Higher-Dimensional Manipulation

Summary
Higher Dimensional Manipulation is the ability to manipulate greater spatial and temporal dimensions than 3-D space, such as 4-D space and above.

According to infinity in projective geometry, and the concept of a Hausdorff dimension, each higher spatial (or added temporal) dimension is a more than countably infinite number of times greater than the preceding number.

An easy way to grasp this concept is as follows:

A 1-Dimensional (line) object only has length.

A 2-Dimensional (plane) object has length and width. The area of a 2-D object = length x width. The width of any 1-D object = 0, so its area = 0, even if its length = infinity.

This works in the same manner with 3-Dimensional space. The volume of a 3-D object = length x width x height. Since a 2-D object's height = 0, it doesn't matter if its length or width = infinity. Its volume, and mass, will still = 0.

"Hypervolume"/the 4-Dimensional volume analogue = length x weight x height x a fourth dimension. Since a 3-D object's fourth dimension = 0, its "hypervolume" and "hypermass" = 0

For a 5-Dimensional volume analogue = length x width x height x a fourth dimension x a fifth dimension. Since a 4-D object's fifth dimension = 0, its 5-D volume analogue, and 5-D mass analogue = 0

Basically, what this means is that, just like an entirely flat, two-dimensional square has a more than countably infinite number of times less volume (and mass) than a three-dimensional cube, the cube also has a more than countably infinite number of times less volume (and mass) than a four-dimensional tesseract, which has a more than countably infinite number of times less volume (and mass) than a five-dimensional hypercube, and so onwards.

Here is a story book that was among the first to take up the subject of different dimensions. Here is an episode of Futurama that does much the same thing.

Here is another Umineko explanation for how objects that appear infinite in lower-dimensional space, relate to ones in a higher dimension.

Users

 * Mister Mxyzptlk, Bat-Mite, Ultimator, Thought Robot, The Primal Monitor, and Mandrakk from DC Comics
 * The Beyonder, The Living Tribunal, The Infinites, and The Beyonders from Marvel Comics
 * Youkai Sages, Bakus, and other powerful entities from Touhou Project
 * Higher layer beings from Umineko and Higurashi
 * The Choushin, The Counter-Actor, The Ds, and Kami Tenchi from Tenchi Muyo!
 * Gan and the Crimson King from The Dark Tower
 * The Lord of Nightmares from Slayers
 * Bill Cipher from Gravity Falls
 * The Godhead and the Amaranths from The Elder Scrolls
 * The Chaos Gods from Warhammer 40,000
 * Outer Gods such as Yog Sothoth and the Nameless Mist from the Cthulhu Mythos
 * Elder God Demonbane, Nyarlathotep, and Liber Legis from Demonbane
 * Marduk, Ashur, and others from I/O
 * The Alternity and Hytherion from Transformers
 * The Six-Fold God from Doctor Who
 * Belldandy from Ah! My Goddess

(Credit to Versus Battles Wiki)