What are the four dimensions in space

The space has four dimensions.

A fourth dimension of space, it will be said, is the fifth wheel on the car. No, I reply, it is actually the fourth wheel on the car, without which it would be of little use. The car needs a fourth wheel to run; it will be seen that the fourth dimension has the same utility for space.

From the outset, of course, I cannot hope to enforce the view of four dimensions of space in two classes of people, those who believe nothing but what they see and those who see nothing but what they believe. By the first I mean the naturalists who rely only on their senses; by the latter, the philosophers.

In order to examine this subject thoroughly, the former will first walk their room lengthways, then crossways. We have two dimensions, they will say. Where's the third one? Since they are not used to looking at the sky, with the exception of the astronomers, who admittedly always look there, but only to see it always upside down according to the nature of their telescope, the third dimension does not come to them so easily , than the other two, and their existence may appear to them from the outset to be less acceptable. In the meantime, they will say, the case compels the body to at least accept it as a hypothesis; at last you will find the hypothesis confirmed by climbing a stepladder. So there is a third dimension but where would the fourth be?

After you have looked around the room for free, you will go outside to repeat the experiment on a larger scale; they will go straight, go right, look up, and be done. You will look at the crystals, the plants, the animals, first with the naked eye, then with the microscope, in a fourth dimension, then anatomize them in order to look for them inside, then cause the government to send out a North Pole expedition, finally little houses with the greatest possible avoidance of the known dimensions in order to gain pure observations about the unknown. After all these ways have proven fruitless, they will say with satisfaction: there is no fourth dimension; as some have said for similar reasons: there is no God.

Well, the philosophers will do it differently. Instead of actually going around and looking around in the thing they want to investigate like the previous ones, they will rather withdraw from it as much as possible and refrain from it, believing that the best way to get behind the essence of things is that they can understand it Turn their backs, and the safest way to avoid contradicting reality is that they don't care about the same thing. Accordingly, they will measure, weigh, anatomize, carefully avoid everything, sit down in their armchair and now expect the pure concept of space to strive for them. He comes. But as he tries to enter, the doorman stops him and says: my lord never counts over three; what is more stays outside. The space, where his being is to be found everywhere, would like to find a place in the head of the philosopher; So, assuming it also has four dimensions, leaves the fourth outside and is now let in. The philosopher counts: it is correct, he says, the concept is threefold, God is threefold, man is threefold, space is also threefold. There is no number in the world but three; the one itself consists of only three thirds; So it always breaks down into such. The one equal to three thirds is equal to the second power of three, that is, equal to nine; nine after the same end eighty-one; So one equals every power of three.

Amazed at how quickly he progressed in this way, he developed from this the principles of absolutely true, pure, higher mathematics, and finding enough to do to refute Newton and Gauss, who always stiffly adhere to the fact that one equals one he forgets time and space so completely that they know nothing better to do than to forget him again.

In short, from these two sides a fourth dimension of space will hardly ever find recognition. More could be hoped on the part of those who relate everything to practical use if we showed them what could be done more with one dimension of space.

How do people now have to shirk and how little is the land enough to comfortably nourish them. If we had one more dimension, the fields would not only stretch out in width, but would also extend in height, and the square of the yield would become at least a cube of the yield. Of course, the sky, which now lies above us in the direction of the third dimension, would have to be relocated to that of the fourth, where it is already for most people. One no longer had to go far, to emigrate, to look for everything one needed and wanted, bread, money, happiness, freedom, equality; but everything could be looked for in the new fourth dimension, which of course would then approach every head just as much as the third dimension now. All current castles in the air would then turn into real castles, and it would still be allowed to build new castles in the higher regions of the fourth dimension, since some people cannot live otherwise than in castles in the air.

Indeed, I do not see why the people, who have been told about this fourth dimension so often under other names, should suddenly stop believing in it if one calls the matter by its right name. The benefits of the fourth dimension are so great and obvious that there is no reason why one should not make up one's mind to accept it as well as to accept so many others who promised far less and yet only because of what it promised it was believed. Where we are to get the fourth dimension is a matter for itself and does not diminish in the least the benefit that one can promise oneself and others from it. In any case, the most convenient thing will be to make them yourself out of the old three dimensions; Yes, basically, since the blessing only starts with the fourth dimension, it would be best to completely eliminate the old three by using them in order to make the fourth the only one. A single dimension also has great practical advantages. You don't need to look around, you always go for a goal. If only an impetus is given, everything must continue in the direction of common progress, and no one can lazily stand aside. General freedom is thereby achieved in the simplest possible way; because everyone can do what he wants, since he can only want one thing, follow the only direction there is. The general equality is there at the same time, because there is now no longer any difference between thick and thin, since everything is as thin as possible. It is undisputed that all friends of progress, freedom and equality in this world will recognize their ideal of one dimension.

In the meantime there are likely to be others who would prefer the old three dimensions, with their comfortable width and thickness, despite all the deficiencies that occur in them, even for the sake of them, than the new dimension of pure progress towards the better. Even the fact that they have room in the three dimensions to avoid the men of that unconditional progress, they will not neglect. Above all, however, she will be shocked to see where in the world of one dimension her stomach should find space, and how thin the sausages have to be if the whole pig is only the thickness of a mathematical line. Nor will they be comforted if you tell them that in the world of progress there is no time or rest to eat at all, that is, a big belly cannot come about anyway. If, however, I add a fourth to those who leave their three dimensions as they are, they are likely to make dubious faces again. It's always something new, they'll say, and there's no end in sight. Let's stick with our good old three dimensions in which we were born and raised; there is too much of the new every day anyway. I hope, however, that I will win them over by promising to restore them from the new dimension the old braids that were cut off at the same time as their heads in the French Revolution; they will have less to do with their heads. I also remind them of the advantages of a stomach of four dimensions. So I think I will get this worthy class entirely on my side. But we let the men of progress go.

Whom I really regret, if a fourth should be added to the three dimensions, are the students who are already frightened when they are to climb from the level of plaimetrie to the mountain of stereometry; now they even see a geometry of four dimensions, a pelion on the ossa, ahead of them. What kind of perspective drawings will they have to be if it is necessary to prove that the prism of four dimensions can be broken down into four pyramids of the same content. But good geometers, who have already crawled through all the nooks and crannies between the old dimensions, will taste like a new snack if you give them something to do with completely new corners after the old sugar candy has become stale. Well, you may at least have your spherical trigonometry ready for the sphere of four dimensions, because now I'm about to bring the fourth dimension.

The way in which I want to try to help space reach a fourth dimension is, of course, peculiar; namely by initially taking one of his three from him.

Imagine a colorful little man in the camera obscura walking around on paper; there you have a being that exists in two dimensions. What prevents such a being from thinking alive. We saw earlier that even a shadow man lets himself think vividly. We do not want to repeat here that it is him: it is enough to have done it once; but you can imagine it. Well, insofar as all seeing, hearing, poetry and aspiration of a being existing only in two dimensions were also included only in these two dimensions, it would of course be just as little able to know anything about a third dimension as we, who only exist in three Live from a fourth dimension. The experimenting shadow or color man would just walk around on his surface and search in vain for the third dimension, just as in vain would he set up microscopes and telescopes for it, as our naturalist would after the fourth; it cannot rise above the surface with its gaze, but only look away in the direction of the surface. And the philosophizing shadow man, since his concepts would undoubtedly be formed in connection with his views, would just as little be able to get beyond the two as our philosopher would be able to get beyond the three. So both would consider it impossible that a third dimension exists, that more than two straight lines at right angles can be drawn through a point. They absolutely had no idea where to put the third one. And yet there is this third dimension. It exists for us who live in three dimensions ourselves.

We are just color and shadow men in three dimensions instead of two. Since we see that there is no cessation in the two, except for beings who themselves cease in the two, it cannot be foreseen why there should be a cessation in the three, except for beings who also cease in the three themselves. Shouldn't the world be able to count over three? Nor is there the slightest reason why she should stop at three; and so I conclude, according to the law of sufficient reason, that it really does not stop there.

Think about it: does the third dimension look a hair different from the second and first? But if there was no greater art involved in creating the third than the second and first, then there will be no greater art in creating the fourth and fifth than the third and second. Where else does nature stop at a beginning, except when you lose strength? But the third dimension is not yet shorter than the other two. You can see that if we only have the fourth dimension, we also immediately have the fifth, sixth, seventh, up to the most infinite dimension; we can truly indulge in dimensions, fabricate them like pins, expand their rafters as far as we want. Otherwise one dimension seemed a strange thing to us; Now the dimensions will be ridiculously cheap, and if one used a new dimension for every hop pole in the whole of Bavaria, and in Austria for every turnpike, and in Russia for every knuckle-rope: there would be no shortage of material for just as many new ones.

The philosophers will of course say: we philosophers are the brains of the world; if in our head, the highest of all, nothing goes beyond the three, then it is already sufficiently proven that nothing in the world goes beyond the three. But I consider the world to be a big hen, of which philosophy, along with all philosophers, is only a wind egg. It is well known that the egg always wants to be smarter than the hen; but because the hen is certainly smarter than the egg, the fact that the egg can only count to three is the best proof that the hen can count further.

In the meantime, even if you are not a philosopher, you might say: Three is always a nice round number, it would be possible for the room to have the adage: All good things must be three, learned in his youth and therefore when he was up to brought the third dimension, tamed the raw urge to go further, the human being, for example, who should also tame himself.

But that is a circular view; for in the space of four dimensions the proverb will of course be: all good things must be four, and in the space of five dimensions: all good things must be five. Incidentally, we want to be frugal and for the time being just stick to the fourth dimension, which we have as good as in our hands, and let the ten or a hundred on the roof fly for it.

If one cannot refute me, one will say: there is no need for refutation; the evidence with the shadow and dummy man is shadow and dummy evidence; just show us 1/100 Line from the fourth dimension and we want to give you a hundred miles or as much as you want.

Well, it is cheap that people do not want to buy the cat in a poke, but want to see at least one end of its tail; Although I could well reply to the philosophers: a poke cat is still better than the sack without a cat that they want to sell to people, and to naturalists that it is best to take a poke cat because if so if we wanted to let her out, she would probably get away.

However, to do my best, I look again at the color man in two dimensions; If I only know how to pack the third in two dimensions, then it must then be all the easier to pack the fourth in three. This is also just a special application of the method that has always been used with fruit, that which cannot be found realiter in three dimensions, in two dimensions, i. H. to look for and find on paper. Lo and behold, it works.

As for the matter: I take the surface in which my dummy man is and lead it through the third dimension, so the dummy man experiences everything that is in this third dimension; it will even change itself by getting into other light spaces, where the rays are arranged and colored differently, and perhaps look pale and wrinkled at the end of the way, while at the beginning of the way it looked red and smooth. Of course, the male never has a piece of the third dimension at once and therefore believes that at every moment he is only in his two dimensions; of the whole movement it grasps only the temporal element and the change that is taking place. But in fact it measures through the third dimension and everything that is in it. Accordingly, the male says: there is a time and in time everything changes, including myself.

Well, we also say: there is a time and in time everything changes, including ourselves. So what is the basis of this? The movement of our space of three dimensions through the fourth, of which movement we only perceive the temporal element and the change that takes place.

Nothing is fundamentally simpler and more natural: our world of three dimensions is an enormous sphere that breaks up into a multitude of individual spheres. Each of these runs; so the great primordial sphere will probably also run; but where would it go if there wasn't a fourth dimension? As it runs itself through this fourth dimension, of course, all balls also run in it, and everything that lives and weaves on these balls also passes through the fourth dimension.

This opens the way to beautiful contemplations.

Actually everything that we will experience is already there, and what we have experienced is still there; our area of ​​three dimensions, because there is nothing to prevent us from speaking of such a space in relation to the body space of four dimensions, it is only through that already through and through this not yet through. So if z. For example, if a person is a child at the beginning, an old man at the end, a man in the middle, one has to imagine that a long bar extends in the direction of the fourth dimension, beginning as a child and ending as an old man Mitte is designed as a man, from which beam the three dimensions always cut off as much as they go into at any moment; that then gives the person who lives in this moment. To make this clear, one thinks of how in some places the cute little mosaics are made, which are used to decorate breast pins, rings, and so on. Long colored pencils are first cemented together in appropriate order and the rods obtained in this way are cut into transverse slices, whereby a number of identical mosaics are easily obtained from one rod. In a similar way, a new person is cut off from the human rod of life at every moment by the progressive intersection of the three dimensions, and the difference consists only in the two circumstances that in the mosaic the intersection has only two, here three dimensions, and that the person in each subsequent cut turns out a little different here than that of the previous one, while the figure in the mosaic is repeated exactly. in the meantime nothing would prevent them from getting the figure of a child in the first cut, and in the last that of an old man, if instead of uniformly continuous pencils one used pencils that changed appropriately in the course of their length.

The foregoing promises very useful results in practice, if only one means is discovered of dividing the life bar of man by cross-sections into disks or short cylinders and placing them next to one another instead of being extended one after the other; then one could cut an entire army of soldiers from a single person and would now have not only uniform skirts but also uniform faces throughout the entire army, the last just a little older than the first; And if one also observed the cleverness of exercising the poles before they were cut up, one would immediately have a uniformly drilled army after the cutting up, whereby nothing prevented the officers from cutting the officers from a special rod as it is now happens. Of course, every soldier would then only live for a short time, insofar as he could only extend into the dimension of time with a fraction of the entire length of a person's life; but what does that do with soldiers who are only there to be shot to death to make way for new people; they would serve their purpose all the faster.

An equally important application of this invention would be that it would save us the entire art of printing.

Every book that an author writes extends into the fourth dimension in the shape of a bar, since it does not immediately disappear from the earth once the author has written it. As described above, however, we can cut any number of copies from it, all of which also have the merit of the author's original manuscript. Of course, each of these copies will only last a short time; but what does that do with books that are only there anyway to write new books afterwards; they would serve their purpose of making this space all the more quickly.

Accordingly, I would recommend setting a price task in relation to this item. Of course, nobody would solve it, but what does that do with price problems that are not there anyway, but only to be asked to make room for new things.

But now there is something more remarkable: namely, that by moving the surface of three dimensions through the fourth, we save ourselves all of our own movement. Then there is no longer any movement in this world. In order to win this beautiful sentence, and with it the eternal rest, which is what the striving of all pious people has always sought, one will of course have to put up with a spiritual motion beforehand.

On the middle o of a white paper, of what AoA an intersection through the middle, see a red and a yellow light beam, or rather the same light bar, the red one or Vertical, the yellow one above directed crookedly against the paper,

so both bars will be in where they stand together on the paper 0to produce an orange spot, as a confluence of a red and yellow spot, but separate from there. Now move the paper vertically against the red bar, so that it z. B. up to the location A 'A' comes, it becomes a red stain r going away seem to rest in the middle of the paper, although the paper keeps cutting new places on the red bar as it progresses; a yellow spot G but move away more and more from the red spot with which it first coincided, come to lie more and more sideways, seem to move briefly over the paper, regardless of the fact that the whole yellow bar remains just as stiff and firm in front of the paper as the red one . It is all due to the skewing of the yellow bar and the movement of the paper.The more skewed the yellow bar is directed towards the paper, the further the yellow spot will move away from the red with a given movement of the paper, i.e. the faster its own movement will appear.

Of course, if something seems to be moving in our three dimensions, this is only due to the fact that the beam which it extends out into the space of four is directed obliquely towards the three dimensions, and therefore as the surface of three moves away Dimensions this always intersects in other places. The more crooked, the faster the movement seems. If the movement is curvilinear, it is only due to a crooked shape of the beam.

This now leads to new fruitful considerations. First of all, one sees that the mathematician no longer has any reason to complain about the increase in work that the fourth dimension does for him, since he is spared the entire theory of motion. Everything is as it is and he does not need to calculate the origin of the world, it always goes its line. In order to calculate the shape of the room with four dimensions, all he needs is his variable t to be regarded as the fourth spatial coordinate. The naturalist, on the other hand, gains beautiful new views of nature. To briefly mention one thing. Let's see a planet going around in circles. So this is only due to the fact that the planet extends in a spiral or corkscrew shape into the space of four. As the surface of three, in which the planet is at every moment, passes through this spiral bar just as the surface of two passes through the light bar before, it certainly looks as if the planet in it is running in a circle. It is clear, then, that the universe is really only to be regarded as a large plant with spiral fibers, and that all of astronomy is only a microscopic part of botany.

Most important, however, are the practical implications. Only now will man see very clearly how he does not gain anything with all his limping and running; he is really not getting anywhere; therefore the Bible already says that running quickly does not help. He doesn’t gain anything more than that he becomes a little crooked, and a person who walks crooked paths only turns himself into a screw. Man is now spared all worry. All bread is already baked to man, what he will eat, he does not even need to open his mouth to eat it, he finds it already opened when the course of the world has led him to the point in question, and then closed again a little further. The shower that someone will hit has actually already hit the front of the beam extension, and a little further on it has healed again; in between lies the plaster. The money that someone will earn is already listed and is only crossed out by crossing out the three dimensions, and if the Jew now runs from house to house for the sake of a penny: he can be sure that if this penny is not already in the front of his pouch, all running and running will only lead him by the way. In short, man can lead the most comfortable life in the world in the future; he always gets where he has to go.

Now all that remains to be resolved is the question of where the whole movement is heading towards the fourth dimension. Two hypotheses can be made about this, one of which leads us in the most natural way to the fulfillment of all that man has ever hoped for in the future, namely to a general resurrection of the dead, a rejuvenation of our bodies, paradise and return in God, with the Jews all getting into Abraham's bosom on the way, to a life with a totally new way of being, which in every respect can be regarded as the complement of the present one and in which the most just retribution that can be imagined takes place. What more can you ask for? Nevertheless, I am concerned that, in his usual immodest manner, when he can see with certainty what he has hitherto only wished or doubtfully hoped, he will begin to wish more or even something different, hence it is always good is to have the other hypothesis ready to satisfy even these immodest ones. Everyone is now watching for themselves.

First I draw your attention to the fact that almost all movements in nature go back and forth. The pendulum swings now and then, the string swings now and then, the aether in the light swings now and then; man also runs now and then; yes every one

Every now and then, the leg swings by itself. It therefore seems more than likely from the outset that the movement of the world will also begin to decline again after a certain time. So that everything that has already happened will happen again in the opposite direction; since otherwise one would have to reproach nature for pursuing only one direction, while it has two at its disposal. Every wheel that rolls forwards can also roll backwards, and it is strange, since one has always spoken of the wheel of time, that one has never thought of this backward movement.

In the course of this declining world order, everyone will now receive from them again what he has now done to others. The shoemaker will receive exactly the same shoes from me that he is now delivering to me, and I will receive the same money from him that I am paying him now; the ox will receive the leather from the shoemaker, what he has delivered to him, and the man from the ox the fodder what he has given to him; the field will get back the grain from man, which he mowed from him, and man from the field the seeds and manure that he brought on it. In short, no one will be able to say that he is almost getting more or less, or better or worse, than what he has done, since he gets exactly the same thing again; which indisputably corresponds to the concept of justice in the most perfect way.

With this, everything promised and hoped would be completely fulfilled. Meanwhile, as said, man will not be satisfied, and now perfect justice is to be part of him, according to which everyone is measured exactly with the measure with which he himself is measured; rather, they seek it in that he is with a greater and measured better, he gets something more and better than he deserves; And since I myself belong to these immodest souls, I leave the previous hypothesis to the men who already want to let the wheel of time run backwards, and I stick to the other hypothesis, that of an infinite progression, in which I am one-sided Direction to which the world falls, so excuse that it is just the direction from bad to better. Do I already see a whip behind every being that drives the same thing, like it or not, in this direction, or, if it is not yet in it, drives it into it. So the whole world itself with its four dimensions may well be nothing else than a large four-legged creature that is driven away by such a whip at the back, while the feeding trough of eternal bliss stands in front, at which it, when it is tired, finally stands will rest comfortably for once and be amicable; whereupon the great wagoner will raise the whip again and drive it a little further forward to a manger that is still full of eternal bliss.

Now one can easily see what one would lose if one wanted to cut off one of its four legs from this animal, but at the same time one will understand why one has so far overlooked the fourth leg. Because the animal always keeps this leg lifted for progress, we beings standing on earth think it is only three-legged at all, and when we look forward we see in the raised leg only a finger pointing upwards, while the elevation is complete is essentially intended to ensure that it goes further below.

Later in 1875: Kant already discussed the possibility of more than three dimensions of space, something I was not aware of when this essay was written (1846); Well-known mathematicians, like Riemann, Helmholtz, and Klein, have entered into speculations about it no less. Furthermore, I remember reading in the advertisement of a paper by Kirchmann that appeared a few years ago, the title of which I no longer remember, that he, indisputably without knowledge of the previous article, the change in the world in a similar way as here happened, only with more philosophical seriousness, sought to replace it with an inventory. Finally, from oral conversations with Professor Dr. Zöllner discovered a very ingenious way of explaining miracles that appear as such in the space of only three dimensions, through the play-in of forces from a fourth dimension, which is of the kind that, if the fact of these miracles could be proven, in it empirical evidence for the existence of a fourth dimension could be found; about which he might well express himself in connection with more general considerations in which this thought has entered.