1-dimensional geometry

Hello. I will now start a series of lectures taking up the challenge of explaining in a very short time the fundamental theories of modern physics : special relativity, classical physics and quantum physics, without any prerequisites.

By making it short, I don't mean doing any superficial work of popularization, that does not really explain the theories, but that in the name of simplification and understanding, actually eliminates any effective understanding, to just roughly describe some mysterious effects.

No. What I will give here is the full, accurate, mathematical expression of these theories. And I will make them much more intuitive and faster to learn than you can find anywhere else.

So that just a short time of these lectures, such as one hour of a new lecture per day for a few days (with the rest of the day re-playing it and caring to understand it), will let you learn essentially as much as by attending the lectures currently provided by any educational institution for a few weeks or even months. Well okay it won't be complete in the same way, I won't explain all the same things and give you the tools to solve the same problems, but it will still be as good, by better explaining things in other ways.

How is that possible ? Because as I explained in the first video, these institutions never really cared to find the best possible ways to teach physics. It isn't so hard to do the best when others, aren't really trying to compete. But anyway, you will see.

Ok, let's start now by a lecture on geometry.

Why introduce geometry to start a series of lectures on physics. Because geometry is the first theory of physics. It is a theory of physical space. And for modern physics, space geometry is somehow just a physical substance among others, in interaction with others, and similar to others. It is the physical substance that we most easily and naturally understand. Its right mathematical expression is very similar to those of other theories of physics, so that its study is the key to understand other theories, both in a purely mathematical sense, and by its way of providing a visual intuition that can help to understand other physics theories too.

The beginning of this lecture will perhaps seem slow and obvious, but be careful, it will then skyrocket and go very fast. You can find a slower and more detailed approach, in my web site (settheory.net). (2 min 30 sec)

So, we start with the geometry that we are familiar with, that is the Euclidean geometry. We seem to live in a 3-dimensional Euclidean space. Still we are also used to do plane geometry : the 2-dimensional Euclidean geometry, easier to draw and to visualize: the geometry of a plane, or of a paper. But why not also consider 1-dimensional geometry. That is, the geometry of a line. It is even simpler.

(show : space, a paper, a wire)

So in fact we have several geometries. But what is a geometry ? A geometry is a mathematical theory. Any mathematical theory is the description of a system of objects. In particular, each geometry describes a system of points. And the same geometry, the same description, can refer as well to several systems of points, that are of the same kind.

For example, the same Euclidean plane geometry can describe the system of points formed by this sheet of paper, or it can as well describe the other system of points formed by that sheet of paper. Because these two systems are of the same kind.

From now on we shall call "space" a system of points described by any geometry, that is not necessarily 3-dimensional, unless specified.

Now, how does a geometry describe a space. Like any mathematical theory, it has a vocabulary. This is the list of names (or symbols) available in the language of this geometry. These names may have different forms of use, to describe different kinds of things : they may describe combinations of points, or differents kinds of figures, giving them qualifications by which some figures will be distinguished from some other figures.

Examples in the language of Euclidean geometry : distance, straight line, square, curve, angle.

But an interesting fact with most geometries as opposed to other mathematical theories, is that their language has clear limits, as there are different figures that cannot be distinguished from each other but remain perfectly similar, no matter the length of the story we may try to tell about them.

For example in Euclidean plane geometry, everything that can be said about this square can as well be said about that other square. (5 min) A description of a figure by the language of geometry cannot specify where this figure is located inside the considered space. And if we have different spaces of the same kind, described by the same geometry, a description of a figure cannot say in which space it is. For any figure in this plane, it is possible to draw a figure in that other plane, that will look the same.

Because if we could not do it, it would mean that these two spaces don't have the same geometry. Because one of them would have the quality that such a figure can be drawn in it, while the other would not have this quality.

Now as both figures in a different space are similar, it means that there is a correspondence between the spaces, we say an isomorphism, that maps every point of the one to a unique point of the other and conversely, that maps one figure to the other, and preserves all structures of geometry, that is, any quality that is attributed to any figure.

But still there is a deep meaning in saying that these 2 spaces, with the exact same properties, are still different spaces. It is that there is not just one isomorphism between them, by which they can be identified, but usually even an infinity of them. Indeed, if we choose a figure here, then we can find a similar figure there, but this does not say which corresponding figure there we should pick up because there are many other figures which are similar as well. So we do not have any way to describe a specific choice of isomorphism to pick up, through which the identification should be made, rather than any other possible isomorphism. So, we do not have a better reason to confuse this figure with that rather than that other figure. But doing both confusions would be wrong because similar figures there must be considered distinct and not confused. So we cannot accept either the legitimacy of any confusion (specific correspondence) between similar figures inside similar but independent spaces. And so the different spaces should be recognized as essentially distinct from each other too.

But if 2 isomorphisms are different, how can we understand their difference ? This difference can be represented by composing one of these isomorphisms by the inverse of the other, that is, as an isomorphism from one of these spaces into itself. An isomorphism from a space to itself, is called an automorphism. In an Euclidean plane or 3-dimensional space there are 2 important kinds of automorphisms: the translations, and the rotations. And there can be other ones depending on the precisely chosen geometry, that will be presented when needed. (7:30)

Now after these generalities that go for any geometry, let us start studying specific geometries, starting with a simple but still an important one :

The geometry of the line.

The geometry of the line is usually not explicitly mentioned in school, even where a space of that kind is actually there behind what is done, as the habbit is to conventionally assume that an arbitrary choice was made of 2 points in this line, to be called 0 and 1, to identify the line with the set of real numbers.

This habbit of expressing the geometry of the line as identified to the set of real numbers, is defended by the argument that once identified the points to real numbers, we can do all the work we need on them, by means of the usual operations among real numbers: the operations of addition, subtraction, multiplication and division. So we can work in this geometry using these operations.

But there is something pitiful in this approach to the geometry of the line : the choices of positions of 0 and 1 constitute supplementary structures that do not belong to the intended geometry of the line : the symbols 0 and 1 are names that distinguish points that were not naturally distinguished in that geometry. Instead, adding them to the language changes the geometry into another theory that is more rigid, and even so rigid that it does not admit any automorphism anymore, except of course the identity function, that is the function that lets every element fixed. In this formalism, every point is identified to a number; and as every real number can be completely specified by its decimal expansion, no real number can ever have all the same properties as another real number.

Instead, the geometry of the line admits similarities between objects. Thus, for a formalization of the geometry of the line to be exactly faithful to its intended meaning, it should directly see as similar the objects intended to be similar, and thus should not involve any choices of 0 and 1. But what are these similarities ?

Let us describe the automorphisms of the line. We said, there is no more automorphisms once two points are fixed, labeled by the numbers 0 and 1. However, we can give these labels to any 2 different points, and any such choice is similar to any other choice. So, any combination of 2 different points is similar to any other such combination.

This means that if we have 2 lines, we arbitrarily choose 2 different points to be labeled 0 and 1 in one line, and another 2 different points in the other line, then there exists a unique isomorphism between both lines that sends the 0 here to the 0 there, and the 1 here to the 1 there. And then, as isomorphisms preserve everything, they preserve the attribution of a real number to a point that is determined by a choice of 0 and 1. So, every point is sent to the point that is represented by the same real number, for the representations determined by these choices in each line. (10 min)

So, what is this representation of points of a line by real numbers ? Since any geometric line can be identified to the set of real numbers as soon as 0 and 1 are chosen, we can just say that the set of real numbers is a geometric line together with a choice of a 0 and a 1. Then, for any other line with a choice of 2 points to be labeled 0 and 1, the identification of points to real numbers is defined as the one given by the only isomorphism between lines that sends the 0 to the 0 and the 1 to the 1.

But what does the use of real numbers have to do with the automorphisms of the line that is not identified with them ? Interestingly, the same operations usually assumed to operate between real numbers, are also the ones that can describe these automorphisms, but for this, they need to be reinterpreted to be used in that different framework.

To describe this in details, let us split the group of all automorphisms into different components, corresponding to a split of the double choice of positions of 0 and 1, as a combination of different structures.

Of course an obvious split would be to say that on the one hand we choose the 0, that we shall also call the

- The distance between points 0 and 1, to be taken as unit of distance.
- The orientation, that is the distinction between left and right, for saying that the 1 is located on the right side of the 0. So it is a choice among 2 possibilities.

As these structures are outside the geometry of the line, they are usually not preserved by its automorphisms. Instead, automorphisms can be measured by their effects on these structures. In this way, automorphisms can be analyzed as composites of 3 transformations, where each transformation can only disturb a different structure:

Name | Formula : x becomes | Modified structure |

Translation by a | x+a | Origin |

Dilation by a positive number k | kx | Distance unit |

Symmetry w.r.t. the origin | -x | Orientation |

- Keeping fixed the 0 and the orientation, the change of distance unit is operated by dilations centered on 0, numerically expressed as the multiplication by a positive number.
- Keeping fixed the 0 and the distance unit, the possible switch of orientation is operated by the symmetry with respect to 0, numerically expressed by the negation (the change of sign).
- Keeping fixed the orientation and the distance unit, the 0 can
be moved to any other point by a translation, numerically
expressed as the addition with a fixed number.

0 a

0 a b a+b

There are 2 ways to interpret this figure.

One way is to say that the horizontal lines are copies of each other, or at least they are lines with the same unit of length and the same orientation, so that we have here an isomorphism between these 2 copies, mapping the (0 a) pair to the (0 a) pair there. And there we have another isomorphism between for these structures but not for the 0, sending this pair (0 a) to the pair (b a+b). So, we obtain a+b by applying on a the translation that maps 0 to

But as the addition is commutative, a+b=b+a, another view is to say that b is moved by the translation of the below line that maps 0 to a. And this translation is somehow symbolically represented by this oriented pair 0 a : having fixed this pair, we can process this translation on any point b below by applying to it the same movement of translation, that sends 0 to a.

Then for multiplication: we have a similar figure but where the lines of correspondence, instead of being parallel, are required to have a common intersection with the line that connects the 1's.

The principle is exactly the same, except that the roles of the 0's, that could be moved, are now played by the 1's, while the 0's now remain always fixed. So, dilations are now allowed and balanced with the required preservation of one more point. But we can also use another graphical representation of the same process, where both copies are no more parallel but intersect at 0, making its preservation automatic, so that now the correspondences between both lines appear parallel again.

Ok. Now as we made some observations about the geometry of the line, and before going further, let us look again at the properties of what we just presented.

Each dilation, that preserves the 0 point, can be described as the dilation centered on this point that is the preserved structure, and with a ratio described by a real number. This description does not involve any specific length nor orientation.

On the other hand, each translation, which preserves both structures of length unit and orientation, can itself be described by... a length and an orientation, but no center. Just like the structures that it preserves. So it resembles what it preserves, except that, here the preserved structures, the length unit and the orientation, are not directly involved. Separate choices of length and orientation are involved instead. But still these new choices can be described using the first ones, by a nonzero real number: that is the ratio between these lengths, with a negative sign if the orientations are opposite. So, any choice of length and orientation provides the language to describe any other choice using a number. This is not the case for a choice of a center: once chosen a point as center, we cannot describe where another point is, by just a number. We need to specify a length and orientation instead.

These properties concerned each of both line geometries that were almost rigidified by choosing structures that only left one possible dimension of movement for the automorphisms. It looks different for the geometry without structure where both dimensions of automorphisms remain available, where a dilation has a center, and a translation has a length and orientation, which were not among structures and could not be specified by numbers.

Now do you think these phenomena are accidental and specific to the geometry of the line, or do you think they are more generally valid for other geometries, because of some logical necessities ?

In fact, much of these properties are not specific to the line, but come from some universal logical necessities. Let us explain this, by coming back to the basic and general definitions. Let us write this down:

An automorphism of a space

is a transformation of this space to itself

that preserves all basic structures of its geometry

And thus also all others definable from them

such as the notion of "automorphism" itself

but not always specific automorphisms among them

is a transformation of this space to itself

that preserves all basic structures of its geometry

And thus also all others definable from them

such as the notion of "automorphism" itself

but not always specific automorphisms among them

What it preserves the notion of automorphism understood as the distinction between which transformations are automorphisms and which are not. So if we take an automorphism and we transform it by another, then the transformation we obtain is also an automorphism, that is similar to the previous one, but maybe different from it.

But how does an automorphism transform another one ?

More generally, if we have 2 spaces, each has its own group of automorphism, but they are similar, so there are isomorphisms between them. So how does an isomorphism between them sends an automorphism of the one into an automorphism of the other ?

For example let us take 2 planes. In one plane, choose an automorphism, for example a rotation, as shown by this drawing : the rotation sending this figure into that figure. Then take a second plane, similar to the first one, so there exists some isomorphisms between them. Now what is the image of this automorphism of this plane when transported into the other plane by that isomorphism between both planes ? It is a transformation of that plane, also a rotation, that we can draw in this way.

So the transformation we get there is indeed an automorphism, and this can be explained in 2 ways. One way is of course that the isomorphism preserves every structure, so that if a transformation here has a property of being an automorphism, then its copy there has the same property. Another way is to say that the transformation we obtain there can be described as a composite of 3 transformations: (...)

Since all 3 of them are isomorphisms, all structures are preserved at each step, so they are also preserved after all steps, so the resulting transformation is an automorphism too.

Now let us ask another question. Take 2 spaces, choose an automorphism inside each space, we will be interested to take similar ones but it does not matter here, and then choose an isomorphism between both spaces. Now the question is, at which condition does that automorphism coincide with the image of this one by the correspondence ? In the case of rotations, they may be the same if they have the same center and angle, or they may be different, if they have a different center, or if the orientation is reversed.

Now to visualize the condition, consider any figure or just any point in one space, and its image in the other. So the condition is that for every 2 points that correspond to each other by the isomorphism, their images by the respective automorphism in each space also always correspond to each other by the isomorphism. We can understand this in terms of the expression we gave of the copy of an automorphism transported by an isomorphism, as a composite of 3 transformation. But we can also express it as the equality between both composites of the isomorphism with an automorphism, that happen in a different order.(20 min)

Now taking the particular case when both spaces are the same, the condition for an automorphism to preserve another one, in other words, for the latter to be invariant by the former, is simply expressed saying that their composite is independent of the order between them, that is the choice of which one we apply before the other. This property is called saying that both automorphisms commute.

So generally for any two bijective transformations f and g of a space, or bijections of a space to itself, we say that f and g commute to mean fog = gof, which is both equivalent to say that f preserves g and that g preserves f.

This has a very interesting consequence: that any automorphism of any space, in fact generally any bijection of any set into itself, preserves itself : fof is obviously independent of the order.

Ok, you may find it not very interesting because this case is easy but exceptional. But what is more interesting is that the same result also applies to any two transformations of a space that may now be different but only required to belong to a same group of transformations that in itself forms a space with 1 dimension, with a shape that may be that of a line or a loop, but that is connected, I mean it is in one piece, it is not several lines.

Now if we represent its dimension in the form of time, this groups appears as a transformation of our space, that depends on time, and the evolution of this transformation along time has the property of always following the same movement. For example : in a plane, we have the 1-dimensional group of all rotations around any given point. They are parametrized by the angle of the rotation, that is proportional to time, so that the progress of this rotation in time is always given by keeping everywhere the same rotational movement.

Another example is a movement of translation, that keeps the same direction and speed along time. Another example is the group of dilations of a line with a fixed center.

So, these transformation groups are such that if we take a small transformation among them, that is one happening in a small time and that is just a little bit different from the identity function, then all others, happening in larger amounts of time, are essentially obtained by composing this small transformation with itself a large number of times. For example a rotation with angle 50 degrees, is obtained from the small rotation of 1 degree around the same center, by iterating it 50 times.

Now in this situation, we can notice that all these transformations commute with each other.

Because, for any choice of a small transformation in this group, as small as we wish, and for any 2 numbers n and p, composing the same transformation with itself, or in other words, repeatedly applying it n times and then p times, gives the same result as applying it p times and then n times, because n+p=p+n as natural numbers.

So, since any 2 transformations in the group can be approximated as precisely as we wish by different numbers of iterations of a common sufficiently small one, and since these approximations of these 2 transformations exactly commute, we deduce that any 2 transformations in the group also exactly commute.

For example, 2 rotations around the same point of a plane, exactly commute: they preserve each other...

Now we have an explanation for the phenomenon that we described earlier: that when we have a space whose group of automorphisms is 1-dimensional and connected, then all these automorphisms preserve each other. So, they are invariant, which means that every one of these automorphism can be specified in the language of the geometry of that space using a number. It does not require to break that poor symmetry of the space, it does not require to choose anything in the space among

Because for an automorphism to be similar to another one, it means it would be transportable into it, by a third automorphism with which it wouldn't commute.

Which is excluded by our assumption that the automorphism group is 1-dimensional and connected.

Notice that if it was just 1-dimensional but not connected, it wouldn't work.

For example, take the geometry of the line together with a unit of distance but no orientation. Its automorphism group is in 2 pieces: one line is the group of translations, and the other line is the set of all symmetries with respect to any center. To specify an translation, we need not only a length, expressible by a number using the distance unit, but also an orientation, which is not expressible here. And to specify a symmetry, we need to choose its center, which is a point, similar to any other point.

Let us now complete the explanations of the above observations.

We took a space, that was the oriented line, whose automorphism group was 2-dimensional. We rigidified its geometry by introducing an additional structure, whose value was picked up inside a given 1-dimensional set of similar objects : in one case it was a point, in the other case it was a length.

When it was a point, the other points remained similar to each other, as they were moved to each other by the remaining dimension of automorphisms with left the chosen point fixed.

And since we explained that these automorphisms could anyway be expressed numerically, it is logically impossible to translate this numerical choice as the choice of another point or length, whose similarity with other objects of that kind makes it impossible to measure in a numerical form.

But when the choice was that of a length, the other lengths were no more similar to each other, as they could be expressed as a numerical multiple of the given length. The automorphisms, that preserved the given length could no more move the other lengths into each other, but they preserve every other length too. Therefore, the property for an automorphism to preserve one given length, is a property that does not depend on which is the length that we ask it to preserve. These automorphisms are the translations, and this notion can also be defined in the initial geometry without any choice of unit of length.

Now in this initial geometry, we have this definition of the distinction of the translations among other automorphisms, and this set of translations is here 1-dimensional. These translations in the line happen to be similar to each other. This isn't logically necessary in general, and we will see later another geometry with a definable subgroup of automorphisms that are not similar to each other, but must be defined by numbers.

Now if the translations are similar to each other, then the choice of a translation constitutes an additional structure, that once added to the language, rigidifies the geometry. From the 2 dimensions of the group of automorphisms, one dimension is removed. We have only one dimension left. But which dimension is this ? It anyway contains the dimension of translations. Why ? Because since the group of translations is 1-dimensional and connected, the chosen one inside is necessarily invariant by all others, which thus remain automorphisms of the rigidified geometry.

This is why the translations of the line are essentially the same kind of geometrical object as those that these translations preserve. That is, a length together with an orientation.

Now let sum up the different concepts that we just described, how they can be expressed by the basic operations, and how can these operations be graphically represented.