Geometry and Space

Henri Poincare
http://pratt.edu/~arch543p/help/Poincare.html

Contents
Geometric space and representative space
Solid bodies and geometry
The law of homogeneity
The number of dimensions
The Non-Euclidean world
The four-dimensional World
Conclusions
Notes

 

 

In an article I published in the Revue générale des sciences, t. ii., p. 774, on the subject of non-Euclidean geometry, I wrote the following sentences:
Beings with a mind made like ours, with the same senses that we have, but without any prior education, could receive from a properly selected world impressions such that they would be led to construct a geometry different from Euclid's and to localize the phenomena of that exterior world in a non-Euclidean space or even in a four-dimensional space.
For us, educated in this present world, if we were suddenly transported into that new world , we would have no difficulty in relating its phenomena to our Euclidean space.
A man who devoted his life to it could perhaps succeed in picturing to himself a fourth dimension. (1)
I did not follow this up with any further clarification and it must have astonished several readers; it seems to me therefore that it is necessary to develop my thought and that I owe some explanations to the public.

Geometric space and representative space. We often say that the images of external objects are localized in space, and even that only on that condition can they be formed. We also say that this space, which thus serves as a frame prepared in advance for our sensations and representations, is identical with that of the geometricians and that it possesses all the same properties.
To all the good people who [think that way], the sentence cited above must indeed have seemed quite extraordinary. But we ought to see whether they might not be under some illusion that a thorough analysis might dispel.
First, what are, strictly speaking, the properties of space? I mean the space that is the object of geometry and which I call geometric space. Here are some of the more essential:

  1. It is continuous;
  2. It is infinite;
  3. It has three dimensions;
  4. It is homogeneous, that is to say that all its points are identical to one another;
  5. It is isotropic, that is to say that all the straight lines passing through a given point are identical to one another.

. . . . .

Solid bodies and geometry. Among the objects that surround us, there are some that frequently undergo displacements susceptible to correction by a correlative movement of our own bodies; they are the solid bodies.
Other objects, whose form is variable, only exceptionally undergo similar displacements (change of position without change of form). When a body moves and in so doing changes its form, we can no longer by appropriate movements bring our sense organs back into the same relative situation with respect to that body; we can no longer, consequently, re-establish the original set of impressions.
It is only later on and after a series of new experiences that we learn to separate bodies of variable form into smaller elements such that each of them moves in approximate accordance with the laws of motion of solid bodies. We thus distinguish "deformations" from other changes of state; in these deformations each element undergoes a simple change of position, which can be corrected, but the modification of the ensemble is more profound and can no longer be corrected by a correlative movement.
Such an idea is already very complex and could only have appeared relatively late. It could moreover not have arisen if the observation of solid bodies bad not already taught us to distinguish changes of position.
If, therefore, there were no solid bodies in nature, there would be no geometry.
Another remark also merits a moment of attention. Let us imagine a solid body first occupying position a and passing afterward to position b; in its first position, it would cause us to receive a set of impressions A, and in its second position a set of impressions B. Now let there be a second solid body having qualities entirely different from the first, for example a different color. Now let us suppose that it passes from position a, where we receive the set of impressions A', to position b, where we receive the set of impressions B'.
In general set A will have nothing in common with set A' , nor set B with set B'. The passage from set A to set B and that from set A" to set B' are therefore two changes which in themselves have in general nothing in common.
And yet we consider both these changes as displacements, and what is more, we consider them as the same displacement. How can that be?
It is simply because we can correct both by the same correlative movement of our body.
It is therefore the correlative movement which constitutes the only link between two phenomena that it would not otherwise have occurred to us to compare.
On the other hand, our bodies, thanks to the number of their joints and muscles, can go through a host of different movements; but not all are capable of "correcting" a modification of external objects; those only are capable of it in which our whole body, or at the very least, all the sense organs involved, move en bloc, that is to say, like a solid body without varying their relative positions.
In summary:

  1. We are first led to distinguish two categories of phenomena: Some, involuntary, not accompanied by muscular sensations, are attributed by us to exterior objects; they are external changes. Others, whose qualities are opposite and which we attribute to the movements of our own body, are internal changes.
  2. We notice that certain changes of each of these categories can be corrected by a correlative change in the other category.
  3. We distinguish, among external changes, those that have also a correlative in the other category; we call these displacements; and likewise among the internal changes we distinguish those that have a correlative in the first category.

We have thus defined, thanks to this reciprocity, a particular class of phenomena that we call displacements. It is the laws of these phenomena that are the object of geometry.

The law of homogeneity. The first of these laws is that of homogeneity. Let us suppose that, by an external change a, we were to pass from a set of impressions A to set B, then that this change were corrected by a voluntary correlative movement b such that we were brought back to set A. Let us suppose now that another external change a' caused us again to pass from set A to set B.
Experience teaches us that this change a' is, like a, susceptible to correction by a voluntary movement b', and that this voluntary movement b' corresponds to the same muscular sensations as the movement b which corrected a. It is this fact that we normally have in mind when we say that space is homogeneous and isotropic.
We can also say that a movement that occurs once can be repeated a second time, a third time, and so on, without its properties varying.
Those readers who know the article I wrote in this journal on the nature of mathematical reasoning will perhaps remember the importance that I attribute to the possibility of indefinitely repeating a single operation.
It is from this repetition that mathematical reasoning draws its strength; it is thanks to the law of homogeneity which it has taken from geometric facts.
To be complete, we should annex to the law of homogeneity a host of other analogous laws into the details of which I do not wish to enter, but which mathematicians subsume into a single word by saying that displacements form a "group".

The number of dimensions. I feel more difficulty in explaining my thought concerning the origin of the notion of point and the number of dimensions: it is markedly different from opinions generally accepted and it is not easy to state it in ordinary language.
We understand displacements in terms of the passage from a set of impressions A to a different set b; but among these displacements we distinguish some such that the initial set A and the final set B conserve certain common qualities. I do not wish to go into further detail nor to seek to determine in exactly what these common qualities consist.
I am satisfied to note that we are led to distinguish certain special displacements such that we may say they leave fixed one of the points of space.
That is the origin of the idea of point.
The set of all displacements constitutes what we call a group; the set of those displacements that leave fixed a point of space constitutes a partial or subgroup.
It is in the relation of this group and subgroup that we must seek the explanation of the fact that space has three dimensions.
The group total is of the order 6, that is to say that any such displacement can be considered as a combination of six elementary and independent movements.
The subgroup is of the order 3, that is to say that any displacement belonging to this subgroup, or, in other words, any displacement which leaves fixed a point of space, can be considered a combination of three elementary and independent movements.
The difference 6 - 3 represents the number of dimensions.

The Non-Euclidean world. If geometric space were a frame imposed on each of our sensations, considered individually, it would be impossible to represent an image stripped of that frame, and we could change nothing in our geometry.
But that is not the case; geometry is only the résumé of the laws in accordance with which images succeed one another. Nothing prevents our imagining a series of representations, in every way similar to our ordinary representations, but succeeding one another according to laws different from those to which we are accustomed.
We can therefore conceive that beings whose education took place in a milieu where these laws were inoperative could have a geometry very different from ours.
Let us imagine, for example, a world enclosed in a great sphere and subject to the following laws:

The temperature is not uniform; it is maximal in the center, and diminishes in proportion as we move away, reducing to absolute zero when we get to the sphere in which the world is enclosed.

I will further specify the precise law according to which the temperature varies. Let R be the radius of the limiting sphere; let r be the distance from a given point to the center of this sphere. The absolute temperature will be proportional to R²- r².
I will suppose, in addition, that in this world, all bodies have the same coefficient of expansion such that the length of any rule will be proportional to the absolute temperature.
Finally I will suppose that an object transported from one point to another whose temperature is different, immediately assumes calorific equilibrium in its new environment.
Nothing in these hypotheses is contradictory or unimaginable.
A mobile object will then become smaller and smaller as it approaches the limiting sphere.
First let us observe that although this world is limited from the point of view of our habitual geometry, it will appear infinite to its inhabitants.
Indeed, when they wish to approach the limiting sphere, they get colder and become smaller and smaller. The steps they take are therefore also smaller and smaller, so that they can never reach the limiting sphere.
If, for us, geometry is only the study of the laws by which invariable solids move, for these imaginary beings, it would be a study of the laws by which solids move deformed by those differences of temperature I have just mentioned.
True, in our world natural solids also submit to variations of form and volume due to beating and cooling. But we neglect these variations in laying the foundation of geometry, for in addition to their being very weak, they are irregular and consequently seem to us to be accidental.
In this hypothetical world, this would not be the case, -and these variations would follow regular and simple laws.
Moreover, the diverse solid particles making up the bodies of the inhabitants would also undergo the same variations ,of form and volume.
I will add one further hypothesis. I will assume that light crosses diversely refracting media in such a way that the index of refraction is inversely proportional to - . It is easy to see that in these conditions, light rays would be not rectilinear but circular.
To justify the foregoing, it remains to be shown that certain changes occurring in the position of exterior objects could be corrected by correlative movements of the sensory beings who inhabit this imaginary world; and in such a way as to restore the original group of impressions felt by these sensory beings.
Let us suppose that an object moves, while changing shape, not as an invariable solid, but as a solid undergoing unequal dilations in exact accord with the law of temperature that I proposed above. Permit me, in the interest of succinct language, to call such a movement a non-Euclidean displacement. . .
If a sensory being were in the area, his impressions would be modified by the displacement of the object, but he could re-establish them by making suitable movements. It suffices that finally the object and the sensory being, considered as forming a single body, should have undergone one of those particular displacements that I have just called non-Euclidean.
Although from the point of view of our habitual geometry bodies are deformed in this displacement and their diverse parts are no longer in the same relative position, nevertheless we shall see that the impressions of that sensory being have again become the same.
Indeed, although the mutual distances of the diverse parts could and did vary, nevertheless the parts originally in contact have returned into contact. Therefore the tactile impressions have not changed.
Furthermore, taking into account the hypothesis set forth above as to the refraction and curve of light rays, the visual impressions will also have remained the same.
These imaginary beings will thus be led like us to classify the phenomena they witness, and to distinguish among them those "changes of position" that are susceptible to correction by a voluntary correlative movement.
If they were to found a geometry, it would not be like ours, the study of the movements of our invariable solids, it would be that of the changes of position that they will thus have distinguished, and which are none other than the "non-Euclidean displacements"; it will be non-Euclidean geometry.
Thus, beings like us, educated in such a world, would not have the same geometry as we.

The four-dimensional World. In the same way as a non-Euclidean world, we can represent a world having four dimensions.
The sense of sight, even with a single eye, joined to muscular sensations relative to movements of the eyeball could suffice to our knowing a three-dimensional world.
The images of exterior objects come and paint themselves on the retina, which is a two-dimensional tabula; these are perspectives.
But, since these objects are mobile, and since the same is true of our eye, we see successively different perspectives of the same body, taken from several different points of view. We notice at the same time that the passage from one perspective to another is often accompanied by muscular sensations.
If the passage from perspective A to perspective B and that from perspective A' to perspective B' are accompanied by the same muscular sensations, we consider them to be operations of the same kind.
Studying afterward the laws according to which these operations combine, we recognize that they form a group having the same structure as that of the movements of invariable solids.
Now, we have seen that it is from the properties of this group that we have deduced the notions of geometric and of three dimensions.
Thus we understand how the idea of a three-dimensional space was able to arise from the spectacle of these perspectives, even though each of them has only two dimensions, because they succeed one another in accordance with certain laws.
In the same way that we can make on a plane the perspective of a figure having three dimensions, we can make that of a four-dimensional figure on a tabula with three (or two) dimensions. It is only a game for the geometrician.
We can even take from a single figure several perspectives from several different viewpoints.
We can easily represent these perspectives since they have only three dimensions.
Let us imagine that the diverse perspectives of a single object succeed one another; that the passage from one to the other is accompanied by muscular sensations.
We will naturally consider two of these passages as two operations of the same kind when they are associated with the same muscular sensations.
Then nothing prevents our imagining that these operations might combine following any law that we wish, for example in such a way as to form a group having the same structure as that of an invariable four-dimensional solid.
In all this there is nothing we cannot represent and nevertheless these sensations are precisely those that a being furnished with a two-dimensional retina would feel if be could move in four-dimensional space.
It is in this sense that we may say that we can represent the fourth dimension.

Conclusions. We say that experience plays an indispensable role in the genesis of geometry; but it would be an error to conclude that geometry is an experimental science, even in part.
If it were experimental it would be only approximate and provisional. And what a crude approximation!
Geometry would only be the study of the movements of solids; but in reality it is not concerned with natural solids, it has as its object certain ideal solids, absolutely invariable, which are only a simplified and very distant image of the natural ones.
The notion of ideal bodies is drawn entirely from our minds and experience is only an occasion that invites us to construct such a notion.
The object of geometry is the study of a particular "group"; but the general concept of group pre-exists in our minds, at least potentially. It is imposed on us, not as a form of sensibility, but as a form of understanding.
Still, among all possible groups, we must choose that one which will be, so to speak, the standard to which we will refer natural phenomena.
Experience guides us in this choice which it does not impose on us; it makes us recognize not what is the truest geometry, but what is the most convenient. It will be noticed that I have been able to describe the fantastic worlds that I imagined above without ceasing to use the language of ordinary geometry.
Indeed, we should not have to change anything if we were to be transported to such a world.
Beings who were educated there would no doubt find it more convenient to create a geometry different from ours, better adapted to their impressions. As for us, in the face of the same impressions, it is certain that we would find it more convenient not to change our habits.
 

Notes:

(1) This selection first appeared as an article, "L'Espace et la géométrie," in Revue de métaphysique et de morale, 1895 t.iii, pp. 631-46. It was especially translated for this volume by William Ryding of the Department of French, Columbia University.

 

 

 

 

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