The Fourth Dimension Simply Explained
The following is part of the preface of the free EBook: Readings of The Fourth Dimension Simply Explained.
The Fourth Dimension Simply Explained is a collection of 22 essays of the many submitted for a contest sponsored by the science magazine Scientific American in 1909. For submission, the articles were restricted to 2500 or less. There were 245 entries form all over the world. Dr. Henry Parker Manning, an expert in non-traditional geometries and algebras like curves spaces and quaternions was chosen to judge the essays from which he edited and compiled a book of the submissions including the winning article —An Elucidation of the Fourth Dimension— and other honorable mentions.
This Datum edition consists of nine of the essays in the original edition of The Fourth Dimension Simply Explained; that’s the reason for the title: Readings of The Fourth Dimension Simply Explained. But in addition to the selected essays I have also included the Introduction to The Fourth Dimension . . . written by Dr. Manning which I consider one of the best readings in this field of mathematics. Manning, an Associate Professor of Brown University, makes important commentaries useful not only to understand the content of the essays but to understand other books about this somewhat esoteric subject. The reader should not skip the reading of his Introduction.
The importance of The Fourth Dimension Simply Explained is the diversity of authors that contributed to it. This is good since the topic of the fourth dimension is not easy to understand due to its abstract nature and out of common sense attributes. Manning states this as “It follows that we shall not find this subject an easy one to understand. It is something that we have to read a little at a time, to read repeatedly and to think over.”
Take the journey, read one article at a time, and enjoy the Readings!
The following are some paragraphs selected from Difficulties in Imagining the Fourth Dimension, by “A Dweller in Three Dimensions” (Mrs. Elizabeth Brown Davis, Washington, D. C.), one of the many articles: of the free EBook.
WE LIVE IN SPACE OF THREE DIMENSIONS. We call these three dimensions length, breadth, and thickness. For example, a line has length, but no breadth or thickness. A square has length and breadth, but no thickness. A cube has all three—length, breadth, and thickness. All the objects which we touch and use have these three dimensions, no more and no less.
Even when we say that a line has length, but no breadth or thickness, in reality we have to exercise our imagination to picture a line absolutely devoid of breadth or thickness. In practice, if we attempted to make such an object of only one dimension, which we could pick up and handle, the nearest approach to it that we could make would be an extremely fine rod or wire, but the most finely attenuated wire that could possibly be manufactured would evidently have some breadth and some thickness, though they might be extremely minute.
If we attempt to manufacture a surface having two dimensions, length and breadth, but no thickness, we will find it equally impossible. Some of the metals are capable of being rolled into extremely thin sheets, but it would not be true to say that they have no thickness at all. We may speak of the surface of a sheet of paper, but we cannot separate this surface from the paper without taking away some of the thickness with it.
Hence we see that the objects with which we are surrounded on all sides and which we constantly use, all have three dimensions. Our own bodies have three dimensions, and we live in a world of three dimensions. The notion of three dimensions is one of our inherent ideas, bequeathed to us by our earliest ancestors. Hence it is difficult for us to conceive the possibility of a world in which there are either more or less than three dimensions.
It is possible, however, to picture in the imagination a world of two, or even of only one dimension, because to do so, it is only necessary to take away, in imagination, from known objects, a portion of themselves, that is, one or two of their known dimensions, and to picture their appearance as it would be under those conditions.
On the other hand, to picture in the imagination a world of four dimensions, or even one object of four dimensions, requires that we add to three dimensions already known, other parts about which we know nothing whatever. It is obviously much easier to imagine a known object stripped of some of its known parts, but whose remaining parts are also known, than it is to imagine that same known object, with all of its known parts intact, and increased by other parts which are entirely unknown, and about which we have no information to guide us.
Moreover, we have no good reason for supposing that a world of four dimensions does anywhere exist. But the question has often been asked, If there are three dimensions, why are there not four, or five, or even more? Why should the number of dimensions be limited to three? Why should it be limited at all? To this there is clearly no satisfactory answer.
Because a condition, or a state of affairs, has never come within our own experience, does not by any means prove it impossible. There are many things in the world around us to-day, even in daily use, which not many years ago we would have declared impossible. We can readily call to mind several instances of this fact.
Hence, if we are not prepared to admit that a fourth dimension is impossible, we must conclude that it may somewhere, under some circumstances, be a possibility. When we have reached this conclusion, the mind eagerly begins to wonder and question what appearance an object of four dimensions would present, and what would be the conditions of life in a world of four dimensions. Since we have no information to guide us, we must look to the imagination for our only answer, and the imagination is ready to respond, as it always is when called upon, though in this case it has extremely meager data.
The best way to approach the solution of this interesting question, is to picture in the imagination beings of two dimensions, living in a world of two dimensions, and then to imagine the relation of our world of three dimensions to theirs. From this we can reason forward, from the known to the unknown, and by analogy, form some notion of the comparison between our three-dimensional space and a world of four dimensions.
A world of two dimensions would lie in a single plane, having length and breadth, but no thickness. Let us suppose this plane to be horizontal, like the flat top of a table. All the objects in it would be absolutely flat, without any thickness whatever. If such a world of two dimensions were peopled by intelligent beings, their bodies also would have two dimensions, length and breadth, but no thickness. They might have straight sides, like squares or triangles, or they might be curved, but whatever their shape, they would be perfectly flat.
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Although the creatures of our hypothetical two-dimensional world would be perfectly flat, they would possess a right side and a left side, just as a person in a photograph has a right and a left side. If we should lift a two-dimensional being from his plane, and replace him in a position that would be from our point of view bottom side up, his right and left sides would be reversed. This may be verified by experimenting with a face card. Hence we may imagine the possibility of any object being lifted from three dimensions into the fourth dimension and replaced in its former position with its right and left sides reversed.
We are told that there are light rays which are invisible to us, solely because our eyes are so constructed as to be unable to perceive them. And we are also told that there are tones so low or so high that we can never hear them, because our ears are not attuned to them. Shakespeare expresses this idea in the Merchant of Venice, when he makes Lorenzo say:
“There’s not the smallest orb, which thou beholdest,
But in his motion like an angel sings.
Such harmony is in immortal souls
But, whilst this muddy vesture of decay
Doth grossly close it in, we cannot hear it.”
And hence, whether or not a fourth dimension does really exist, it might be that causes similar to those just mentioned, that is, the limitations of some of our senses, would operate to render us unable to perceive it. But just as we may enjoy in imagination the “music of the spheres,” though we cannot hear it, so we may take pleasure in exercising our ingenuity in picturing the different properties of the fourth dimension.
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