818... by the constant  e'=(\/3/2)pi = 2.7206990... in those relationships (so that e/pi is replaced by e'/pi), then one has a compelling connection with the geometry of a tetrahedron inscribed in a sphere.   It is certainly true that this constant e' has very interesting connections with that geometry.  Hoagland and Torun point out the following connection. If a regular tetrahedron T is inscribed in a sphere S, then it turns out that the ratio of the surface areas of the sphere and the tetrahedron is exactly equal to e':

e' =  (Surface Area of S)/(Surface Area of T) ..

Here is another connection. If two regular tetrahedra of the same size are inscribed in a sphere so that each vertex of one is diametrically opposite to a vertex of the other, then the eight vertices (four vertices from each tetrahedron) form the vertices of a cube C inscribed in the sphere. Then the number e' turns out to be exactly equal to the ratio of the volumes of the sphere and the cube:

e' = (Volume of S)/(Volume of C).

But this interesting ("tetrahedral") constant e' does not actually occur in any significant way in the numerical relationships found by Hoagland and Torun. It is not the number that Hoagland and Torun single out as the key.   The number that they single out is \/3/2.  It is this number that occurs a few times in the Torun-Hoagland Model ("redundantly", as Hoagland likes to say).  The link between these two numbers is that they are related by a factor of pi.:..e'.=.(\/3/2)pi.
               Now the number \/3/2 occurs in quite a few contexts and would itself bring to mind many associations to anyone knowledgeable about geometry (without any need to multiply it by the number pi).  Geometry related to a tetrahedron inscribed in a sphere would be somewhere down the list.   Here are some of the associations for \/3/2 that come to my mind:  (1) It's equal to sin(60^o) and cos(30^o).  (2) It's the length of an altitude of an equilateral triangle whose side has length 1.   (3) It's the ratio of the radius of the circle inscribed in a given regular hexagon to the radius of the circle circumscribing the regular hexagon.  (4) It's the radius of the circle circumscribing a rectangle whose sides are of length 1 and \/2. (This rectangle will be mentioned later in connection with the "Mounds" of Cydonia.)  (5) It's the radius of the sphere which circumscribes a cube with side of length 1.  Only the last one in this list has a connection with the geometry of a tetrahedron inscribed in a sphere, a somewhat indirect connection resulting from the fact that the eight vertices of a cube turn out to be the vertices of two regular tetrahedra (each of which has four vertices) and the sphere which circumscribes a cube also circumscribes those two regular tetrahedra.
               Richard Hoagland has lost me in the giant leap that he makes from the constants occurring in the Torun-Hoagland Model to the idea that, sometime in the past, a Martian architect decided to encode a subtle message about tetrahedral-spherical geometry in the design of a five-sides pyramid on his planet.  It is an interesting story, but doesn't correspond very well to those constants.  I can see only an extremely flimsy connection.   But then he loses me completely by making another, even larger, giant leap, suggesting that this encoded tetrahedral-spherical geometry (which, for me, is just part of old-fashioned Euclidean three-dimensional geometry) is really a message encoded by that remarkable architect about some kind of Physics involving more than three dimensions and that this Physics furthermore can somehow explain the locations of volcanos on Earth and other planets as well as the Dark Spots on Jupiter and Neptune.
 

        4.   Hoagland's elaborate interpretation seems absurd for other reasons too.   With the generous level of accuracy that Hoagland and Torun give themselves, there are just too many numerical relationships.  The very number and variety of such numerical relationships would make it impossible to find any clear message.  It would be almost like a mathematical Rorschach test.  Any  "messages" found in the numbers would be a reflection of the mindset of the individual taking the test.  Just as a simple illustration, consider the rather striking set of relationships involving the numbers  \/2,..\/3,..\/4,..\/5,..\/6,..\/7  which I presented above. Those numbers might suggest the curve y=\/x.  This curve is part of a parabola and that might bring to mind many associations. Instead of finding a message involving "hyperdimensional physics," as Hoagland does, perhaps some other individual might regard these relationships as the basis of a message involving classical Newtonian physics, where parabolas occur in a variety of ways.  That interpretation is just one of many that could be imposed on those numbers. It is certainly as tenable as Hoagland's interpretation,  perhaps even more so in light of the discussion in 3.
             As an illustration of how easy it is to impose an interpretation on a bunch of numbers, the reader should visit Tetrahedral Geometry in the Seattle Telephone Directory.  As I was writing the page describing my Telephone Number Experiment, I decided to look for connections with the geometry of a regular tetrahedron inscribed in a sphere (motivated by Hoagland's apparent fascination with that geometry).  It was rather amusing to find that the numbers that come up in that experiment (as ratios, etc.) seem to have an even more compelling connection with that geometry than the numbers occurring in the Torun-Hoagland Model for the D&M Pyramid.
             Even the numbers themselves that occur in the Torun-Hoagland Model are ambiguous.  For example, as Hoagland and Torun are aware,  the numbers  e/pi  and  \/3/2 (which is e'/pi) are quite close to each other.  As another example,  \/3 and 1/gamma are also quite close to each other.  Thus, in any numerical relationship involving one of these numbers, one could also give another numerical relationship involving the other number, with only a slightly different percentage error.  If these relationships are intentional, which of the numbers is what the architect had in mind?
            Any artificial structure will exhibit a large number of unintended relationships in addition to whatever intended relationships might have been chosen by the builders or architects. The Great Pyramid of Giza is an example. One can find quite a number of fairly accurate numerical relationships exhibited by the Great Pyramid. (See my page Some Elegant Numerical Relationships.) It seems likely that only one (or at most one) of those relationships was actually intentional. With an object like the D&M Pyramid, even if one assumes that it is artificial, how can one realistically guess the intentions of the architect.  Those intentions might have absolutely nothing to do with ratios of angles.  Perhaps the architect was more concerned with relationships involving lengths or areas or volumes associated with the object. Such relationships are also bound to occur. (See my page Secrets of a Certain Pentagon for a description of another numerical experiment illustrating this point.)   Even if the architect's intentions did involve ratios of angles, only a subset of all of those relationships pointed out by Hoagland and Torun, or the additional ones pointed out by me, or others that no one has yet noticed could be intentional (as I have argued in 2 above).  But which subset would that be from among the incredibly large number of hypothetical possibilities?
            I am really making two objections here. First of all, among the plethora of relationships that one could find in any artificial object (even with a fairly high degree of accuracy),  it is virtually impossible to decide which ones are intentional (unless there is solid documentation available concerning the design).  Second of all,  even if it were certain that specific mathematical constants were intentionally incorporated in the design, how can one possibly guess what those numbers meant to the architect without knowing much more about the ambient culture and its traditions.

        5. The above comments have concentrated on the Torun-Hoagland Model for the D&M Pyramid, which is just part of what Richard Hoagland calls the "Geometry of Cydonia." In The Monuments of Mars , Hoagland discusses various alignments of objects such as the D&M Pyramid, the Face, etc.  He discusses at length the probabilities of those specific alignments occurring just at random.  He points out various occurrences of angles such as 19.5^o (which is Hoagland's favorite angle and represents arcsin(1/3)=19.4712206...^o).  He points out that the latitude of the D&M Pyramid is related to e/pi.  He points out various triangles which are formed by taking certain formations (the so-called "mounds" ) as vertices.
            The kinds of probability calculations that Hoagland presents are meaningless. He and his associates and followers seem to make this mistake over and over again. Tom Van Flandern explains the issue very well in an essay that I will briefly discuss later.

  "...... 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 last paragraph, in particular, is quite relevant. All of the probabilities that Hoagland calculates are "a posteriori," and so have no real significance. Improbable events are probable! That seemingly paradoxical statement is illustrated fairly well in some of the experiments that I have described elsewhere on this site and referred to above.  Just to choose one of many examples,  consider the two relationships mentioned on my page Secrets of a Certain Pentagon involving the Golden Section phi. One of those relationships is completely intentional, just part of the way the pentagon considered on that page is constructed. But the other relationship is unintentional - an accidental consequence of Horace Crater's random choice of the angle D. The accuracy of that relationship is impressive and the probability for that specific relationship occurring so accurately is extremely small.  But since I did not make an a priori choice to look for that specific relationship, or even the Golden Section as a specific number, there is nothing miraculous about finding such an extremely improbable relationship.
        As far as the occurrence of the angle 19.5^o here and there throughout the Cydonia region, this is fully to be expected because of the incredibly large number of angles that are formed by various objects that Hoagland and others have singled out for attention.  To give the reader some intuition about this, suppose that one considers just 12 points. If you draw all possible line segments between such points, you will have 66 such line segments. The number of triangles that you can form using these line segments as sides (i.e.  using any three of the 12 points as vertices) turns out to be 220.  Each such triangle has three angles and so one gets a total of 660 angles. (If it happens that three points are on a single line, then some of the angles may coincide.)  If instead one starts from 25 points, then it turns out that one can form 2300 triangles and therefore one gets a total of 6900 angles, all between 0^o and 180^o.  In fact, Hoagland has more than 25 points that he could potentially have used as vertices to form angles.  With so many possibilities, it is almost inevitable that quite a number of these angles will be very close to 19.5^o, 45^o, 60^o or any other angle that one might be interested in looking for.  I cannot see any significance in the fact that Hoagland has found several 19.5^o angles, especially because they just occur scattered about and form no systematic or symmetrical pattern.
        Hoagland finds significance in the fact that the (hypothetical) apex of the D&M Pyramid is located at a latitude of approximately 40.87^o N.    As he points out, tan(40.87^o) is approximately e/pi. The level of accuracy is quite high.  Undoubtedly, Hoagland carefully looked at the latitudes of the other objects in the Cydonia region.  Since he mentions nothing about the latitude of the Face itself, it would seem that he found nothing that he considered significant about it. The latitude of the center of the Face, which I will denote by the letter L,  is between 41.18^o N and 41.19^o N (based on a map in Hoagland's book).  Actually,  there is a very nice relationship for that latitude too: tan(L)=7/8. The level of accuracy is again quite high. If the latitude of the D&M Pyramid is significant, why shouldn't the latitude of the Face also be significant?  Just as with the angle-relationships for the D&M Pyramid, Hoagland seems to be rather selective - emphasizing some relationships, and ignoring or not noticing others.
         There is a theory that the axis of rotation of Mars has dramatically shifted over time and that some shift has even taken place in the relatively recent past.  One can find some papers in well-known scientific journals on this topic.  Such a shift would change the latitude of almost everything on the surface of Mars. Tom Van Flandern has a theory that Mars was once a moon of a planet that was destroyed by an explosion.  A dramatic shift in the axis of rotation would be an almost inevitable consequence of such a catastrophic event.  Van Flandern suggests that the Face may have once been in the vicinity of the Martian equator before such an explosion.  If Van Flandern is right (and Hoagland seems to be quite supportive of Van Flandern's theory), then  it would seem that the present location of the D&M Pyramid at a latitude of 40.87^o N  is merely an accidental consequence of natural events.  If so, then both of the above very accurate relationships involving that latitude as well as the latitude of the Face would be nothing more than coincidences- more illustrations of the "Power of Randomness" (as are all of the other relationships pointed out by Hoagland in my opinion).
 
 
 

              When I challenged Richard Hoagland to a debate back in May, 1998, I stated my position as follows:   "You have often stated that the numerical relationships exhibited by various objects on Cydonia provide strong evidence that these objects are artificial. My contention is that virtually all of this evidence is fallacious and should be discarded."  The reason that I used the word "virtually"in my statement is because of a paper by Horace Crater and Stanley McDaniel entitled Mound Configurations on the Martian Cydonia Plain.  At that time, I had looked at their paper, but had not thoroughly studied it. My initial reaction was a combination of skepticism about their approach and curiosity about how strong their evidence really was. On the surface, it seemed that their evidence might possibly be quite interesting.  During June, 1998, I carefully studied their paper.  It might be worthwhile for me to discuss my opinion of their work which, as the reader will see, turns out to be somewhat inconclusive.  For that, turn to another page:  Cydonia Mound Geometry
 
 
 
 
 
 
 

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