4D

rotating around the up-forward plane (or in the kata-right plane) ]]
 rotating on the 4D axis.
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**4D**, meaning the **common 4 dimensions**, is a theoretical concept in mathematics. It has been studied by mathematicians and philosophers since the 18th century.  Mathematicians who studied four-dimension space in the 19th century include Möbius, Schläfi, Bernhard Riemann, and Charles Howard Hinton.

In geometry, the fourth dimension is related to the other three dimensions  of length, width, and depth by imagining another direction through space. Just as the dimension of depth can be added to a square to create a cube, a fourth dimension (ana/kata) can be added to a cube to create a tesseract.

By analogy, the universe of a 2D (two dimensional) being is a plane, but this plane can be said to be in a 3d space. This means there’s an extra direction that someone in the 2d world cannot move through or see. One dimension higher, a 3d space could be in a 4d world, and that 4d world would have a direction that cannot be seen or touched. Like with the 2d case, this is because the new direction is not anywhere in the 3d creature’s 3d space. 

If we try to show a 3d cube spinning around to a 2d world’s creatures, they would see a distorting set of lines due to perspective, and if the cube was face-on, they’d see a square in a square, due to perspective. The same thing can happen with 4d objects in 3d space: the smaller cube is the furthest 4d face (the furthest cell), the squares between them are like 4d cube edges (ridges), and so on.

Going into theoretical physics, if we look with high energies, we may be able to see higher dimensions that are looped. The LHC and other particle colliders have tried to do that, but haven’t found any extra dimensions. However, because the loops are so small, these higher dimensions would not be noticeable, like how a paperclip has a 2d surface but is a lot like a 1d line.

4d is also an important idea in physics, developed in the 20th century. In physics, it refers to the idea of time as a fourth dimension, added to the (3D) spatial dimensions. Albert Einstein developed the idea of spacetime by connecting space and time together. The difference is that spacetime is not a Euclidean space, but instead is called "Minkowski spacetime", because distances with time are different, which also means rotations are different.

4d also has a new way to turn things, named double rotations. A double rotation is a way to turn all parts of a 4d shape except a single point. Another completely new thing in 4d related to double rotations are isoclinic rotations, which can turn a 3-sphere like turning a 2d circle. Everywhere on a 3-sphere would move at the same speed, like with the circle. These rotations come in two types (left-handed and right-handed), and cannot be flipped in 4d by looking up-side down (although in 5d and higher they can be flipped), just as a clockwise rotation in 2d cannot be rotated to look counterclockwise without going into 3d.

In 4d, a single rotation happens around a plane. Rotations happen in planes in all dimensions: in 2d, they happen in the whole 2d plane, in 3d, they happen in the plane perpendicular to the axis, and this happens in 4d, but since a plane is 2d, there are now 2 dimensions left for rotation around. This is why rotations can also be said to happen around planes, and why double rotations exist (an extra rotational plane can be added where the first rotation’s “axis” is).

## Other websites
*The Cube shown in Five Dimensions