▀ Is The Book of Sand a book from the fourth dimension?

This is the fourth article in the series about the popular short story The Book of Sand of Jorge Luis Borges.

I finished the previous article with the following paragraph:
"Despite of this book being infinite, as Borges admits (and transfinite —according to my interpretation) he also suggests that it can be equally finite. Is this possible?"
The goal of the current article is to show that the book Borges bought from a Bible salesman was a mysterious object that was more than infinite in content: it was also a multidimensional and extra-real object, a supernatural book that was capable of existing in several dimensions, and being finite and infinite at the same time.

The Book of Sand is a hyperbook

What are some of the hints that Borges give us that point toward the book extending to other dimensions beyond our 3-dimensioned space? There are two:
  • At the beginning of the story, Borges mentions his initial intention of starting his story as a geometrical plot. We see this at the very first paragraph where he writes: "The line is made up of an infinite number of points, ... , the hypervolume of an infinite number of volumes". And then proceeds "No, unquestionably this is not —more geometrico— the best way of beginning my story". With this brief introduction he follows the geometrical tradition of Charles Hinton (1853 - 1907) with his books and essays about the fourth dimension like What is the Fourth Dimension, and Scientific Romances, and Edwin Abbott Abbott (1838 - 1926) with his Flatland: a Romance of Many Dimensions —a journey to the 1 and 2-dimensional space and a journey to the fourth dimension. Borges, like them, constructs the dimensioned space as a series or set of infinite points, planes, and volumes, but nowhere else in the story he makes a reference again to the concept of dimensionality. However, and in analogy with those concepts, we will shortly see that he implicitly assumes that a hyperbook can be made up of infinitely many books.
But, what is a hyperbook? Let us first review what is hypercube. Similarly to the line, the plane, and the cube, a hypercube is a cube that is the result of moving or dragging our ordinary cube to the 4th-dimension. But where is the fourth dimension? Although difficult to imagine, the fourth dimension is supposed to be "perpendicular" to our ordinary three dimensions. We (as Borges) arrive at this conclusion by simple induction: the plane is generated by moving a straight line perpendicularly to its length; a solid is obtained by moving a plane perpendicular to its surface. Note that in each case the motion is perpendicular to the dimensions of the previous object.

The hypercube is similarly obtained: by moving a cube perpendicular to the three dimensions it has. Visualizing the hypercube is not an easy task because we cannot imagine where is a dimension that is perpendicular to our daily 3-dimensional world. Note that according with the previous assumptions a plane is a "hyperline", and a cube is a "hyperplane". Furthermore, even a cube can be considered as a "hyperhyperline", but since this concept is little more than meaningless, we can plainly say that the cube is also a "hyperline".

Returning to our hyperbook, let us simplify the shape of the book assuming that it is like a cube (a book can be cubical). Then, a hyperbook is a 4-dimensioned book such that any reduction of its 4-hyperspace to a 3-dimensioned space results in an ordinary book.

Mathematicians not only speak of hyperspaces an hypercubes, but also of hyperspheres, so Borges —who was also related with modern mathematics— simply extended this concept to the books: if mathematicians could conceptualize such hyperobjects, for him was also very easy to conceptualize the hyperbooks.

Attributing to Borges the idea that The Book of Sand is a hyperbook —and not a simple solid— is not a sound reasoning if we cling to the above arguments alone. But there is another passage in his story that reinforces my argument that for him The Book of Sand is not a "mere" infinite book: it is also a book from other dimensions.

Let's quote again the passages to where I'll make reference:
I turned the leaf; it was numbered with eight digits. It also bore a small illustration, like the kind used in dictionaries —an anchor drawn with pen and ink, as if by a schoolboy's clumsy hand.
It was at this point that the stranger said. "Look at the illustration closely. You will never see it again."
Borges noted the place and closed the book, but once he reopened it he never found again the illustration of the anchor.

However, later he found another illustration: a mask. But there was a curiosity among them and he explicitly narrates it for us:
The small illustrations, I verified, came two thousand pages apart.
A book with illustrations every 2000 pages? Take notice that he is not saying that the next illustration is so far: he is saying that all illustrations are so far apart. He is not explicitly referring to those two illustrations, the anchor and the mask; so we can safety assume that he is writing about all the figures of the book.
Isn't this crazy? He says he verified this fact of the illustrations separation, but what type of book of earth has this particularity? Why exactly 2000 pages apart?

Borges used an alphabetical notebook to record the pictures he found:
I set about listing them alphabetically in a notebook, which I was not long in filling up.
The book was somehow full of illustrations because —as he says — it didn't take him too much time to fill the notebook, despite "Never once was an illustration repeated".

The Book of Sand is a dictionary

Remember the quote: "... a small illustration, like the kind used in dictionaries"? From this quote we obtain the second hint: The Book of Sand is a dictionary! This marvelous book —this hyperbook— is a dictionary because:
  • It never repeats an illustration: every instance of a "book of sand" is just a definition of an object! This is what dictionaries do: they show a single small plate or diagram and then a short or long explanation of what is this object. Every time the book is reopened, a new random definition with its corresponding pictogram appears.
  • Only a single picture picture per book "instance" appears. Every time he opened the book the whole book is dedicated to the description of the picture he found. Sadly, he didn't understand the accompanying definition definition and prose because "The script was strange to me".
The dictionary was unreadable for Borges because it was written in a foreign language or dialect: "It seems to be a version of the Scriptures in some Indian language, is it not?" he asked to the salesman, but the answer was "No ... I acquired the book in a town out on the plain in exchange for a handful of rupees and a Bible. Its owner did not know how to read. I suspect that he saw the Book of Books as a talisman".

The Book of Sand is a dictionary: what a surprise! No wonder the figures were 2000 pages apart; each picture definition was 2000 pages long so that each "copy" or "instance" of the Book of Sand is dedicated to one particular object or thing. Now we understand why the figures were never repeated: because each "book of sand" consisted of a 2000-page long definition.

The Book of Sand as an object of parallel universes

Is the possibility of multiple instances of the same object in multiple dimensions coexisting together an insane idea? Not at all, because with the triumph of the modern physics the concept of multiverses is just one of the many hard ideas to digest.

But, how and why so many "instances" could occupy the same time-space in such a manner that the book could be held in Borges' hands? The answer is simple: remember that a hypercube is a cube surrounded by cubes in every possible dimension; therefore, in a similar manner, The Book of Sand is a hyperdictionary surrounded by dictionaries in every possible dimensions. In this way, every time Borges opened the book he was opening a dictionary in other dimensions. This is the reason why he could hold a multidimensional infinite dictionary in his hands: he was holding only a 3-dimensional instance of and infinite-dimensional dictionary. All other "copies" or chapters, or "definitions" or "instances" were in very near dimensions: just touching the "real" one, but inaccessible at the same time, as the the figure at right shows.

The notion of parallel universes is not more insane than the notion of a single an unique universe; both extremes are hard to understand. I leave the reader with two simple questions related with the parallel universes idea: What law of physics states that there should be a unique 3-dimensioned space? What law of physics states that there cannot exist more than three dimensions?

Articles in this series
1 - ▀ Nobody understands the infinite so well as Borges
2 - ▀ The Book of Sand of Borges and the Continuum of Cantor
3 - ▀ The Book of Sand is a Transfinite Book
4 - ▀ Is The Book of Sand a book from the fourth dimension?