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Considerable thought has been given to the concept of what Dimensional Space is, and what is meant by the concept of different dimensions. We have chosen to go with a variant of the Many Worlds theory. The Dimensional Sea is a higher-dimensional space within which many universes exist. If it is finite, no edge or boundary has yet been found, though it remains unproven that none exists. Dropping out of Dimensional Space into a dimension places you somewhere within it in real space. There are a limited number of dimension-templates, caused by branching timelines.
Earth, for example, is a Type-1 world. There are many Type-1 worlds in Dimensional Space, and in the area that the TSAB exist, they are the most common type of world. The entry point into Type-1 worlds is in the vicinity of the third planet out from a Class-G yellow star, in a system of between eight and twelve planetary bodies, depending on classification.
Rarely, dimensions will "bud", a single dimension splitting into two separate ones. The frequency of these events is very low, and the causes that lead to it are not well understood. It is generally accepted that there are a multitude of factors which interact in complex ways, and it is an ongoing topic of research amongst dimensional physicists. The split is violent, often causing dimensional quakes and severe turbulence in the Dimensional Sea during the process. Generally, while the daughter dimensions will be identical immediately after the split, divergence will render them noticeably separate on planet-wide scales within seventy to one hundred years, as chaos theory takes hold and events in the surviving populations proceed differently, different children are born to different couples, etcetera.
"One of the most common questions from children when they find out about dimensional space, and how most planets are merely dimensional twins of at least one other world, is "Is there another me out there?". This isn't a question that merely fascinates children, either. Brilliant scientists have devoted their lives to trying to answer it. And in the end, the only answer we have is "probably not".
"Let me lay down the facts. Whatever makes dimensional spaces bud appears to be fairly rare, though our understanding of it is still sadly incomplete. Once a branch has occurred, chaos theory means that with a few generations, populations are completely different. Then it simply becomes a question of mathematics. For example, Type-1 worlds, which are where human beings evolved, are one of the most common overall, and the most common in the area of the Dimensional Sea in which the TSAB operates. At the moment, thirteen Type-1a worlds, where anatomically modern humanity has evolved are recorded, with Alhazred strongly believed to have been one - the origin of modern dimensional-space civilisation. Only thirteen such worlds, over all that time, shows that it is rare. That means that they must have budded less than a few million years ago, although, of course, there is no way of telling how old a dimensional space is. Notably, some of the humans have notably different late evolutionary history than the dominant lineage, and so we have mutual genetic incompatibility. Indeed, some argue that the classification of 1a is over-general, but I believe that such an approach is somewhat chauvinistic. Certainly, 1b worlds, where an ecosystem has appeared, but which was colonised by humanity from elsewhere (ancient Belka being an example of one of these worlds) are much more common. 1c worlds, where the third planet has no life, are also common, though less so than the 1b worlds, as they had a shorter space of time in which to branch before the oxygen crisis. These, however, are generally no more useful than any other uninhabited world.
"So... I do appear to have got a little off topic, but I think that explains why you don't have a dimensional twin out there. There are so few worlds that bud like that, and as soon as chaos theory gets involved, identical timelines become a near impossibility. And it is just as well. A dimensional budding is hypothesised to involve massive area dimensional quakes around nearby spaces, and hideous disruptions in dimensional space itself, drastically altering the positioning of dimensions relative to one another in the Sea."
-- On the Absence of "Alternative Me", Yuuno Scrya, 071.
This situation leads to the fascinating case that multiple worlds can bear claim to the development of intelligent life. The above described world template; Type-1 worlds, host more of the population of Dimensional Space than any other, as they are both numerous and habitable. Intelligent life developed on a Type-1 world, but intriguingly, it then underwent several dimensional splits after the evolution of hominids, meaning that they continued their development, in parallel, on several different worlds. By the point it split, their evolutionary path was relatively set, and the divergences between what came to be called Type-1a worlds were small enough that the vast majority of humans in Dimensional Space are interfertile (although there are a few groups who aren't, likely from the earliest splits). Thirteen Type-1a worlds are on record, with UA-97 (Earth) being the last Unadministered one. It is also believed that the lost world of Alhazred was a Type-1a world.
Far more numerous are the Type-1b worlds, which split after the oxygen catastrophe and the development of a biosphere, but prior to the development of intelligent life. These worlds are perfect for habitation and terraforming, and the day cycle of these worlds, as they house the majority of the TSAB-aligned planets' population, are what set the TSAB New Calendar. The homeworld of Ancient Belka was a Type-1b world. Finally, Type-1c worlds split before the oxygen catastrophe, and are no different to any other non-viable planet, such as Mars. Barring unusual activity such as Lost Logia stored there, these are generally ignored, as they are no easier to terraform than any other unsuitable planet.
It is interesting to note that in rough terms, the maps of most Type-1 worlds are broadly similar, with the continental shapes being recognisable to anyone from any other Type-1 world. Geological activity is relatively predictable, and the timescales of change are such that divergence is very, very slow. No other form of intelligent life has ever been encountered. There are archaeological traces of a species of highly intelligent bird-like creatures approximately 9000 years before the present day, but it is unclear as to whether they were actually sapient, and they all abruptly went extinct around 4800 BCE (it is interesting to note that this matches relatively closely to the period during which the Alhazredians were first expanding across Dimensional Space). Otherwise, humans appear to be the only sentient creatures in Dimensional Space, and it is theorised that this may be because we evolved early and then spread so fast that nothing else had a chance to evolve.
Example Worlds: Earth
Type-2 worlds are colder, possess slightly higher gravity (about 1.1g) than Type-1 worlds and are technically moons, orbiting the nearest gas giant to their red dwarf. Due to the dimness of their stars, the red sun in the sky is notably larger than a Type-1's one. They are tidally locked to their gas giant, so the day is over a week. The tides are nearly non existent on such worlds. As a result of this colder-than-Earth climate, most civilisation on these worlds is equatorial, on several continents (including one thin one that lies roughly along the equator), and there are very large ice caps which cover both poles. In some instances, ancient terraforming efforts have put more CO2 in the atmosphere, and made the equatorial regions temperate, but those were not always well advised, and on two cases, the ecological catastrophe caused has required planetary resettlement.
By contrast, Type-3 worlds are hot. Like the Earth, they orbit Type-G stars, but they are hotter despite lower atmospheric CO2, and have only a single supercontinent like historic Pangea (which makes things even hotter around anywhere but the edges). The native life has innate abilities from a structure homologous to the linker core (although technically it comes via a completely different evolutionary pathway and isn't related at all), and appear similar to 1-life insects. This is the world-type seen in A's, where the tentacled worm-things live, which the Wolkenritter were draining. There are actually some thriving shallow-sea and floating communities on these worlds, because due to the shallowness of some of the seas, plant life (on a large number of such worlds) grows from the sea-bed to rise as trees and results in "false continents" and entire ecologies built around what end up more like salty mangrove swamps. They have a high level of ambient mana, which is the reason that the native life have adapted to use it, and the flying islands that can be found there are in fact biological organisms, vaguely crab-like, which use their innate magical abilities to hover out of the reach of predators, earth and plant life building up on their backs as they grow.
Example Worlds: Pihroea
Mid-Childa, the current base of operations for the TSAB, is a Type-4 world, the second planet out from its star, with two moons, two major supercontinents and a smattering of island chains. One of the supercontinents is situated directly over the south magnetic pole, and the interior of that one is a colder-than-the-Antarctic desert, but the edges are habitable, if rough. At least one Type-4 world has civilisation there, centered around an Alhazredian ambient-magic-to-heat plant, and in others, the Alhazredians and Belkans alike built strange, now-wrecked computer arrays in the cold. In the modern day, quite a few research labs usually get put there, well away from populated areas and water supplies.
Example Worlds: Mid-Childa
On Type-5 worlds, the gravity is fractionally higher than Type-1 standard, but a much more radical difference is the nature of the Type-5 sun, which is an A-type blue-white star considerably more massive than Type-1 stars. Type-5 worlds exist in the outer system, with no fewer than four super-Jupiters much more close in, but still have more incident radiation than Type-1 worlds. The atmosphere is nitrogen-oxygen, with an oxygen level of about 16%, meaning that humans require breathing apparatus (or an appropriately set up Jacket) to walk around outside of sealed environments. The world is volcanically active, especially among the Haan Ridge, which runs down the centre of one of the continents where the major plate is tearing itself in half. There are no polar caps.
From orbit, the worlds appear shining, for the native plant life incorporates metallic elements into their leaves. Rather than the Type-1 method of photosynthetic absorption, the "plant-life" of Type-5 worlds are technically thermosythnetic. Sprawling root systems deep in the earth create temperature gradients in the plants, and the metal layer the plants create over their surface protects their cells from the ultraviolet radiation. As a result, the world has an exceptionally high albedo. Patches of shining plants are broken up by salt flats, and the world's oceans. The pollen of the native plantlife is allergenic and toxic to humans. Macroscopic life exists native to Type-5 worlds, with the dominant clades possessing both a protective exoskeleton and a structural endoskeleton and a decapodal structure. The female produces eggs which hatch inside her, producing an analogue to live birth, and the young feed off superfluous flesh the female secretes from her exoskeleton. They are mutually toxic with Type-1 life.
Example Worlds: Natrium-Blüte
To summarise, Type-1 worlds are wonderful for human life, and common. Type-4 are also relatively inhabitable, at least in the cases where there has been biosphere replacement (which the Alhazredians and later civilisations seem to have done as standard; the modern TSAB somewhat disapproves of wiping out entire ecosystems, down to the restructuring of soil, and replacement with 1-life imported species). Other world-types, especially those from 5 onwards, are less conducive to 1-life and are rarely used except in cases where terraforming has already left a usable world ready for habitation.
The full explanation of the details of dimensional cartography are far too complex to explain in the limited space available here. Some of the brightest minds in Dimensional Space spend their lives trying to expand on the knowledge available on it, and geniuses have died in frustration having accomplished little. The human mind is simply not designed to be able to handle the seven dimensional coordinate manifold which is the current best model for the structure of the dimensional sea, and must resort to allegory, metaphor, and mathematics.
Under the current paradigm, each dimensional frame (colloquially referred to as a "dimension", though this is strictly inaccurate) can be represented as a point in a seven dimensional locally flat framework, meaning the relative position of any world to any other can be described using a seven dimensional coordinate. With the already-existing four dimensions within each frame, that gives a total of ten dimensions of space and one of time, by which any given coordinate in any frame can (or, at least, should) be able to be described. The coordinates are denoted with the first seven letters of the hebrew alphabet, while the classical dimensions are still given by latin letters. The geometry of these higher dimensions is complicated, with regions with positive and negative curvature at a larger scale; for example, the so-called Alhazredian Rift is a sparse area of the Sea with a pronounced negative curvature. It is unknown if the universe is closed, flat, or open at the macroscale, how old it is, or many other details.
Dimensional coordinates are quantised such that each dimension is denoted with an integer; the system measures how many dimensional frames are between it, and the reference point along that axis. The way a larger "distance" exists as a concept is due to the fact that due to the physics of dimensional space, one can only enter a frame relatively close to a gravitational body. As a result, there are "empty" dimensions which cannot be entered, which are those with no large gravitational bodies close to them. This serves to separate the dimensional clones of the stars which happen, for as yet unknown reasons, to be close enough that with the still-absolute light speed applying, a dimensional cruiser can move parallel to the skein of the frame to reach it. As an extreme example, Type-9 worlds are almost half a light year displaced from the position of Type-1 worlds, and as a result only one such world has ever been explored.
As a result, the following details use metaphors and comparisons to the normal human scale of things - the very fact that the dimensional sea is referred to with nautical terms is an example of this. This is not, strictly speaking, true, but such allegorical descriptions are useful in their own way. It is not true to say that the vast majority of the known fraction of the Dimensional Sea is "east" of the Alhazredian Rift, but it is more useful to the colloquial reader than saying that under the Heimholtz convention (used by the TSAB for navigational cartography), the known areas have have a coordinate of ה =x-108 or greater.
Most importantly for humanity, the known Type-1a worlds once formed a rough seven-dimensional hypersphere, around a central radial point, believed to have been produced by a large-scale budding. This event appears to have been around the time of the Toba eruption, which is a common occurrence on all Type-1a worlds with that origin. There are known frames where mankind did not survive that occurrence. Known human civilisation is clustered around the eastern side of this shell, which intersects with a background of mostly Type-1b and Type-4 worlds. This is, as might be seen, exceptionally useful from a habitability viewpoint, and the Alhazredians spread across the habitable Type-1a and 1b worlds, and terraformable Type-4 worlds rapidly, reaching down as far as almost to Unadministered World 97 in the "bottom", to Praové in the "south", Unadministered World 912 in the "clockwise", and so on (as one might see, the number of terms that had to be repurposed to describe the extra dimensions is notable).
This hypersphere was ruptured by the Fall of Alhazred, and broken in the dimensional quakes that followed, and as a result, there are fractured shell remnants where clustered stayed together. The TSAB has centred around three clusters, the so-called core worlds. By and large, they have been inhabited since Alhazredian times, have a median population in the single digit billions, and have been extensively terraformed. Mid-Childia is one of these worlds, and stands as a classic example; despite its status as a Type-4 world, its biosphere has been completely replaced by Type-1 life, with only a few remnants of what was once there existing in the form of Type-1 life engineered to look like the Type-4 creatures they replaced.
Imaginary space is the hyper-dimensional layer "under", for lack of a better word, the Dimensional Sea. Theorised to be infinitely large and "deep", it is nonetheless rarely encountered, and generally stays hidden until and unless Dimensional Space is torn or ruptured somehow, generally by titanic amounts of highly concentrated mana, or by effects that somehow damage or rend open the fabric of spacetime (such as the Arc-En-Ciel, or the destruction of the Garden of Time). Conventional magic does not work in Imaginary Space, as the chaotic fluctuations create an effect similar to an extremely powerful AMF - and indeed, it was from encounters with Imaginary Space that the concept of an AMF was developed in the first place. If a large enough rent in spacetime is created; objects, structures or even entire worlds can drop into it, as Alhazred did. As far as is known to modern scholars, nothing lost to Imaginary Space has ever been retrieved in any form.