# Fractal Space

Fractal Space: Fractal space, or **F-Space**, was considered to be nothing more than a mathematical theory until 2408. The conceptualization of Fractal Space, however dates back to 1890, and was first put forward by Guiseppe Peano. The exact properties of F-Space are unmeasured, although reinforced wave motion has been observed in the Tch'Kairn fractal spacial communications signal.

Fractal space is best visually described by the **Hilbert curve** (also known as a **Hilbert space-filling curve**), which is a Geometrically continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling curves discovered by Giuseppe Peano in 1890.

Because it is space-filling, its Hausdorff dimension is **2** (precisely, its image is the unit square, whose dimension is of course 2 in any definition of dimension; its graph is a compact set homeomorphic to the closed unit interval, with Hausdorff dimension 2).

**H_n** is the ** nth ** approximation to the limiting curve. The Euclidean length of ** H_n ** is ** 2^n - {1 \over 2^n} **, i.e., it grows exponentially with **n**, while at the same time always being bounded by a square with a finite area.

For multidimensional databases, Hilbert order has been proposed to be used instead of Z order because it has better locality-preserving behavior.

There exist at least three other major layers of Space, and an almost infinite number of sub-levels including fluidic space. Each major layer of Space possess different quantum resonance signatures. It is there fore possible to detect the origin of an object by scanning it's quantum resonance frequency. The chart at the left shows a very basic concept of various quantum resonance states as defined by their place of origin.

Travel between or through the D-Branes that separate the various layers of space is possible. One means of doing so is through the use of an Interphasic Rift, a Wormholes, Phase Variant Modulator, or through Subspace Rifts.

The principles of the Warp Drive operates by breaking down the D-Brane between N-Space and S-Space. The result of this is a relative reduction in mass, so much so, that the ship is capable of being propelled beyond the the speed of light, making interstellar travel possible.

The other major known layers of space are:

Normal Space, or **N-Space**, is the universe in which we exist. It follows the fundamental rules of physics and must adhere to the rules of conservation of energy. It is the universe in which most known life forms exist and is the universe in which the majority of beings inhabit throughout their lives.

Subspace: subspace, or **S-Space**, is a layer of space immediately beneath Normal Space. It is characterized by the bending of several fundamental rules of conventional physics, including the absolute limits of velocity. It is Subspace that allows the utilization of Warp Drive.

Subspace is an integral part of the space-time continuum, distinct, yet coexistent with normal space. Subspace and normal space are confluenced together. However, there are in some regions an interfold layer forms between the two realms. (VOY: "Real Life") Subspace has an infinite number of domains. Geordi La Forge compared it with "...a huge honeycomb with an infinite number of cells". (TNG: "Schisms")

Tetryon particles are a form of subatomic particle which can only exist naturally in subspace. If they are released into normal space, they exhibit erratic properties. This can only occur when the fabric of subspace comes into contact with normal space in some manner.

*Exactly what subspace is has never been revealed on screen, though many theories, both fan-based and scientific, have been put forward. Explaining subspace would be hard, since it is used as a solution and cause of various problems throughout the series. It is implied to be the medium through which faster-than-light travel and communication is possible, similar to hyperspace.*

Phase Space: Phase space, or **P-Space**, is comprised of an infinite number of dimensional membranes which are only separated from the other layers of space due to quantum resonance. it is possible to shift one's quantum resonance into another layer of P-Space, through the use of a Phase variant modulator.