**Take a piece of A4-sized paper and fold it over three times. Unfold it again, put the piece of paper in front of you and mark six points like on the photo here on the left. Stick it to your window and you’re done. You have just transmitted a message with deep mathematical propositions to your neighbours and the people walking by your house: about prime numbers, the Pythagorean theorem, points of tangency from the tetrahedron in its circumscribed sphere, and even something about electron spin! Don’t you believe me? Does it sound like someone from another planet talking to you? Good, then you’re thinking in the right direction.**

### The Mounds of Cydonia

In November last year Horace Crater, Stanley McDaniel and Ananda Sirisena published an article in the *Journal of Space Exploration: *The Mounds of Cydonia: Elegant Geology, or Tetrahedral Geometry and Reactions of Pythagoras and Dirac? These three scientists, unknown to me before reading their article, apparently have been investigating a number of small hills on Mars. These hills or mounds could contain a message of an alien civilisation, which is coded in the mathematical relationships between their positions. The mounds are situated not far from the so-called ‘Face on Mars‘, a mountain which looks like a face on the fairly unsharp photos made by the Viking orbiters during their missions in 1976. During later missions more detailed photos became available and the resemblance was gone.

Less known are the rock formations close to the Face on Mars in the Cydonia region, which according to some people, point to artificial constructions. Twelve mounds have been identified that show up more clearly than other mounds (higher in albedo) on the Viking pictures. By drawing numerous triangles with these points it was found that a lot of the angles were more or less straight or whole multiples of a ‘base angle’ of 19.5 degrees. This was discovered by Richard C. Hoagland, who was also responsible for the popularisation of the ‘Face on Mars’.

### Imaginative mathematics

There is some rationale behind the idea that mathematical or geometrical constructions can be used as a means of communication with aliens. Mathematics doesn’t really depend on language, the meaning of a figure of Pythagoras’s theorem will probably be recognisable for any advanced enough civilisation. You could also use the prime number sequence, like in *Contact *by Carl Sagan.

It is a bit easier to talk about the discoveries of Crater and his co-authors when we have look at the following figure which is an idealised version of the pattern they think to see (the labels ‘r’, ‘s’ and ‘t’ I’ve added for easier reference). They talk mainly about the pentad formed by the points B,D,E,G and A. Point P is the sixth point that fits the pattern well, they argue.

Now you actually only have to know that the sides of the big rectangle (PtBr) have the ratio √2:1. If you look at a half of this rectangle, for instance the rectangle tBsG, then you’ll find the same ratio – and this pattern continues if you keep on halving the rectangles. This is characteristic of the paper sizes we know as A0, A1, A2, A3, A4, A5, A6, etcetera. OK, nice so far. But Crater and his buddies have only just started.

We’ll look for some prime numbers now. Take the area of the smallest rectangular triangle DAB to be unit 1. Then triangle EDA has area 2, triangle DAG area 3, pentad BDEGA area 5 and the figure with vertices BDEPGA has an area of 7. So according to the authors we now clearly have the start of the prime number sequence 1, 2, 3, 5, 7, … A mathematician like me will probably read the rest of the paper with some hesitation now that 1 is called a prime number, but also because they seem to be rather selective in picking the areas of interesting figures. Why didn’t they also take the tetragon DEGA with area 4 (which they do mention but leave out of the sequence without any argument) and pentad DEPGA with area 6?

The numbers 2, 4 and 6 are being used however by these mathematical geniuses to state the following:

Then starting with the prime number 3 from the sum 1+2, all of the prime numbers from 5 through 89 can be obtained by adding the three even numbers 2 or 4 or 6 corresponding to the squares of the sides of the middle sized right triangle (which of course satisfy 2+4=6). So, 3+2=5, 5+2=7, 7+4=11, 11+2=13, 13+4=17, 17+2=19, 19+4=23, 23+6=29, 83+6=89.

At 89 this ‘generating feature’ stops, because the next prime, 97, is 8 more that 89. What is the message the authors want to convey with this? You can always write something down like this when you have found the numbers starting from 1 to 7, for instance in *Miffy’s Counting Book*. And why did they use 2, 4 and 6 here? Well, first look at the smallest rectangular triangle BDA and take side BD as 1. Then AB is equal to √2 and AD equals √3 (using Pythagoras’s theorem). The triangle AEG is similar in shape (the ‘middle sized triangle’), only a factor √2 bigger. Its sides have lengths √2, √4 en √6. Et voila, there we have our 2, 4 and 6! The logic in this is unmistakable, isn’t it?

But things get even worse. The √2 which we found as the ratio of the sides of our A4-sized piece of paper you can also find in three-dimensional figures, like the tetrahedron. With a little trial and error, it is not difficult to find parts of the pattern in cross-sections of this tetrahedron. Or you can even find the ‘base angle’ as the “tetrahedral latitude because when a tetrahedron is embedded in a sphere, its base marks that latitude on the sphere.” Duh. You could now start to think about what this all means, but the authors already have another miracle to show: with the √2 in mind it is also possible to get a projection of a model for electron spin which fits the pattern. I’ll spare you the details.

I do understand however why the authors want to draw the attention to the tetrahedron. The Platonic solids, of which the tetrahedron is one specimen, are mentioned like the prime numbers as clearly recognisable mathematical material, which would make the solids suited as a purposeful signal to aliens. But I think that’s something quite different than hiding √2 somewhere in your pattern.

### How could someone create this?

Based on the high-resolution images a geologist was able to explain the authors about the likely origins of the mounds, they are probably the remnants of mud volcanoes. These have a natural origin, but the authors point at the disaster at Sidoarjo, Indonesia, where since 2006 huge amounts of mud are flowing out of a borehole. If we on Earth can cause such an eruption of mud by accident, it is perceivable that an even more advanced civilisation could do this on purpose on a location at will, isn’t it? And if a number of mounds were already almost in the right place for such a message to start with, maybe only a couple would have to be added tot complete the pattern. At least that’s the far-fetched theory of the authors. If true, the aliens have been quite lucky that in the meantime no other mounds have formed naturally in the area of the design, which of course would have made the message unreadable …

### How accurate is the pattern?

Eventually, that’s the main question when we want to evaluate the claims in the article. In the Viking photos, the smallest details have a real size of about hundred meters. The newer material has a far higher resolution: one pixel = 5 meters. I think it is good to have a look yourself to get some feeling for the level of detail. Below is an image taken by the Mars Orbiter in 2012 (click the image for an enlargement).

Pepijn writes: “I think it is quite difficult for people who have seen the images of 1976 before the others to objectively look at the newer images.” That may well be true but all at NASA are also photographing geologic structures that they have seen before, so I am not sure what your argument is here. It doesn’t invalidate the fact that I can see a face in these images.

Before answering the final item of your critique, let us review the properties of the pentad of mounds and later all 12 mounds that were chosen from Viking image number 35A72 orbiting Mars in 1976. From this image we obtained 10 triangles amongst the pentad of mounds (A,B,D,E,G) Taking the smallest triangle to have area one all of the other triangles have either area one, two, or three. The area of the whole pentad of mounds has the self referent value of five. The pentad of mounds has three different sized right triangles that are similar. If we take the smallest side of the smallest right triangle to be one unit of length, then the middle sized side of the middle sized right triangle has length of two and the large size, that is the hypotenuse, of the largest similar right triangle has length three. All three of these triangles have an idealized ratio between the sides of the triangle which duplicates precisely that of the magnetic moment of a spin one half particles precessing in a constant magnetic field, be it electron or quark or proton. These properties of area, length, and fundamental physics followed directly from the pentad placement of mounds and are not independent of that. Within the pentad there is the isosceles triangle ADE. As explained in our papers this triangle has a direct physical interpretation in terms of the addition of angular momentum of two spin one half particles to give a total spin of one. Again, this property is intrinsic to the property of the pentad and is not separate from it.

One can infer from this pentad a more general geometrical figure, that of the square root of two rectangle. Indeed there is another mound which we call P which forms a right triangle that is congruent to the middle sized right triangles of the pentad. From the position of this hexad one can infer a larger version of the square root of two rectangle. Mound P forms one of the remaining seven mounds of the 12 mounds complex. From the position of mound M, one finds an isosceles triangle PMA that is precisely similar to the smaller isosceles triangle ADE of the pentad. There is an additional mound O which to a fairly high degree of accuracy forms with mounds P and G an equilateral triangle. This equilateral triangle has an area that has a ratio to the area of the triangle ADE which is precisely that of the area of the sides of a tetrahedron to the cross-section as described earlier. This property of O, does not follow as a logical consequence of the pentad but is an independent feature. There is the further remarkable property of the the connection between inferred square root of two rectangle and the tetrahedron which we have described earlier an item #6. Again we have to remember that all of this is represented by city block size features on the planet of Mars in the region of Cydonia.

Most of our reply to your statement in the URL below has to do with our response to the criticisms of mathematician Ralph Greenberg.. He claims in essence that by focusing on the angle 19.5° we may be discounting the effects of other geometries for example that related to the Golden section or 30° etc.. He also states that the geometry itself that is to say that defined by 19.5° has a self replication property that may account for the large number of occurrences we find in our search of the angles among the Cydonia Mounds. The details found in the URL

http://spsr.utsi.edu/files/wr9.pdf

in essence show that there are effects of other geometries but they are very small compared with the effects of the geometry in 19.5°. The details also show that there is indeed a self replication property as we and Ralph Greenberg also hypothesized. However the self replication property does not come anywhere near accounting for the number of occurrences of the isosceles and right triangles we found in our analysis. The details are a little tedious but rather straightforward to follow

This URL is to file with picture missing in # 6 your write and our reply

http://spsr.utsi.edu/files/wr6.pdf

You write:#7

How could someone create this?

Based on the high resolution images a geologist was able to explain the authors about the likely origins of the mounds, they are probably the remnants of mud volcanoes. These have a natural origin, but the authors point at the disaster at Sidoarjo, Indonesia, where since 2006 huge amounts of mud are flowing out of a borehole. If we on Earth can cause such an eruption of mud by accident, it is perceivable that an even more advanced civilization could do this on purpose on a location at will, isn’t it? And if a number of mounds were already almost in the right place for such a message to start with, maybe only a couple would have to be added tot complete the pattern. At least that’s the far-fetched theory of the authors. If true, the aliens have been quite lucky that in the mean time no other mounds have formed naturally in the area of the design, which of course would have made the message unreadable …

Our commnent (from paper)#7

It is evident that there are repetitions of inter-angles with values of 45°,60°,70°,90°; and 120° within 1° and at all scales. Although these inter-angles do not include the angles between the mounds of the pentad, they do show consistent values between other mounds that have significant separation (many tens of kilometres apart) as the mounds of the pentad as well as

with smaller-scale inter-angles at mounds and interior to them. Inter-angles between joints and faults tend to repeat both locally and regionally on earth as well. So, the geology supports repetitive values of well defined inter-angles. The angles forming the triangles of the mounds of the pentad are 35.3°, 54.7°, 70.5° and 90°. Only the last of these angles is ubiquitous for the inter-angles involving various geological markings. Thus, although the repetition of similar angles appears throughout the region, none of these are associated with explicitly identifiable lineaments and such with the pentad mounds (FIG. 28).

You write:#6

But things get even worse. The √2 which we found as the ratio of the sides of our A4-sized piece of paper you can also find in three-dimensional figures, like the tetrahedron. With a little trial and error, it is not difficult to find parts of the pattern in cross-sections of this tetrahedron. Or you can even find the `base angle’ as the “tetrahedral latitude because when a tetrahedron is embedded in a sphere, its base marks that latitude on the sphere.” Duh. You could now start to think about what this all means, but the authors already have another miracle to show: with the √2in mind it is also possible to get a projection of a model for electron spin which fits the pattern. I’ll spare you the details.

I do understand however why the authors want to draw the attention to the tetrahedron. The Platonic solids, of which the tetrahedron is one specimen, are mentioned like the prime numbers as clearly recognisable mathematical material, which would make the solids suited as a purposeful signal to aliens. But I think that’s something quite different than hiding √2 somewhere in your pattern.

Our Reply:#6

The tetrahedron not only has the indicated cross-section but more fundamentally the sides of the tetrahedron are congruent equilateral triangles. What elevates this from the “duh” status is that the ratio between the area of the equilateral triangular sides of the tetrahedron and the cross-section of the tetrahedron is exactly the same as the idealized ratio between the area of the isosceles triangle formed from the mounds ADE and the equilateral triangle OPG. See the discussion near the bottom of page 15 of our paper. Our discussion about the electron is not an opinion but a fact related to procession of the electron’s magnetic moment in a constant magnetic field. That pressession is precisely modeled in the right triangles that we discussed in our paper. The six mound or hexad image as well as your folded A4 image both infer the square root of two rectangle.

To answer your question about the connection between the square root of two rectangle and the tetrahedron do the following. Take a plane that intersects one of the vertices of the tetrahedron and also intersects one of the opposite sides perpendicularly. That plane will form the tetrahedral cross-section we have been discussing by its interior overlap with the tetrahedron. That tetrahedral cross-section is an isosceles triangle. Furthermore, the sqrt2 rectangle is simply this isosceles triangle cross-section with its component right triangles moved to share a common hypotenuse. Thus we see how in a natural way the √2 rectangle is embedded in the tetrahedron. Below we take from McDaniel’s article at the SPSR website. It shows more clearly how the square root of two rectangle is related to the tetrahedron cross-section.

[Figure]

The right image is simply the isosceles triangle identical to the cross-section of the tetrahedron bisected at the vertex D. The component right triangles when moved to share a common hypotenuse. produce the √2 rectangle in the figure on the left. The angles p,r,s are respectively 90 degrees, 45+t/2=54.75 degrees and 45-t/2=35.25 degrees. And t= 19.5 degrees.

(I will put in link to figure at a later time)

I do not argue that sqrt(2) is uninteresting, my main point is that you can think of all sorts of interesting facts in which this ratio plays a role. That doesn’t mean these interesting facts are communicated through the diagram you start with. If a four year old is able to write down all letter of the alphabet, does he also now about the sentence “The quick brown fox jumps over the lazy dog” which contains all these letters? I don’t think so.

The sqrt(2) rectangle is very rich in it implications both in elementary and fundamental mathematics and in elementary and fundamental physics.

it is not the ration itself that is so interesing, it is the connection between the square root of two rectangle and the tetrahedron which I find particularly fascinating.

You write: #5

The numbers 2, 4 and 6 are being used however by these mathematical geniuses to state the following:

Then starting with the prime number 3 from the sum 1+2, all of the prime numbers from 5 through 89 can be obtained by adding the three even numbers 2 or 4 or 6 corresponding to the squares of the sides of the middle sized right triangle (which of course satisfy 2+4=6). So, 3+2=5, 5+2=7, 7+4=11, 11+2=13, 13+4=17, 17+2=19, 19+4=23, 23+6=29, 83+6=89 At 89 this `generating feature’ stops, because the next prime, 97, is 8 more that 89. What is the message the authors want to convey with this? You can always write something down like this when you have found the numbers starting from 1 to 7, for instance in Miffy’s Counting Book. And why did they use 2, 4 and 6 here? Well, first look at the smallest rectangular triangle BDA and take side BD as 1. Then AB is equal to √2 and AD equals √3 (using Pythagoras’s theorem). The triangle AEG is similar in shape (the `middle sized triangle’), only a factor √2 bigger. Its sides have lengths √2, √4 and √6. Et voila, there we have our 2, 4 and 6! The logic in this is unmistakable, isn’t it?

Our reply:#5

2,4,6, is simply 2(1,2,3) in which one to three are the areas of the three different sized similar right triangles. We thought that the prime number sequence is just an interesting sidelight worth mentioning in a footnote.

A footnote that makes it into the abstract …

The footnote refered to the extended sequence of primes up through 89. The abstract refered to those primes that correlated with the geometry of the triangles of the pentad.

It’s also worth noting that we cannot have any numbers without number 1. Something that ancient Indian and Arabic mathematicians pointed out a long time ago.

You write:#4

Imaginative mathematics

There is some rationale behind the idea that mathematical or geometrical constructions can be used as a means of communication with aliens. Mathematics doesn’t really depend on language, the meaning of a figure of Pythagoras’s theorem will probably be recognisable for any advanced enough civilisation. You could also use the prime number sequence, like in Contact by Carl Sagan.

It is a bit easier to talk about the discoveries of Crater and his co-authors when we have look at the following figure which is an idealised version of the pattern they think to see (the labels `r’, `s’ and `t’ I’ve added for easier reference). They talk mainly about the pentad formed by the points B,D,E,G and A. Point P is the sixth point that fits the pattern well, they argue.

Fig: Now you actually only have to know that the sides of the big rectangle (PtBr) have the ratio √2:1. If you look at a half of this rectangle, for instance the rectangle tBsG, then you’ll find the same ratio — and this pattern continues if you keep on halving the rectangles. This is characteristic of the paper sizes we know as A0, A1, A2, A3, A4, A5, A6, etcetera. OK, nice so far. But Crater and his buddies have only just started.

We’ll look for some prime numbers now. Take the area of the smallest rectangular triangle DAB to be unit 1. Then triangle EDA has area 2, triangle DAG area 3, pentad BDEGA area 5 and the figure with vertices BDEPGA has an area of 7. So according to the authors we now clearly have the start of the prime number sequence 1, 2, 3, 5, 7, … A mathematician like me will probably read the rest of the paper with some hesitation now that 1 is called a prime number, but also because they seem to be rather selective in picking the areas of interesting figures. Why didn’t they also take the tetragon DEGA with area 4 (which they do mention but leave out of the sequence without any argument) and pentad DEPGA with area 6?

Our reply#4:

We include only the areas of the similar right triangles. Of course there are other areas but they do not involve right trianglesI. I have heard that mathematicians have demoted 1 from prime number status. I would be interested in knowing why. (Shades of Pluto’s fate?)

Then please explain why tetragon DEGA does not comply to your criterium, while pentad BDEGA does.

For a number of reasons, see for instance: https://primes.utm.edu/notes/faq/one.html

I get that one is a special number more special than just the prime designation. However it seemed like Occam’s razor would come into play here. Prime numbers are those whose only divisors are the unit number one and the number itself.

I taught my grandniece about prime numbers beginning with one before I learned about it more extended complicated definitions. She started with one and was able to figure out on her own mostly all of the “prime numbers” from one through 97.

Horace

Of course what numbers we call ‘prime’ is a matter of definition. But when you don’t exclude 1, most theorems in which primality plays a role would have to be reformulated, where you would now read simply ‘for all primes’ you would have to write ‘for all primes greater than 1’, starting with the Fundamental Theorem of Arithmatic. So if you want to decide on this matter with Occam’s razor …

You write#3:

Less known are the rock formations close to the Face on Mars in the Cydonia region, which according to some people, point to artificial constructions. Twelve mounds have been identified that show up more clearly than other mounds (higher in albedo) on the Viking pictures. By drawing numerous triangles with these points it was found that a lot of the angles were more or less straight or whole multiples of a `base angle’ of 19.5 degrees. This was discovered by Richard C. Hoagland, who was also responsible for the popularisation of the `Face on Mars’.

Crater and McDaniel have published a couple of articles about this formation of mounds. They focus on five of the twelve so-called `Mounds of Cydonia’ and at a later stage try to fit a sixth point in the pattern they claim to have found.remarkable whole number relations between the areas?

Our Reply #3

Hogland only dealt with a few of the 12 mounds. Although he discussed with Errol Torun the 19.5° angle and its connection to various features in and about Cydonia he did not discover the connection between that 19.5° and the triangles he drew not to mention the connection between 19.5° and the various triangles we have uncovered amongst the 12 mounds. There is no doubt that the sixth mound is there and within measurement uncertainties it leads to an additional right triangle that fits the ideals that appear amongst the pentad. Given the ideal placement, the areas (not to mention the lengths of the various sides of the three similar different sized but right triangles) are logical deductions

You write#2

The Mounds of Cydonia

In November last year Horace Crater, Stanley McDaniel and Ananda Sirisena published an article in the Journal of Space Exploration: The Mounds of Cydonia: Elegant Geology, or Tetrahedral Geometry and Reactions of Pythagoras and Dirac? These three scientists, unknown to me before reading their article, apparently have been investigating a number of small hills on Mars. These hills or mounds could contain a message of an alien civilisation, which is coded in the mathematical relationships between their positions. The mounds are situated not far from the so-called `Face on Mars`, a mountain which looks like a face on the fairly unsharp photos made by the Viking orbiters during their missions in 1976. During later missions more detailed photos became available and the resemblance was gone.

Our reply#2:

The resemblance was gone for a very good reason. In the original face on Mars image the shot (1976) was taken from directly overhead. Amongst the more so-called detailed photos the first and the one that was touted to the public is debunking the face was a shot (1998)i from an oblique angle and one that was poorly enhanced. In the figure below we have those two images placed side-by-side. To the left is the original face on Mars image and to the right is the photo that was compared unfavorably with the original face on Mars image.

[Figure]

11 years later in 2009 a shot was taken that, unlike 1998 shot, had orbital parameters that were consistent with the in 1976 shot. As you can see in the second image with the 2009 shot placed side-by-side with the 1976 shot they compare favorably. In fact the 2009 shot displays two details not seen but expected for a higher enhancement. The first is the eye has more detail, displaying a pupil. The second is that the mouth displays teeth. Both of these details were hinted at but not clearly evident in the 1976 shot. They are more evident in the similarly oriented 2009 shot. Most importantly one can just imagine the difference in impression to the general public if the orbital parameters and enhancement of the 1998 shot were the same as those of the 2009 shot.

[Figure]

Okay I see that the software does not reproduce figures. If you are interested in these figures please email me at hcrater@aol.com and I will send you the figures.

The photo’s you mean are probably found at the NASA website, you could just share a link. Or you could upload the images to imgur.com for instance and drop the link here.

Are there more recent images than the HiRise ones from 2007? https://hirise.lpl.arizona.edu/PSP_003234_2210

There is so much imagery at this moment, that it is hard to persist that the Face is not pareidolia. You could maybe argue that the first images (from 1998) presented to debunk this idea were not convincing enough, but who cares today?

https://nssdc.gsfc.nasa.gov/photo_gallery/photogallery-mars.html#controversy

https://nssdc.gsfc.nasa.gov/planetary/mgs_cydonia.html

https://science.nasa.gov/science-news/science-at-nasa/2001/ast24may_1

Have a look at these two images:

http://viewer.mars.asu.edu/planetview/inst/ctx/B01_010143_2216_XN_41N009W#P=B01_010143_2216_XN_41N009W&T=2

http://viewer.mars.asu.edu/planetview/inst/ctx/P22_009642_2216_XN_41N009W#P=P22_009642_2216_XN_41N009W&T=2

Ok, these images appear more like the Viking photo because of the lighting. But would we be talking about a Face on Mars if these photos had been available first? I doubt that. That we see the resemblance with the older Viking photos and the Face is likely influenced by the fact that we know that older images well. Of course, this could be tested by showing the newer images to people who have no knowledge about the ‘controversy’ and ask them if they notice something remarkable.

The Face may be pareidolia.

You ask: “Who cares today”?

Shouldn’t scientists care about what we see and discover on other planets?

My remark “Who cares today?’ is about the question whether the 1998 photo is a proper debunk of the Face on Mars as seen on the Viking photo.

We don’t know if the face is an artificial construct or not. That’s the whole point of research and study – to find out, not to dismiss immediately. I am impressed enough with Dr. Crater’s analysis to say, “Let us look at this ground pattern on Mars. Let us see if the pentad/hexad and any other pattern arising from it is of significance in our SETI.” I am open-minded enough to look, to ask questions and to try and understand sceptical viewpoints such as yours. So I am glad you took the time to study the paper. Oh, and I like the title of your detailed essay, “Far-fetched….”. Yes Mars is very far away. I would like to go there to check out these mounds myself; maybe someday……..

Suppose that instead of the distorted 1998 image NASA had available in 1998, the 2008 image that Ananda showed you was available, I doubt very seriously people would have viewed that as debunking the Viking image. In fact the added detail to the Viking image that the 2008 image gives would likely have spurred more interest in this feature, instead of squashing interest as the actual one did.

I think it is quite difficult for people who have seen the images of 1976 before the others to objectively look at the newer images. As a small test, I showed my kids (11 and 13 years old) apart from each other the picture of 2008 and asked them if they saw something in that image.

The younger saw a face

en profile, something which I hadn’t expected at all. And the older saw something like a sheep. I outlined the images on the photo:Although I just asked two people, I think it tells us something. there are some features on this rock formation which we see easily as an eye (not surprising). The rest of a face we can find by adding features to that first impression, but there are too many ways to do that to conclude that there is an actual face hidden in the formation (artificially or just pareidolia).

http://spsr.utsi.edu/files/wr2.pdf

this file contains images missing from our #2 reply comments

We are responding to your paper item by item (#1-#8). Thank you very much for the time you spent in writing your fair critique of our papers. We assume the reader has access to your figures and diagrams.

You write#1:

Take a piece of A4-sized paper and fold it over three times. Unfold it again, put the piece of paper in front of you and mark six points like on the photo here on the left. Stick it to your window and you’re done. You have just transmitted a message with deep mathematical propositions to your neighbours and the people walking by your house: about prime numbers, the Pythagorean theorem, points of tangency from the tetrahedron in its circumscribed sphere, and even something about electron spin! Don’t you believe me? Does it sound like someone from another planet talking to you? Good, then you’re thinking in the right direction.

Our reply#1:

All of what you say about the square root of two rectangle is true. The question that remains unanswered is whether or not the pentad of mounds were placed in their positions on Mars by extraterrestrials. We pointed out the connection between the square root of two rectangle and the various properties that follow from that in our paper (see Sec 2.3Footnot)we could regard this as a mathematical fluke but then it repeats with the discovery of mound P and mound O and mound M, which as seen in our first paper with McDaniel are of course independent of the five mounds of the pentad.

Footnote: We paraphrase here what we write there :The remarkable geometrical and prime number properties of the pentad and hexad follow from the corresponding geometrical properties of the square root of two rectangles. Those geometrical and prime number properties are a logical outcome of the (idealized) relative placement of the mounds and are not independent of those placements. This would hold true for any subsequent theoretical discovery related to those placements. For example, the connection between the pentad and electron spin discussed later in the paper is such a theoretical discovery and a consequence of an already discovered property of the relative placements of the mounds (and of course your folded A4) . By contrast, the placement of mound P has both a new and supportive consequences for the properties of the pentad of mounds. It is new in that it involves a mound separate from the five mounds of the pentad. It is supportive in that it not only leads to a coordinated fit with a fifth right triangle similar to the four right triangles of the pentad, but it is also placed in such a position as to accentuate the square root of two rectangles inferred from the pentad.

We also have obtained coordinated fits that involve mound M and all six mounds of the hexad (P,E,D,B,A,G). That coordinated fit reveals a larger isosceles triangle PMA that is similar to the triangle ADE.

I meant to say: The European paper sizes conform to an aesthetic design of paper. It is self-referential and every time you fold it you get the same ratio of the sides.

Also view this short film:

https://www.youtube.com/watch?v=JjmePR-3ojg

What this paper shows is what has been discovered on Mars. After all, research is about something that has been discovered. You folded an A4 sheet of paper – very good because that shows that A4 paper conforms to the square-root two grid, which is what professor Stan McDaniel observed in the mound layout. What you fail to mention is that the A4 paper size on Earth is a result of “intentional design”. The European paper sizes conform to an aesthetic design of paper. It is self-referential and every time you fold it you get the same ration of the sides. The British ‘foolscap’ paper size and American paper sizes do not display this self-repeating pattern. Let us think about this. Thank you for your analysis.

The main issue I wanted to raise by folding the A4 paper and colouring some points on the pattern, is that pattern is not something remarkable considering the ratio of the paper you start with. Is it intentionally? Yes, but I wouldn’t really call this ‘design’, we just picked this size because of the nice properties, but it was already there in a sense. So ok, the pattern is interesting, but the rest of the things you decided to mention as possible messages like the “prime numbers, the Pythagorean theorem, points of tangency from the tetrahedron in its circumscribed sphere, and even something about electron spin!” goes way beyond that basic geometrical ratio of 1 to sqrt(2). Even if someone is willing to accept that ‘they’ hid this ratio in this pattern, it doesn’t mean that ‘they’ also had the other ‘features’ in mind.

So what remains to be discussed as far as I’m concerned is whether that ratio which you see – with a certain precision – is a more than a statistical fluke. I have strong doubts about that, but I guess that was already clear from my blog 😉

Hello Pepijn, the choice of the A4 paper size was a conscious decision made years ago in the German DIN 476 standard. It was chosen precisely because of its self-replicating property. You seem to agree that the six mounds on Mars forming the pattern analysed by Dr. Horace Crater are done toa certain precision. Professor McDaniel’s observation that the layout fits a square-root 2 grid is an astute one. I have a question for you: if this pattern had been observed on Earth, would you state that the pattern was an intelligent layout. Because the pattern exists on Mars, do you dismiss it as being impossible? As far as it being a “statistical fluke”, I’ll let Dr. Crater reply to that.

My question still stands unanswered:

“If this pattern had been observed on Earth, would you state that the pattern was an intelligent layout?Because the pattern exists on Mars, do you dismiss it as being impossible? ”

What is your answer to this question please?

You do say now, “Ok, these images appear more like the Viking photo because of the lighting. “

I don’t think I used the fact that this configuration of mounds is found on Mars and not on Earth in any of my arguments. I don’t think it matters that much.

If you like a Bayesian approach you could argue that finding such a pattern on Earth would deserve higher priors for the intelligent design option because there is proof that there is at least intelligent possible on the planets surface, but I doubt that it would change much. A civilisation which would be able to construct this, might be able to go to a planet just to leave this as a puzzle without staying there long.

Pepijn writes: “I don’t think I used the fact that this configuration of mounds is found on Mars and not on Earth in any of my arguments. I don’t think it matters that much.

If you like a Bayesian approach you could argue that finding such a pattern on Earth would deserve higher priors for the intelligent design option because there is proof that there is at least intelligent possible on the planets surface, but I doubt that it would change much. A civilisation which would be able to construct this, might be able to go to a planet just to leave this as a puzzle without staying there long.”

The Face on Mars is a subjective impression. The Mound Configuration on Mars is an objective fact – the precision of Dr. Crater’s study and analysis shows this to be the case. If I found this pattern of mounds on Earth I would send a task force to do more work. Why not on Mars then? Are you sure that there is “proof” there is intelligent life on Earth? I would classify it as a backward intelligence in space…….

As I mentioned in the paper even if you consider the pentad by itself a statistical fluke, the existence of supporting features, i.e. mounds P, M and O that are independent of the pentad, cut out any remaining support that the pentad in conjunction with the others is just a fluke.

We replied to Greenberg’s critique in our 2007 JSE paper. We pointed out the connection between the square root of two rectangle and the various properties that follow from that in our paper. One could regard this as a mathematical fluke but then it repeats with the discovery of mound P and mound O and mound M, which as seen in our first paper with McDaniel are of course independent of the five mounds of the pentad.

I think your critique would carry more weight if you took the time to try to find mounds on other portions of the Martian surface or on Earth’s surface for that matter with relations with the same accuracy as in our papers to the square would of two rectangle features found amongst the pentad and other mounds.

Thank you for the time you spent in writing your fair critique of our paper.

If you are interested in some of my physics publications see Two Body Dirac equations which are reviewed in Wikipedia and associated references.

I may reply more later on when I have more time.

Best regards

Horace Crater