Lobachevsky Space

Lobachevsky space, or L-space, refers to the geometrical properties of several parallel dimensions discovered after the excavation of the Azada Complex in PL 232, and released to the world after Sacrosanct’s destruction in PL 1091. Due to its unique properties, it has been used extensively for building and storage.

## Properties

L-space is a hyperbolic space defined as a 3-dimensional space possessing of a constant negative Gaussian curvature. Its defining property is that the fifth postulate of Euclidean. However, it may be locally Euclidean, where its curvature number is low enough to approximate Euclidean geometry with small quantities, much like the elliptical geometry of a planet. Beyond the distances where this happens, however, the effects of a hyperbolic space begin to take shape. This can be defined simply as the exponentiality of equidistant points: as the distance from a point increases, the number of points with that distance from the origin point increases exponentially. This can be seem in the equation of a circle’s circumference: while it is proportional to the square of the radius on Earth, it is proportional to the exponent of the radius in L-space beyond a certain distance. This distance is known as the breakdown radius, and can be used to obtain the Gaussian curvature.

The curvature of L-space is geometrically embedded, which means that moving through Gaussian space, one experiences the world “warping” around them rather than feeling as if they are walking across a curved surfaces. Objects seem to change shape as they are approached, with the most distant objects shrinking exponentially in size. Thus it feels as if the world is “compacting” around an individual when that is in fact not the case.

Since advanced astrophysics sometimes defines the universe as a gigantic hyperbolic space generalised into two dimensions, it has been retroactively changed to refer to such spaces with a breakdown radius not exceeding 100km.

## History

Experiments on reaching parallel dimensions had been conducted in the Old World since the 2300s, and mathematical theories on their nature existed for centuries prior. However, no actual breakthrough was made until around 2640, when the Azada research complex in Iran finally succeeded in opening a rift for several microseconds to a so-called other dimension. This rift was reopened multiple times but was never stable, leading to the program being terminated and the technology stored. Due to the security of the storage silos, the Azada documents and samples were kept safe while the Phage nearly exterminated humanity.

During the Second Scouring in PL 232, the Azada Complex was opened for extermination. After several weeks of vicious hand-to-hand fighting in the dark corridors, the Complex was finally purged of all Phage presence. The technical documents recovered from within were scrutinised by Sacrosanct agents embedded in the expedition, and deemed too dangerous for humanity. As such, they were stolen and stored in Sacrosanct’s main data bank, where they sat unused for nearly a century.

After Sacrosanct was finally destroyed in PL 1091, all of its knowledge was released to the world. Scientists and researchers immediately began work on the wealth of new information released to them, and two years later a rift was replicated and held stable. The first expeditions into this L-space came back disoriented and nauseous, reporting of a “warped world” beyond the gate. Mathematicians confirmed that it was indeed a hyperbolic space.

Over the next few decades, colonisation of L-space began, as humans saw the potential applications for fast transport and compact building. Other L-spaces were also discovered with differing curvatures, totalling 9 separate spaces: Alpha, Firenze, Yukon, Enkidu, Somath, Ur, Danzin, Klatz and Pontiq.