In essence, there are infinite higher spatial dimensions in the Mythos, with everything in a lower-dimensional space being derived from an infinitesimal portion of something in a higher dimension, the latter of which corresponds to "substance and reality," while the former is "shadow and illusion." This hierarchy, as mentioned before, has no end, and extends up to the "dizzy and reachless heights of archetypal infinity."
"The waves surged forth again, and Carter knew that the BEING had heard. And now there poured from that limitless MIND a flood of knowledge and explanation which opened new vistas to the seeker, and prepared him for such a grasp of the cosmos as he had never hoped to possess. He was told how childish and limited is the notion of a tri-dimensional world, and what an infinity of directions there are besides the known directions of up-down, forward-backward, right-left. He was shewn the smallness and tinsel emptiness of the little gods of earth, with their petty, human interests and connexions—their hatreds, rages, loves, and vanities; their craving for praise and sacrifice, and their demands for faith contrary to reason and Nature."
"Then the waves increased in strength, and sought to improve his understanding, reconciling him to the multiform entity of which his present fragment was an infinitesimal part. They told him that every figure of space is but the result of the intersection by a plane of some corresponding figure of one more dimension—as a square is cut from a cube or a circle from a sphere. The cube and sphere, of three dimensions, are thus cut from corresponding forms of four dimensions that men know only through guesses and dreams; and these in turn are cut from forms of five dimensions, and so on up to the dizzy and reachless heights of archetypal infinity. The world of men and of the gods of men is merely an infinitesimal phase of an infinitesimal thing—the three-dimensional phase of that small wholeness reached by the First Gate, where 'Umr at-Tawil dictates dreams to the Ancient Ones. Though men hail it as reality and brand thoughts of its many-dimensioned original as unreality, it is in truth the very opposite. That which we call substance and reality is shadow and illusion, and that which we call shadow and illusion is substance and reality."
These higher dimensions are not necessarily properties that span the whole cosmology, however, and a single spacetime continuum can easily be construed as having an indefinite number of higher-dimensional planes within itself.
"Any being from any part of three-dimensional space could probably survive in the fourth dimension; and its survival of the second stage would depend upon what alien part of three-dimensional space it might select for its re-entry. Denizens of some planets might be able to live on certain others—even planets belonging to other galaxies, or to similar-dimensional phases of other space-time continua—though of course there must be vast numbers of mutually uninhabitable even though mathematically juxtaposed bodies or zones of space.
It was also possible that the inhabitants of a given dimensional realm could survive entry to many unknown and incomprehensible realms of additional or indefinitely multiplied dimensions—be they within or outside the given space-time continuum—and that the converse would be likewise true. This was a matter for speculation, though one could be fairly certain that the type of mutation involved in a passage from any given dimensional plane to the next higher plane would not be destructive of biological integrity as we understand it. Gilman could not be very clear about his reasons for this last assumption, but his haziness here was more than overbalanced by his clearness on other complex points. Professor Upham especially liked his demonstration of the kinship of higher mathematics to certain phases of magical lore transmitted down the ages from an ineffable antiquity—human or pre-human—whose knowledge of the cosmos and its laws was greater than ours."