A coworker of mine at Google sent me a link this morning to an interesting piece of crackpottery: a guy who calls himself "the Soldier of the Truth" who claims to have proved Euclid's parallel postulate; and that therefore, all of non-Euclidean geometry, and anything in the realms of math and science that in any way rely on non-Euclidean stuff, is therefore incorrect and must be discarded. This would include, among numerous other things, all of relativity.

Let's start with a bit of background. Euclidean geometry is one of the earliest mathematical systems to be axiomatized. What that means is that it consists of a simple set of logical statements called *axioms*, and given those axioms, anything else within the system can be proven using those axioms and the basic rules of predicate logic.

Euclid had five axioms. The first four are very straightforward, and it's clear that none of them can be proven using the others:

- Given any two points, there is one line segment that connects them.
- Given any line segment, you can extend it infinitely in either direction.
- Given a point and a radius, there is exactly one circle around that point with that radius.
- All right angles are equal to one another.

The fifth axiom is where things get problematic. It can be expressed in different ways, but ultimately, they all mean the same thing. In modern notation, what it says is: given a line L and a point P, there is exactly one line parallel to L which crosses through P.

(As an aside, there are actually more axioms; Euclid requires some basic arithmetic axioms, but since they're numerical rather than geometric, they're generally taken as a given.)

Euclid, and mathematicians for thousands of years after him, thought that there was something fishy about the fifth axiom. That is, it seemed like it *shouldn't* be an axiom; it should be a rule provable by the other axioms.

The problem turned out to be one of definitions. If you take the axiomatic definitions of line, point, circle, and parallel from Euclid, they *don't* actually mean what everyone thought they meant. That is, the definitions don't say anything about the surface on which you're doing geometry. They *assume* that it's a flat plane. But that's *not* part of the axioms!

You can take the first four axioms, and apply them to the surface of a sphere. If you do that, you get a system which works - only it has the property that parallel lines *don't exist!* Or you can apply them to a hyperbolic surface --- in which case given a line L and a point P, there are *many* lines parallel to L that cross through P. Or you can apply it to irregular surfaces, in which case the fifth axiom applies in *some* places, but not in others.

So, back to our crackpot friend. He claims to have assembled a suite of forty different proofs of Euclid's fifth postulate. If this were true, it would make this weeks announcement of a proof that P != NP look like chicken feed. It would make Andrew Wiles' work proving Fermat's theorem look trivial in comparison.

Unfortunately, it's a pile of foolish, shallow, amateurish rubbish.

Remember what I said just a couple of paragraphs ago? The four axioms were built on the assumption that they uniquely described planar geometry. But that's *not* the case. They describe geometry, but they can describe not just geometry on a plane, but also geometry on a variety of other surfaces.

What our crackpot friend does is, very simply, assume that we're talking about planar geometry, and uses the basic properties of geometry on a plane as implicit axioms in his proof. In other words, he assumes that the fifth axiom is true by restricting himself to a plane; then he uses it to prove that the fifth axiom is true on a plane; and from that, he concludes that the fifth axiom is universally true and that only planar geometry exists.

Let's look at one example - his favorite of the proof, which he named after the village where he was born: the Theorem of Ehmej. About this proof he says:

I hope that the theorem of EHMEJ, my Birth-place, carries the Mathematical Community to reject the Non-Euclidean geometries and to recognize the theorem of EHMEJ as the only true foundation of the geometry. I am ready to defend it before any jury, and to prove that its deductions are infallible. Anyone not convinced must detect a flaw in the theorem of my Birth-place.

Now, I'm not particularly good at geometry. It is, by far, the area of math where I'm weakest. But I'll take on this challenge from our Soldier of the Truth.

He's using a more traditional statement of the fifth axiom. It's usually stated constructively. So, roughly, take any two straight lines. Draw a third line which intersects both of the original two lines. Now, look at the angles at the point of intersection with the two lines. Either both angles will be right angles; or on one side of the intersections, the sum of the two angles will be *smaller* that the sum of two right angles. In the latter case, if you extend the lines infinitely, they will eventually intersect on the side where the sum of the angles was less than two right angles. (See why I like the modern version of it better?)

Ok. So - this proof of his works using this second form. He says that he's going to prove that "a straight line cuts all the coplanar lines of different direction". "Different direction" is just another way of saying the "two right angles" thing above: two lines go in the same direction if the sum of the angles made by any crossing line add up to the sum of two right angles.

So, here's his proof.

By one given point B, outside of a given straight line (D) in a plane surface, let's draw any straight line (S) that cuts (D) in A. Take the bundle (F) of all straight lines around A of which each forms a determined angle with (D). The straight line (F1) that superimposes on (D) forms an angle equal to 0⁰, and the straight line (Fn), that superimposes on (S), forms an angle α.

When A and (Fn) translate on (S), the straight line of (F) keep their respective angles with (S) constant, therefore their directions remain fixed, and consequently their angles with (D) remain constant. In sweeping the plane surface, only the positions occupied by (F1) don’t cut (D), while all the positions of the other straight lines of the bundle (F) cut it.

In particular, when A coincide with B, the straight line (D’), occupying the position of (F1), is the only straight line that does not cut (D).

We conclude:

In the plane surface, by one given point, passes only one parallel to a given straight line.It is what it was necessary to demonstrate.

So can I do it? Can I find a flaw in his infallible proof, in the *only true foundation of the geometry*?

Heh. Of course I can. Even someone as bad at geometry as me can find the flaw, right in the very first sentence.

The problem is right there, in the very first line: *... in a plane surface*. The first four axioms of Euclid *don't give you a plane surface*. All of the reasoning in this proof relies on that little assumption: "in a plane surface".

Yes, in a plane surface, the fifth axiom does hold. But the first four axioms *do not* specify the plane surface - and in fact, they do hold in non-plane surfaces.

In fact, in real world situations, we do tons of geometry involving non-plane surfaces - that is, we do geometry where the fifth axiom *does not hold*. For example, when we calculate orbits of satellites, we're doing geometry over an essentially spherical surface. Even map-making is thoroughly non-euclidean!

While not a huge issue with your critique, I've seen the term hyperbolic plane used. I've also read descriptions of hyperbolic paraboloids as saddle-shaped planes. He may mean plane as the class of all such surfaces. Of course, that'd make the real issue with his proof that he assumes that the transformation of the 'bundle' produces no new parallel lines with D.

If he's using the word plane in that informal sense, then his proof doesn't work. None of the things that his proof asserts hold for anything but the traditional, flat, euclidean plane.

It has been like this for well over a hundred years now: every few days some brainiac who doesn't know the first thing about mathematics comes around with a proof of the fifth postulate.

Poincaré remarked on this in Science and Hyptothesis, saying that since the breakthroughs of Bolyai and Lobachevsky the academy "has received only a handful of proofs to the contrary every year."

The more things change, I guess...

M.Firionel

Rachid Matta MATTA knows very well the first things about mathematics, and especially geometry.

Rachid Matta MATTA

5-5-2012

I love the way the crackpots think every mathematician who ever lived was overlooking the "truths" that are so glaringly obvious to the crackpots.

M. Tualha

The content of any proposition of Modern mathematics is not saying the truth about the real world where we live. For this reason I said that all the Modern mathematicians overlooked the truth.

About the Modern mathematics, Richard Courant and Herbert Robbins wrote in the second edition of their book:

«What is Mathematics?»

the following:

«If this description were accurate, mathematics could not attract any intelligent person»

Rachid Matta MATTA

27-04-2012

What is the meaning of a line on a non-planar surface? And do the first two axioms restrict the kinds of surfaces that can be considered?

For example, there isn't a single unique "line" that connects the two points on opposite ends of a sphere along the surface. So it doesn't make sense to talk about axiom #1 on a spherical surface. The first four axioms must be more restrictive than you were letting on.

The axioms *define* what a line is, by saying that given any two points, there is exactly one line that connects them. That's the real catch: the axioms define what it means to be a line. But when we talk about lines and line segments, the image that we get in our mind has more properties than those actually specified by the definition from the axioms.

On the surface of a sphere, the axioms taken together mean that lines are all great-circles. For any two points on the surface of a sphere, there is only one great-circle line connecting them.

No, that's exactly what the axioms

don'tdo!The definitions come before the postulates, and can largely be dispensed with. The axioms have absolutely nothing to do with what the objects of discourse

are, but only with how theyrelate to each other.As Poincaré said, the theorems of geometry should remain true if everywhere the words "point", "line", and "plane" are replaced with "table", "chair", and "beer mug".

Pouincaré also said that every axiom is essentially a definition in disguise....

You are of course basically right, but I don't think that contradicts Mark's point at all. Hyperbolic geometry (to pick a random example) is a set of statements about abstract entities (which we happen to call point and line and so on) and by some extraordinary stroke of luck (well, it's not really luck - but that's for another discussion) these happen to map rather nicely on certain geometrical models.

And so

in a sensethe original axioms determine what, say, a point is, because every system fulfilling the axioms will beisomorphicto some geometric model.Only for certain theories. All planes are the same, so if you include the fifth postulate then yes, all models are isomorphic.

However, once you leave it out there are many,

manynon-Euclidean geometries -- not just those of constant curvature either.And how many non-isomorphic models are there for the axiomatically-defined theory of groups?

A point of view with which Hilbert disagreed believing it is possible to define mathematical objects through the precise use of axioms. In everday mathematical instruction Hilbert's (in my opinion incorrect) view prevailed, see for example the introduction of the real numbers.

Actually, I was wrong: Hilbert was the one I was quoting, not Poincaré.

So.... yeah, he didn't disagree.

And so what fault do you find with the definition of the real numbers as the unique (up to isomorphism) Archimedean complete totally ordered field?

M. John Armstrong

David Hilbert said: geometry should remain true if everywhere the words "point", "line", and "plane" are replaced with "table", "chair", and "beer mug".

Rachid Matta MATTA

27 -04- 2012

M. John Armstrong

Yourself you have corrected the citation attributed firstly to Poincare.

Rachid Matta MATTA

M. John Armstrong

I ask you to kindly reply to the following question:

Why David Hilbert didn't develop a geometry with the words

"table", "chair", and "beer mug"?

Why Hilbert's followers don't do this kind of geometry?

Rachid Matta MATTA

27-04 -2012

M. John Armstrong

I agree with you that the definitions come before the axioms. The definitions must reveal the true nature of the geometric objects defined by giving their essential proprieties.

The axiom is a proposition evident by itself and its truth is accepted by the mathematicians. As you wrote, axioms relate the objects of discourse to each other. I give an example: Two figures which coincide are equal.

The modern mathematicians do not differentiate between axiom and postulate, and this attitude is not logic, because the axiom is an evident truth, while postulate is a demand to do something, for example: Draw a straight line from a locpoint to another. I mean by locpoint the location of a point in the space.

Rachid Matta MATTA

26-4-2012

"For example, there isn’t a single unique “line” that connects the two points on opposite ends of a sphere along the surface."

The problem is that you need to also redefine what you mean by a point. In the sphere model a "point" is actually a pair of antipodal points.

Also, the great circles are the natural "lines" on the sphere because they have the following property. If you travel along a great circle at a constant velocity then your acceleration is always perpendicular to the surface of the sphere. On the sphere the great circles are the only paths with this property. Further, when you wish to define the geometry of more general surfaces, then this property is the way you distinguish the "lines." It is actually equivalent to requiring that the lines minimize length (measured within the surface) locally, but that is more difficult to state ... .

If you were to place that "proof" in different context -- as in an appendix of a respected math textbook -- and read it quickly, it sounds correct. I think most of us tend to think in 2 dimensional planar terms naturally.

His main failure here (aside from being wrong) is to be wrong in such flamboyant fashion which draws sharp criticism. On the off-chance that the original author of that proof happens to read this blog and happens to read my post, I simply suggest that future "proofs" be submitted more humbly and invite criticism.

As an aside, I was blessed in 6th-8th grades with a terrific math teacher who stretched everyone in the class well beyond the required course material. I, for example, was introduced to trig functions in 7th grade as part of his answer to me about ways I could draw circles on my Atari 800XL in BASIC. One of the discussions we got into in his class was regarding non Euclidean Geometry. He didn't call it that -- he just said "what if..." and proceeded to fundamentally change theorems we had accepted as truth one by one, and then asked us to work out how these changes propagated through everything we had ever learned in math. It was quite fun.

I thought the big deal about the fifth axiom is that most mathematicians believed it was not an axiom, but could be derived from the other four axioms. That's why for centuries people have been trying to prove the fifth axiom that way.

Because all these efforts seemed pointless, some mathematicians then decided to prove the axiom wrong. This is what gave birth to non-Euclidean geometry.

What I think (and I have limited knowledge of this) the problem is with the proof above is that he is assuming parallelism between D and D' while you can't have parallelism without the fifth axiom.

See also the related wikipedia page:

http://en.wikipedia.org/wiki/Parallel_postulate#Logically_equivalent_properties

I figure I should know this, but I'll ask anyway. Why would parallel lines not exist on a sphere? Is this a problem with my definition (two lines that don't have any points in common)?

I just imagine two horizontal circles spaced apart, seems like they are parallel to me.

Because a line on a sphere is a great circle. By axiom, for any two points, there is exactly one line that connects them. To make that work, the lines are circles whose radius is the full radius of the sphere. There's no way to have two of those that don't intersect.

That does make sense (after looking up great circle, trouble of learning mathematics in a different language). I suppose now I just don't see why you can't make it all match up with small circles, i.e. there's no way to satisfy the axioms with both, so you need to limit what is allowed to be a line and what is not, but I don't quite see why only great circles can serve as lines (ostensibly they're the logical choice as continuations of line segments and you can probably prove that if you use anything else your definition of line segment would introduce several non-unique line segments between two points).

I was never any good at geometry, mind.

Like Mark said: you need to satisfy the axiom that between any two points there's a

uniqueline. If you allow any circle on the sphere to be a "line", then you can draw an infinite number of "lines" between two points.It seems to me, a sphere isn't a very good example - as someone already pointed put, if you choose two *opposite* points, uniqueness falls apart because there's infinitely many great circles that go between them. (Think poles and meridians.)

And as someone

elsepointed out, a "point" in the spherical geometry is a pair of antipodal points on the surface of the sphere.Actually, "spherical" geometry is taking place on the real projective plane.

"And as someone else pointed out, a “point” in the spherical geometry is a pair of antipodal points on the surface of the sphere.

Actually, “spherical” geometry is taking place on the real projective plane."

That is true, only I don't think that anyone actually calles that "spherical geometry". What you describe is projective geometry. In proper spherical geometry points are simply points on a sphere. (But the price is that not just one, but several familiar axioms don't apply.) So there's a destinction.

Well, you have to define what you mean by a "line".

Usually, a line is the shortest route between any two points. If you use that definition, then on an ordinary 3-d sphere, lines are great circles.

Another definition of "line" is this: pick one great circle on a sphere. A "line" is any circle whose centre lies on that, and points reflected in that circle count as the "same"point. That definition gives you hyperbolic geometry.

I actually had this come up in NORBERT WIENER: WOLFE IN SHEEP’S CLOTHING, novel #3 of a Trilogy of science fiction novels I'm trying to complete (technically, I'm deep in Book #3 of two different trilogies at once).

The man, woman, Neanderthal, and wolves walked to an intersection of five corridors. “I should have pointed this out at the previous intersection of degree five,” she said.

“It seems that 5 squares can meet at one corner, on this side of the Singularity.”

“Indeed,” said the CalThaum man, "that’s a measure of the angular deficiency, negative or positive, of the Lovecraftian geometry of this alternate universe. It gives us an estimation of the Gaussian curvature over quadrilateral meshes based on the Gauss-Bonnet theorem.”

“I guess you're further along in Mathemagics than me,” she said. “Can you translate?”

“Sure. The way that Liu and Xu put it, in many application areas, such as image processing, surface processing, computer aided geometric design and computer graphics, the geometric data are usually available as polygonal meshes, typically triangular and quadrilateral meshes. Therefore, the estimation of several intrinsic geometric quantities of a surface, such as the normal vector, the mean curvature and the Gaussian curvature etcetera has been a signiﬁcant task.”

“I haven't been trying to draw a map of the parts of the labyrinth that we've gone through,” she said. “I could hardly draw it on graph paper, where there can only be four squares meeting at each corner. The glowing footprint spell with have to do, for getting us out of here.”

“So who are you going to believe,” he joked. “Me, or your lying eyes?”

“My eyes,” she said. “I can see the angles at the each intersection.”

“Well, then. In the literature, many approaches for estimating the Gaussian curvature of triangular surface meshes have been developed. Several of these approaches are established based on local surface ﬁtting techniques, including paraboloid ﬁtting, quadratic ﬁtting, and circular ﬁtting. The approximate Gaussian curvature is calculated from the ﬁtting function at the surface point. Some other approaches are based on differential geometric theorems and formulas, such as the Gauss-Bonnet theorem, Euler theorem and Meusnier theorem. In short, the Gaussian curvature at the given point on a surface is the limit of the area of the spherical image of the region around the point divided by the area of the region as the region shrinks around the point. So this universe is very hyperbolic.”

“And that's no hyperbole,” she said.

The wolves, one at a time, doubled back and sniffed the scent of the young man and young woman. They seemed curious, dubious, but willing to accept the third-party evaluation of them by the Neanderthal as honorary parts of the pack.

“So what do I need to know about Lovecraftian Geometry here,” she said.

“In a nutshell,” he said, “hyperbolic geometry (also called Lobachevskian geometry or Bolyai-Lobachevskian or Bolyai-Lovecraft geometry) is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced.”

“The parallel postulate in Euclidean geometry is equivalent,” she said, “to the statement that, in two dimensional space, for any given line L and point P not on L, there is exactly one line through P that does not intersect L, i.e., that is parallel to L. Right?”

“Right. But in hyperbolic geometry there are at least two distinct lines through P which do not intersect L, so the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid.”

“Because there is no precise hyperbolic analogue to Euclidean parallel lines, the hyperbolic use of parallel and related terms varies among writers. In this article, the two limiting lines are called asymptotic and lines sharing a common perpendicular are called ultraparallel; the simple word parallel may apply to both.”

“I don't care what you call it,” she said. “True names matter for Magic, but this is an intrinsic property of where we are, and there’s not anything we can do about it.”

“Yup. Magico-Physics was needed to make the Lovecraftian wormhole, but the manifold we jumped into is what it is. What I meant about the angles, to answer your question, is simple, whatever you call the theorems. A characteristic property of hyperbolic geometry is that the angles of a triangle add to less than a straight angle (half circle). In the limit as the vertices go to infinity, there are even ideal hyperbolic triangles in which all three angles are 0°.”

“So how did Lovecraft come up with this? And did he visit Chthuloid universes himself, or what?”

“Lovecraft wasn't the first to study this stuff,” he said. “A number of geometers made attempts to prove the parallel postulate by assuming its negation and trying to derive a contradiction, including Proclus, Ibn al-Haytham (Alhazen), Omar Khayyám, Nasir al-Din al-Tusi, Witelo, Gersonides, Alfonso, and later Giovanni Gerolamo Saccheri, John Wallis, Johann Heinrich Lambert, and Legendre. Their attempts failed, but their efforts gave birth to hyperbolic geometry.”

and so forth... 😉

“It seems that 5 squares can meet at one corner, on this side of the Singularity.”

That's not right: you don't get 5 right-angles meeting at a point, but you can get a 5-sided figure with a right angle at each corner. The area of the figure gives you the curvature.

Tonight this posted on the arXiv:

arXiv:1008.2667 [pdf, other]

Title: Did Lobachevsky Have A Model Of His "imaginary Geometry"?

Authors: Andrei Rodin

Comments: 31 pages, 8 figures

Subjects: History and Overview (math.HO)

The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to provide a sound reading of Lobachevsky's geometrical works. Although the modern notion of model of a given theory has a counterpart in Lobachevsky's writings its role in Lobachevsky's geometrical theory turns to be very unusual. Lobachevsky doesn't consider various models of Hyperbolic geometry, as the modern reader would expect, but uses a non-standard model of Euclidean plane (as a particular surface in the Hyperbolic 3-space). In this paper I consider this Lobachevsky's construction, and show how it can be better analyzed within an alternative non-Hilbertian foundational framework, which relates the history of geometry of the 19th century to some recent developments in the field.

Actually, I would go so far as to say that Euclid effectively had a sixth axiom. Namely Proposition 4 where Euclid "proves" the so called Side-Angle-Side theorem. The only trouble is that his proof here wasn't very good, relying as it did on "moving" the triangles about—despite the fact that "moving" a triangle hadn't been defined. Even worse, as it is written the theorem doesn't work at all for irregular triangles which are mirror images of one another.

Now, opinions on the proof may vary (Bertrand Russell was not kind to it), but one of the disappointing this about modern books on geometry is that they

explicitlytake this proposition as an axiom. Now, it may be that taking an additional axiom is unavoidable, but in my opinion simply asserting the Side-Angle-Side theorem is a very, very strong statement and it's a pity most geometers take this rather than an alternative route.All that aside, if you're looking at a good subject to Good Math/Bad Math blog about, Going through the first book of Euclid's elements (though not in strict order) might make for an interesting read.

As I understand it, the hyperbolic plane of constant negative curvature can't be embedded in less than 5 Euclidean dimensions; however, I haven't been able to find anything describing how that works (either how the embedding works, or why it won't fit into 3 or 4 dimensions).

Oh, you can fit it into 3, if you have a few spare hours. Make yourself a whole swag of equally-sized paper squares when it's a slow day at work, cut off the corners, then sticky-tape them together edge to edge, systematically putting five squares around each (missing) corner.

(You can leave the corners on if you like, on the understanding that they are not really "part" of the surface.)

You'll need a ... surprising number of squares. A square is a point. A row of squares (joined opposite edge to opposite edge) is a line. Pick a line. Pick two points next to the line and on the same side of it. Identify the lines coming off those two other points parallel to the first line. They will intersect at some point. Thus - that intersection point has two lines running through it, both parallel to the first line.

It's endless fun, and somewhat worrying.

The reply of the Soldier of Truth

M. MarkCC, you have well done in saying « Now, I'm not particularly good at geometry. It is, by far, the area of math where I'm weakest. But I'll take on this challenge from our Soldier of the Truth», because what you have said before this statement is wrong, and what you have expose after the same statement is also wrong.

If you don’t know that parallel lines and Euclid’s fifth postulate are valid only in the plane surface containing two straight lines intersected by a third straight line (called transversal) and forming with the two straight lines the interior angles lesser than two right angles, you have missed the whole of the geometry written in the «Elements» of Euclid, and used by all true mathematicians. In this case, you are not qualified to discuss or to judge about the foundation of geometry.

I invite you to read what had written Sir Thomas Heath about Euclid’s Elements.

The Soldier of the Truth is my nickname. My name is Rachid Matta MATTA and you can contact: rachidmattamatta@hotmail.com for more information.

Rachid Matta MATTA

April 25, 2012

Please add the letter d to the word expose in the statement

what you have expose after the same statement is also wrong.

Who can reply?

Why David Hilbert didn't develop a geometry with the words

"table", "chair", and "beer mug"?

Why Hilbert's followers don't do this kind of geometry?

Rachid Matta MATTA

21-05 -2012

The real and true entities

Rachid Matta MATTA affirms that without the Euclidean entities: point, straight line, plane surface... ,David Hilbert's geometry will not be comprehensible.

26-5-2012

To show the importance of Euclidean geometry, I put in French the quotation of Galileo:

La philosophie est écrite dans ce très grand livre qui se tient constamment ouvert devant les yeux (je veux dire l'univers), mais elle ne peut se saisir si tout d'abord on ne se saisit point de la langue et si on ignore les caractères dans lesquels elle est écrite. Cette philosophie, elle est écrite en langue mathématique ; ses caractères sont des triangles, des cercles et autres figures géométriques, sans le moyen desquels il est impossible de saisir humainement quelque parole ; et sans lesquels on ne fait qu'errer vainement dans un labyrinthe obscur.

Rachid Matta MATTA

July4,2012

M. Matta

Kindly Translate in English

Robin

The philosophy is written in this great book always open before our eyes (I mean the universe), but we cannot understand its language if we ignore the characters in which it is written. The mathematics is the language of this philosophy and its characters are triangles, circles and other geometric figures. Without these characters, it is impossible to understand any speech and we err in an obscure labyrinth.

Rachid Matta MATTA

October 18, 2012

rachidmattamatta@hotmail.com

Please Reply!!!

From the followers of David Hilbert Who can develop a geometry with the words "table", "chair", and "beer mug"?

Rachid Matta MATTA

May 30, 2013

The real space is of three dimensions and There is not a space of higher.

Rachid Matta MATTa

8 June2013

The real space is of three dimensions and There is not a space of higher dimension.

Rachid Matta MATTa

8 June2013