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. And here is a simple explanation of how even a 4-dimensional character relates to our 3-dimensional world.

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
 * 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 Azathoth from the Cthulhu Mythos
 * Elder God Demonbane, Nyarlathotep, and Liber Legis from Demonbane
 * Marduk, Ashur, Divine Presence and others from I/O
 * The Alternity and Hytherion from Transformers
 * The Six-Fold God from Doctor Who
 * Belldandy from Ah! My Goddess