Higher-Dimensional Existence

## Introduction

A given object or entity is referred to as being higher-dimensional when they exist as part of a system with a number of coordinates axes greater than our own, or in layman's terms, if they possess more than three dimensions.

Speaking in simple terms, a "dimension" can be considered an axis of movement and measurement, through which beings and objects can displace themselves throughout a given physical system. 1-dimensional beings would only be able to dislocate themselves to left and right across a single direction, while 2-dimensional beings would have an additional spatial axis inherent to them, and thus be able to move up and down as well. Finally, 3-dimensional beings like ourselves are able to displace themselves left and right and up and down, as well as forwards and backwards, encompassings parts of additional axes of space.

Following this same principle, a 4-dimensional being would be capable of moving through an additional, perpendicular direction incomprehensible to lower-dimensional beings, and this can be generalized unto any higher number of dimensions.

## Mathematical Intuition

Speaking in more rigorous terms, a “dimension” in a system can be most effectively equated to the real number line, which is itself the most basic example of a 1-dimensional coordinate space (commonly denoted as simply as ℝ), and contains bounded subsets (smaller, finite lines if you will) whose exact measurements can be taken by “pointing” at a real number represented as a specific point in the line (Example: A line with length 3 is a subset of the real number line, whose length is infinite)

In this context, adding another dimension to a given space is effectively adding another real number line in a different, perpendicular position, which is mathematically represented by continuous cartesian products of ℝ. For example, 2-dimensional space is represented by ℝ^2 (ℝ x ℝ), 3-dimensional space by ℝ^3, and so on and so forth.

In this context, measurements of positions in space are once again obtained by directing oneself at arbitrary numbers contained in the axes of the given expanse, thus obtaining one's own coordinates in it. Simplifying this case, one could also informally state that the “Dimension” of a space is the exact number of coordinates needed to specify any position within it.

Likewise, any n-dimensional object can be thought of as being a subset of a real coordinate space of corresponding dimension. Here is a good illustrative example of that.

The term “Higher-Dimensional Existence” refers to objects and entities that exist in more than the regular 3-dimensional space, with at least one additional dimension. It is important to note that certain criteria must be met for an object or entity to be considered Higher-Dimensional.

* Simply viewing 3-dimensional objects, entities, or constructs as fiction does not qualify them as Higher-Dimensional, as they are still portrayed as regular 3-dimensional beings.
* Ontological differences over 3-dimensional objects, entities, or constructs, with three or more dimensions, are often a measure of power and do not necessarily indicate the presence of an extradimensional axis.
* As far as being larger than infinitely-sized objects or spaces goes, one must analyze the context of the feat in question to determine if it truly qualifies for Higher-Dimensional Existence. In terms of volume (Or, more generally, measure), the only way to be truly bigger than an object of infinite size is to have a non-zero size in a space of more dimensions than the object in question. However, portrayals of more expansive realms containing infinitely large things within themselves are not necessarily indicative of such.
	+ A good construction to exemplify this is the topological space known as the long line. In essence, it is a space obtained by taking an uncountably infinite number of line segments and “gluing” them together end-to-end, and so it is in some sense much longer than the real line, which is comprised of only a countably infinite number of such line segments. Nevertheless, they are both 1-dimensional spaces.
	+ The long line itself can also be generalized into 2-dimensional and 3-dimensional analogues, and as such the same principle holds for higher dimensions as well.
* Stating that something is Extra-Dimensional simply means it comes from outside of the regular 3-dimensional space. It does not necessarily mean that it has an extradimensional axis in contrast to 3-dimensional objects, without further context.
* Stating that something transcends space or space and time does not necessarily imply that it has an extradimensional axis or that it pertains to the geometry of the object.
* Simply stating that something is from a Higher Plane or a Higher Existence does not necessarily imply the existence of an extradimensional axis in relation to 3-dimensional entities or objects. Statements that something is "higher-dimensional" also need to be interpreted in context, as authors at times use the term figuratively.
	+ Additionally, usages of “Higher Dimension” should be treated with scrutiny, as “Dimension” is often times used to refer to places, and not directions in space, and as such the term can easily be used in the same sense as "Higher Realm/Plane" and similar verbiage. Context should be used to determine whether it truly refers to increased dimensionality.
* Characters who embody or are a literal timeline / spacetime continuum (The entire past, present and future of 3-dimensional space) on a universal scale can also be considered for Higher-Dimensional Existence (4-D). However, simply being called a universe does not count without proper evidence of the entirety of the past, present and future of the timeline being mentioned.

## Applications

Keep in mind that this is merely a very general list of the abilities that a higher-dimensional entity would possess, and is not necessarily applicable to all the fiction.

**Pseudo-Invulnerability:** Due to the additional axis spatially inherent to them, higher-dimensional objects are comprised of more information than what can normally be described in lower-dimensional spaces, and thus may only be capable of interacting with them through lower-dimensional "slices" or cross-sections of their bodies.

**Pseudo-Teleportation:** Higher-dimensional entities may be capable of seemingly phasing in and out of sight by moving through additional axes imperceptible to lower-dimensional beings, granting them a notable advantage in overall range and mobility.

**Unconventional Physiology:** Although much of the same physical laws and principles would theoretically apply to higher-dimensional worlds, they would likely be instantiated very differently, resulting in matter behaving in strange ways compared to our own Universe.

**Large Size:** Due to the additional angles possessed by them, higher-dimensional objects and beings would be much larger than lower ones.

## Additional Notes

Due to the the wide range of such an ability, as it encompasses anything from a singular additional dimensional axis to an infinite amount of dimensional axis please list the degree of HDE that the character qualifies for the profile in question. For example, "Higher-Dimensional Existence ([5-D] {Explanation of the rating})" would be acceptable, but just HDE itself would not be.