Dimensions

Information
It is evident that not a lot of people understand spatial and temporal dimensionality. Thus, this page is also created in order for you to be able to clarify doubts and strengthen your knowledge on this certain topic. Keep in mind that although a character has a higher dimensional tiering, they are not necessarily stronger. You still have to prove their strength via feats. Reason for this is that although a spatial dimension increase mass-energy, temporal dimensions do not, and even if we know for a fact the ones added are all spatial dimensions, hypermass and 4D and above spatial constructs are literally impossible to calculate. As such, We use feats we can see and measure to scale their AP. However, higher dimensional being can interact with the structures of the same level of dimensionality or lower but not vice versa. Hence if a character of a lower dimensionality that cannot interact with a higher dimensional being has higher AP/DC fought a higher dimensional being with paper AP/DC; it would likely end as a stalemate as neither side could damage one another.

What is a dimension?

 * https://mathworld.wolfram.com/Dimension.html
 * The dimension of an object is a topological measure of the size of its covering properties. Roughly speaking, it is the number of coordinates needed to specify a point on the object. For example, a rectangle is two-dimensional, while a cube is three-dimensional. The dimension of an object is sometimes also called its "dimensionality."Dimensions.gif


 * https://arxiv.org/pdf/gr-qc/9702052.pdf
 * This paper is written by Max Tegmark, a Professor and Researcher with a PhD in Physics. In said paper, the following is stated.  "When the partial differential equations of nature are elliptic or ultrahyperbolic, physics has no predictive power for an observer. In the remaining (hyperbolic) cases, n > 3 may fail on the stability requirement (atoms are unstable) and n < 3 may fail on the complexity requirement (no gravitational attraction, topological problems)." (p. 1)  Most of this is simply arguing that spacetimes with fewer dimensions than ours wouldn't be fall short in the complexity needed to give life like us and that spacetimes with more dimensions than ours are too complicated and ruin stability.  However, what's important here are two acknowledgements. Firstly, that there are two separate kinds of dimensions. Spatial and temporal. Spatial are ones that involve, as you may have guessed, space. The more spatial dimensions a construct is made of, the larger they physically are. As you and I are 3D spatial beings, we have length, width and depth. Temporal dimensions are those that involve changes in constructs. As our universe only has one temporal dimension, our time is a timeline, moving in only one direction. The future. The second important acknowledgement is the possible existence of spacetimes with fewer and higher spatiotemporal dimensions than our 4D one. As such, there could be spacetimes with 2 dimensions of space and 6 dimensions of time, an 8D spacetime, or even a spacetime of 1 dimension of space and 1 dimension of time, a 2D spacetime.

First Dimension

 * https://en.wikipedia.org/wiki/Length
 * The 1st Dimension is the X-Axis. You can only move in 2 directions. Left and Right. Being 1D makes you have length and nothing more. You’re not even a line. Because at least a line has SOME width. You're even less than a line.

Second Dimension

 * https://en.wikipedia.org/wiki/Height
 * The 2nd Dimension is height. With this, you have both length and height and can be much more than what a 1D being is. You can also move in both the X and Y axes simultaneously, meaning up, down, left, right, and diagonally.

Third Dimension

 * https://en.wikipedia.org/wiki/Three-dimensional_space
 * Something with depth alongside height and length gives us a 3rd Dimensional being and/or object. Whatever is around you is 3rd Dimensional and even you yourself are a 3D being, who is able to move around in a 3rd Dimensional Space.


 * https://en.wikipedia.org/wiki/Three-dimensional_space#Coordinate_systems
 * "In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z" Thanks to being a 3D being, you can move in three axes simultaneously. X, Y and Z. Up, Down, Left, Right, Diagonal in any direction, Forwards and Backwards.

Fourth Dimension

 * https://en.wikipedia.org/wiki/Four-dimensional_space#Orthogonality_and_vocabulary
 * "In the familiar three-dimensional space in which we live there are three coordinate axes—usually labeled x, y, and z" We 3D beings can move on the X-Axis, Y-Axis and Z-Axis simultaneously and are therefore 3D. "Comparatively, four-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w."
 * We 3D beings can move on the X-Axis, Y-Axis and Z-Axis simultaneously and are therefore 3D.
 * "Comparatively, four-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w."


 * http://eusebeia.dyndns.org/4d/vis/01-intro
 * “right is positive X, left is negative X, forward is positive Y, backward is negative Y, and up is positive Z, and down is negative Z. We shall refer to these directions as +X, -X, +Y, -Y, +Z, and -Z”
 * “In a 4D world, there is another directional axis which is perpendicular to the X, Y, and Z axes. We shall label this axis W, and call the direction along this axis the fourth direction. This new axis also has positive and negative directions, which we shall refer to as +W and -W.”
 * What I should note is that this is for 4D space, not 4D spacetime. 4D spacetime is what our universe is given it's the combination of 3D space and 1D time. This means that spacetime uses the T axis instead of the W.


 * https://math.la.asu.edu/~surgent/mat267/examples/xyz_coordinate_system_2.pdf
 * If you don’t deem the two previous sources credible, then I will link this source, from Professor Scott Surgent, who was an author on Calculus and Its Applications. In the source, he says, “The XYZ coordinate axis system is denoted 𝑅 3, and is represented by three real number lines meeting at a common point, called the origin. The three number lines are called the x-axis, the y-axis, and the z-axis. Together, the three axes are called the coordinate axes.”


 * https://en.wikipedia.org/wiki/Spacetime
 * "In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum."


 * http://eusebeia.dyndns.org/4d/vis/01-intro
 * “Einstein's theory of Special Relativity postulates that space and time are interrelated, forming a space-time continuum of 3 spatial dimensions and 1 temporal dimension. While it is possible to visualize space-time simply by treating time as time and examining “snapshots” of space-time objects at various points in time, it is also useful to treat space-time geometrically.
 * For example, the distance between two events is the distance between two 4D points. The light-cone also has a particular shape that can only be adequately visualized as a 4D object.”
 * Spacetime is the combination of the X, Y, Z and T Axes. The three initial spatial dimensions and the fourth temporal dimension. It is the construct of our very own universe. Being able to interact with Spacetime or move in its direction would mean moving in those 4 axes simultaneously. We 3D beings cannot do such a thing as we're restricted to the 3rd Dimension. 4D beings have no restriction. Moving across the 4 axes simultaneously includes traveling through Spacetime to go to the past or future as well as stopping the flow of Spacetime.


 * Again, if you don’t agree with these sources, deeming them not good, let me get a source from John D. Norton, a professor that wrote a thesis on Spacetime.
 * https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/spacetime/index.html

“Take snapshots of the earth orbiting the sun in the three dimensional space around the sun in the course of a year, which will look like:



Now, we stack them up into a third dimension:



When we clean things up a little, we have a spacetime:


 * So far we have described how a two dimensional space is combined with the one extra dimension of time to generate a three dimensional spacetime, such as shown above in the figures. Our space is three dimensional. So when we add the extra dimension of time we generate a four dimensional spacetime.”


 * http://www.sci.osaka-cu.ac.jp/~kawauchi/UniverseTopology.pdf
 * This 2019 paper by Kawauchi Akio, a Mathematician and winner of the Geometry Prize, a prestigious award in Japan for significant or long-term research in the field of Geometry, says the following.
 * “Every 4D universe and every 4D punctured universe are open 4-dimensional manifolds.”
 * Simply put, the universe is a four dimensional spacetime continuum, i.e. a fourth dimensional manifold with a third dimensional manifold inside of it, namely Euclidean space.


 * https://mathworld.wolfram.com/Manifold.html&#x20;&#x22;
 * The basic example of a manifold is Euclidean space, and many of its properties carry over to manifolds. In addition, any smooth boundary of a subset of Euclidean space, like the circle or the sphere, is a manifold. Manifolds are therefore of interest in the study of geometry, topology, and analysis."
 * Note how Kawauchi called it an open 4-dimensional manifold.
 * This brings us to the shape of the universe and subsequently, this NASA article.


 * https://map.gsfc.nasa.gov/universe/uni_shape.html


 * “The WMAP spacecraft can measure the basic parameters of the Big Bang theory including the geometry of the universe...We now know (as of 2013) that the universe is flat with only a 0.4% margin of error. This suggests that the Universe is infinite in extent; however, since the Universe has a finite age, we can only observe a finite volume of the Universe. All we can truly conclude is that the Universe is much larger than the volume we can directly observe.”
 * Thanks to the WMAP spacecraft, we know with a 0.4% margin of error that the universe is a flat one and that it has been expanding infinitely so for billions and billions of years given that


 * https://www.scientificamerican.com/article/how-old-is-the-universe/
 * According to this article written by astrophysicist Sabrina Stierwalt, as she says,
 * “Our universe is 13.8 billion years old.” Which brings us to the Grapefruit argument.


 * You need to be 4D to be universal but you don't need to be universal to be 4D. Similar to how all Squares are Rectangles but not all Rectangles are Squares.


 * https://www.abc.net.au/science/articles/2013/08/06/3818354.htm
 * “Very soon after the Big Bang was the 'inflationary epoch'...It began about one millionth of one millionth of one millionth of one millionth of one millionth of one millionth of a second after the Big Bang...The universe expanded from much much smaller than a proton to about the size of a grapefruit...The next notable stage was 3 to 20 minutes after the Big Bang.”
 * The universe went from <<< the size of a proton to the size of a Grapefruit around the moment of its inception, more specifically, three minutes. Therefore, it’s possible that a 4D construct is smaller than the 93 billion light year wide universe we know today. Subsequently, suggesting that you can destroy a universe because you can destroy a grapefruit-sized 4D construct is analogical to suggesting that because we can crush an apple, a 3D construct, we can crush a star or a galaxy, both also being 3D constructs.


 * Our universe is 93 billion light years in diameter and subsequently contains 4x10^69 J of Mass Energy. https://web.archive.org/web/20140819120709/http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/980211b.html
 * This Mass-Energy is calculated using the formula E = mc2. As I established before, 4D spacetime continuums can be different in size. And so, I ask, do you think that our spacetime continuum when it was, let’s say, 20 million light years in diameter, has an equivalent amount of mass-energy to our current 93 billion light year-sized one?


 * https://www.nationalgeographic.com/science/space/dark-matter/
 * Saying that it is is simply wrong as National Geographic states,
 * “Scientists now think that the accelerated expansion of the universe is driven by a kind of repulsive force generated by quantum fluctuations in otherwise "empty" space. What's more, the force seems to be growing stronger as the universe expands.”


 * http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/dareng.html
 * And according to An Introduction to Modern Astrophysics, a book written by Dale Ostlie and Bradley Carroll, both physicists,
 * “Yet extended uniformly throughout the entire universe, this dark energy becomes the dominant influence on the expansion of the universe in this era.”
 * As the universe grew from 20 million light years to 93 billion light years in diameter, the dark energy grew stronger. As dark energy is extended uniformly throughout the entire universe, there’s no reason to believe that those two sizes have equivalent mass-energy values. So why would destroying any construct be Universal?
 * While one could make the argument that via the law of conservation of energy makes it so an expanding universe would have an equivalent amount of energy whether it’s expanded or not, that’s actually not true.


 * https://www.scientificamerican.com/article/energy-can-neither-be-created-nor-destroyed/
 * As this Scientific American article states, “As space expands, it releases stored up gravitational potential energy, which converts into the intrinsic energy that fills the newly created volume.” That energy is what we call Dark Energy.


 * https://science.nasa.gov/astrophysics/focus-areas/what-is-dark-energy
 * “The first property that Einstein discovered is that it is possible for more space to come into existence. Then one version of Einstein's gravity theory, the version that contains a cosmological constant, makes a second prediction: "empty space" can possess its own energy...As more space comes into existence, more of this energy-of-space would appear. As a result, this form of energy would cause the universe to expand faster and faster.”
 * Note. Having complete control of a timeline is a universal level ability.


 * https://einstein.stanford.edu/content/relativity/q411.html
 * "It is believed to be a 'continuum' because so far as we know, there are no missing points in space or instants in time, and both can be subdivided without any apparent limit in size or duration. So, physicists now routinely consider our world to be embedded in this 4-dimensional Space-Time continuum, and all events, places, moments in history, actions and so on are described in terms of their location in Space-Time."
 * As we have determined to be no missing points in spacetime, we call it a continuum and consider it impossible for it to happen.
 * Stopping time on a planetary or anything less than universal level would create a massive hole in spacetime as what should be there at that point in time and space would not be there due to it having been paused for x amount of time.
 * Reversing or accelerating time does the same thing. It pushes back the direction in which the continuum continues in. Doing so on anything less than universal would separate the section of spacetime from the rest of the continuum. Something that has never been observed nor confirmed to be possible. Hence the use of the term continuum.


 * https://web.mit.edu/bskow/www/research/temporality.pdf
 * This is a paper written by Bradford Skow, PhD, in which the differences between space and time are talked about.
 * "In four-dimensional Minkowski geometry, whether time like vectors have positive or negative lengths, time is one-dimensional and space is three-dimensional. Perhaps it is because it is one-dimensional that time is time. Before examining this thesis I’ll say something about what it means to say that time is one-dimensional. Intuitively speaking, to say that time is one-dimensional is to say that we can represent time as a line, and that all events that occur in time can be assigned a position on that line...Each three-dimensional submanifold is a time, and the sequence gives their temporal ordering. Since every event occurs somewhere in spacetime, every event occurs somewhere in this one-dimensional sequence. So time is one dimensional in the intuitive sense in these spacetimes because it divides up in this way" (9-10)
 * Time/the fourth, temporal dimension is one-dimensional as explained here.
 * "There is another role that time like directions play in some familiar laws that spacelike directions do not: quantities like mass, charge, and energy are conserved in time like directions, but not in spacelike directions." (15)
 * It's stated here that temporal dimensions do not change the mass, charge and energy of constructs within spacetime continuums. In other words, adding temporal dimensions doesn't change your mass-energy in any way.

Fifth Dimension

 * https://bethbeck.wordpress.com/2012/09/06/5th-dimension-imagination-space/
 * In Physics, 5D is outside of spacetime. Via Hausdorff Dimensions, 5D beings are infinitely beyond and independent to any Spacetime continuum and the laws it has. Because they are beyond the laws of Spacetime, they are above the concepts of Distance and Time, making the equation,


 * http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
 * Velocity = Distance/Time, obsolete to them as they are infinitely above Distance and Time making their speed infinite.

Extra Notes

 * Hypermass

Hypermass is the hypothetical trait a construct with higher spatial dimensions would have.

1D constructs have Length.

2D constructs have Length + Width.

3D constructs have Length + Width + Depth.

4D space constructs have Length + Width + Depth + Something else.

Mass relates to volume. For example, let's take a 3D cube. The formula for volume of a cube is Length x Width x Depth. So a 10 m cube would have a volume of... 10 × 10 × 10 = 1 000 cubic metres or $$m^3$$ Let's take the density of rock to keep things simple.

https://books.google.ca/books?id=s0BwEETyfo0C&pg=PA74&lpg=PA74&dq=granite+mass+2700+kg/m3&source=bl&ots=k5oYbDn5GC&sig=4ocxfuGLndXkrntbRE2ERTrpBGk&hl=en&sa=X&ved=0ahUKEwii2tme8eTUAhXEzz4KHaneDx4Q6AEIMDAC#v=onepage&q=granite%20mass%202700%20kg%2Fm3&f=false

Density of Granite/Earth = 2 700 kg/m3 1 000 × 2 700 = a mass of 2 700 000 kg While 2D constructs have area and 3D constructs have volume like the cube above, 4D space constructs would have a higher tier measurement and this would result in a higher tier of mass, Hypermass, and as mass is used in both Potential and Kinetic Energies, this subsequently results in a higher tier of energy.

Lastly, always keep in mind that this hypermass is only viable with spatial dimensions and not temporal. Therefore, you still have to base their AP/DC via their feats.