"Do you mean," Sir Giles said, "that the thing never gets smaller?"
"Never," the Prince answered. "So much of its virtue has entered into its outward form that whatever may happen to it there is no change. From the beginning it was as it is now."
"Then by God, sir," Reginald Montague exclaimed, "you've got the transport of the world in your hands."
Summary: I think to understand what is going on in Many Dimensions, one must begin with organic law, a constant theme throughout the book. Organic law is law that constitutes the 'organs' or instruments of governance themselves. This may be stricter or looser, depending on the state of law. For instance, in the United States, which has a fairly stable and clearly defined organic law, organic law is constituted by four documents: the Declaration of Independence, the Articles of Confederation, the Northwest Ordinance, and the U. S. Constitution and its amendments (to which are added, in the case of the states, the state constitutions). Each of these set up important instruments or organs of governance, and taken together and in proper order (e.g., taking into account the fact that the Constitution revises the others) they constitute the organic law of the United States (and, although I don't know if it's actually done any more, they used to be printed as such at the beginning of every copy of the U.S. Code). Because organic law sets up the means of government themselves, the government has no legitimate action except so far as it acts consistently with organic law; and nothing the government passes can be a law if it conflicts with organic law, unless it is revising organic law in the way organic law itself provides. To have means of government at all implies some organic law, although it will often not be as straightforward as the American version, which has the simplicity of not being parliamentary. American organic law is quite rigid, whereas British organic law, to take the obvious example, is quite fluid.
The two protagonists of Many Dimensions, Christopher Arglay and Chloe Burnett, are working on a book, Survey of Organic Law. Lord Arglay is Lord Chief Justice of Britain and really knows his law, which he would have to in order to be writing a book concerned with British organic law. Chloe is his secretary and research assistant. Both of them take the task very, very seriously. One of Arglay's key character traits is that he really does see himself as serving Law and (crucially) would never manipulate it for his own purposes. Miss Burnett approaches the matter as a student, and, indeed, is referred to in one of the chapter titles as "The Pupil of Organic Law". References to organic law are found throughout the book, and the key action that resolves the problems of the book is itself called in the chapter title, "The Process of Organic Law".
This is important because it explains what is going on with the Stone of Suleiman. Having been acquired through some rather crooked dealings by Sir Giles Tumulty, it soon becomes clear that the Stone has rather miraculous properties. You can divide it without diminishing it -- split the stone and you will have the Original and its Type, perfectly identical and both looking exactly like the Original. With the Stone you can manipulate time and space to travel anywhere or anywhen, and you can even project yourself into the minds of others. The Stone also makes possible extraordinary healing. None of this is arbitrary. The reason the Stone can do these things is that it is the organic law of creation itself, miraculously contained in a stone through the power of the Ineffable Name. All of the properties of the Stone reflect features of organic law generally: organic law is one and indivisible although shareable, it constitutes the means of governance and thus all of the government is implicit in it, laws must be interpreted in light of it, and it provides remedy in particular cases. The Stone is repeatedly referred to as the First Matter, but it is the First Matter of creation in the sense that the constitution of a nation is the first matter of its government. The Stone of Suleiman is the Constitution of the Cosmos; it has absolute supremacy over the laws of time, space, and thought because it is what grounds and constitutes those laws in the first place.
This sets up the major conflict in the book, which is not merely maneuvering over a magical relic but a genuinely moral question of one's relation to organic law. The difference between Lord Arglay and Chloe Burnett on the one hand and people like Sir Giles Tumulty on the other is that for the latter the Organic Law of the Universe is merely something to be manipulated for his own ends; whereas Lord Arglay and Chloe Burnett both see it as an end in itself, indeed, the End, the End of Desire. It is Law as such, and the heroes are marked by their refusal to treat it as something merely to be used. Indeed, no less than three chapter titles highlight this point. It necessarily gives the plot a somewhat curious structure: the difference between the heroes and the villains is that the heroes repeatedly refuse to further the plot, and their heroic action consists entirely of letting the Stone determine how things should happen.
Williams also plays with the meaning of organic law, and the semi-paradoxical nature of the phrase: vital spontaneity with rigorous necessity. This serves also as a symbol of what differentiates the heroes from the rest, since Lord Arglay and especially Chloe live under law, and ultimately it is Chloe's self-sacrificially expressing Law through her life that makes possible the resolution of the problems of the book.
This play on Law is one of the strengths of the work, and it is impressive how Williams is able to show Law as itself a semi-mystical thing through the Hajji's obscure and riddling comments on the Stone. Everything the Hajji says about the Stone, however mystical it may sound, is true in some sense of law in general, including very ordinary human laws. The only difference is scope.
He looked out, and in the sky itself there was a change. There was movement between him and the heavens; the chimneys and clouds and sky took on the appearance of the Stone. He was looking into it, and the world was there, continents and cities, seas and their ships. The Stone was not these, yet these were the Stone---only there was movement within and beyond them, and from a point infinitely far a continual vibration mingled itself with the myriad actions of men. And then, in the foreground of that vastidity, he saw rising the Types of the Stone, here and again there appearing and through all those mingled colours rushing swiftly together. Loosed from their cells and solitudes upon earth, living suddenly in conjoining motion, closing within themselves the separation which men had worked on them, those images grew into each other and were again made one. For a moment he saw the Unity of the Stone at a great distance within the Stone which was the world, and then the farther Mystery was lost in the nearer. Colour and darkness were a great background for her where she stood; they concentrated themselves upon her; through her they poured into the Stone upon her hands, and behind her again appeared but the sky and the houses of a London street.
Recommendation: This isn't Williams's strongest work, and I find Chloe Burnett a bit annoying, but as an idea novel it is marvelously constructed, and the basic story is quite readable. Recommended if it comes your way.