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Cosmogony Part 1: Dimension

By Revold April 29, 2020 1 Comments

Definition

Dimensions (of a mathematical space) are the parameters needed to define a point within the mathematical space.

For example, in a graph where the x-axis exist, all points on it have an x-coordinate, and the resulting space (finite or infinite) is said to have an x-dimension.

An axis is an imaginary reference line (i.e. 1-dimensional). An axis has one dimension, but it is not a dimension itself.
A coordinate is one of the numbers that specify the 0-dimensional point. It is not a dimension either since it's just a value of the parameter, not the parameter itself.

The "space" here doesn't necessary mean "spatial dimensions", but just a generic meaning of being occupiable or empty.

For simplicity, anything that can have dimensions (e.g. empty space, solid objects or even abstract constructs) will collectively be called figures.

Measurements
In daily lives, the more common use of the word dimension is about the size of an object. The dimensions of this box are length, width and height. ✓ The dimensions of this box are 3 cm by 3 cm by 3 cm. Suppose you are asked to "specify the dimensions of the box". By common sense, most will give the first answer. This is the value of dimension, contrasting with the name of dimension in the second answer. If we discard common sense and speak strictly, the supposed answer would be the name if the question does not specify name or value. In shorter terms, the value of dimension is also called measurement, comprised of a magnitude (i.e. 3) and a unit (i.e. cm). The string's length is 3 cm. The string's length has a value of 3 cm. ✓ To illustrate this, here's an more obvious example. As you can see, the more accurate description will be statement 2. However for brevity, we often just get straight to the point in common language.

Measurements

In daily lives, the more common use of the word dimension is about the size of an object.

The dimensions of this box are length, width and height. ✓

The dimensions of this box are 3 cm by 3 cm by 3 cm.

Suppose you are asked to "specify the dimensions of the box". By common sense, most will give the first answer. This is the value of dimension, contrasting with the name of dimension in the second answer. If we discard common sense and speak strictly, the supposed answer would be the name if the question does not specify name or value. In shorter terms, the value of dimension is also called measurement, comprised of a magnitude (i.e. 3) and a unit (i.e. cm).

The string's length is 3 cm.

The string's length has a value of 3 cm. ✓

To illustrate this, here's an more obvious example. As you can see, the more accurate description will be statement 2. However for brevity, we often just get straight to the point in common language.

0-Dimensional "Point"

A point is the 0th dimension. ✖

A point is 0-dimensional. ✓

A point has 0 dimensions. ✓

First off, I want to note a mistake that almost everyone make: there is no such thing as the 0th dimension. There are, however, 0-dimensional figures that have no dimensions.

"0th dimension" and "0-dimensional" illustrates the clear distinction between dimension and dimensionality.

Dimensionality
Roughly speaking, it is the number of coordinates needed to specify a point on the object... The dimension of an object is sometimes also called its "dimensionality." [Wolfram Mathworld] This box is 3 dimensions. ✖ This box has a dimension of 3. ✖ This box has a dimensionality of 3. ✓ This box is 3-dimensional. ✓ This box has 3 dimensions. ✓ Not to be confused with dimension, dimensionality is a property of figures, the number of dimensions a figure has. Most people's first intuition of dimensions are along the lines of length, width and depth. Do not limit your understanding to just that, or even to just x, y, z axes. There are other coordinate systems (i.e. ways to define a point) such as spherical polar coordinates or cylindrical coordinates, which does not involve linear axes but angular ones. However, whichever system you choose, all of them requires 3 independent parameters for a 3-dimensional space, even though the dimensions can be completely different. This illustrates the fact that dimensions are arbitary, but the dimensionality isn't.

Dimensionality

Roughly speaking, it is the number of coordinates needed to specify a point on the object... The dimension of an object is sometimes also called its "dimensionality." [Wolfram Mathworld]

This box is 3 dimensions. ✖

This box has a dimension of 3. ✖

This box has a dimensionality of 3. ✓

This box is 3-dimensional. ✓

This box has 3 dimensions. ✓

Not to be confused with dimension, dimensionality is a property of figures, the number of dimensions a figure has.

Most people's first intuition of dimensions are along the lines of length, width and depth. Do not limit your understanding to just that, or even to just x, y, z axes. There are other coordinate systems (i.e. ways to define a point) such as spherical polar coordinates or cylindrical coordinates, which does not involve linear axes but angular ones.

However, whichever system you choose, all of them requires 3 independent parameters for a 3-dimensional space, even though the dimensions can be completely different. This illustrates the fact that dimensions are arbitary, but the dimensionality isn't.

The reason why people always confuse dimension with dimensionality is because that is in fact the definition of dimension in mathematics. This is because the intuitive sense of "dimension" doesn't exist once we dive deeper, so we might as well just define use it in place of the word dimensionality since it's shorter.

But here in fiction, we don't care about any strict sense of the word any more than human intuition. Therefore the distinction between dimension and dimensionality is very important.

1-Dimensional "Line"

Lines are one-dimensional figures that are straight and infinitely long.

Line Segments are closed intervals corresponding to a finite portion of an infinite line.

Lines are one-dimensional figures. ✓

Straight Lines are one-dimensional figures that are straight. ✓

Infinite Straight Lines are one-dimensional figures that are straight and infinitely long. ✓

In geometry, lines are straight and infinitely long by definition, and its finite counterpart instead called line segment. However in common language, lines are simply any one-dimensional figures, and can be curved or finitely long.

Curves are one-dimensional figures.

Curves are curved one-dimensional figures. ✓

Since lines are already defined as a specific type of one-dimensional figure in geometry, one-dimensional figures in general are defined as curves. However, this is counter-intuitive as even straight lines, which by definition are not curved, are considered as curves.

Curvature
Curved lines are 2-dimensional because it requires an extra dimension to describe the line. ✖ Curved lines are still 1-dimensional. We only need one axis along the line to map to every point on it. What the above statement is describing is not the line itself, but the space in which the line exist in. It has to be at least 2-dimensional, but if the line spans across 3-dimensions or more, it can honestly be any number of dimensions.

Curvature

Curved lines are 2-dimensional because it requires an extra dimension to describe the line. ✖

Curved lines are still 1-dimensional. We only need one axis along the line to map to every point on it.

What the above statement is describing is not the line itself, but the space in which the line exist in. It has to be at least 2-dimensional, but if the line spans across 3-dimensions or more, it can honestly be any number of dimensions.

Homogeneity of Dimensions
Lines have only one dimension: length. It has no width and no height. ✖ Lines have only one dimension. ✓ While lines are defined as one-dimensional, it can be any dimension or type of dimension (need not be space). It can be length, width, height, depth, or even time! A line with the dimension of time is called a Timeline, or World Line when the entire 3-dimensional world is imagined to be a 0th-dimensional point. The 4th dimension is time. ✖ There are 3 dimensions of space and 1 dimension of time. ✓ This also illustrates the fact that there are real order for any number of dimensions. For example, time can just as well be the 1st dimension or the 3rd dimension.

Homogeneity of Dimensions

Lines have only one dimension: length. It has no width and no height. ✖

Lines have only one dimension. ✓

While lines are defined as one-dimensional, it can be any dimension or type of dimension (need not be space). It can be length, width, height, depth, or even time! A line with the dimension of time is called a Timeline, or World Line when the entire 3-dimensional world is imagined to be a 0th-dimensional point.

The 4th dimension is time. ✖

There are 3 dimensions of space and 1 dimension of time. ✓

This also illustrates the fact that there are real order for any number of dimensions. For example, time can just as well be the 1st dimension or the 3rd dimension.

Distance
Unlike in physics class, the difference between length and distance is quite apparent, as length is a dimension, and distance is not. Distance is defined as the separation between space. While it can be measured with any spatial dimension, length is most commonly used. Duration is the separation of time. Time not as in the dimension, but the dimension type in general. Interval is the separation between spacetime or any type of dimensions in general.

Distance

Unlike in physics class, the difference between length and distance is quite apparent, as length is a dimension, and distance is not.

Distance is defined as the separation between space. While it can be measured with any spatial dimension, length is most commonly used.

Duration is the separation of time. Time not as in the dimension, but the dimension type in general.

Interval is the separation between spacetime or any type of dimensions in general.

2-Dimensional "Plane"

Planes are flat 2-dimensional figures that is infinitely large.

Hyperbolic planes are surfaces in which the space curves away from itself at every point.

Planes are 2-dimensional figures. ✓

Similar to the definition of line, the definition of plane in geometry requires it to be flat and infinitely large. This is not only counter-intuitive in this case but also self-inconsistent.

Surfaces are

Multi-Dimensionality and Co-Dimensionality
The Universe is a multi-dimension. ✖ The Universe is multi-dimensional. ✓ A structure is multi-dimensional if it has 2 or more dimensions. A property of some figures. Codimension is a term used in a number of algebraic and geometric contexts to indicate the difference between the dimension of certain objects and the dimension of a smaller object contained in it. Codimension is a relative concept: it is only defined for one object inside another. Like dimension and dimensionality, we also distinguish codimension and codimensionality. For example with respect to a 4-dimensional figure of dimensions (x, y, z, w), the 3-dimensional figure of dimensions (x, y, z) has a codimensionality of 1, and a codimension w.

Multi-Dimensionality and Co-Dimensionality

The Universe is a multi-dimension. ✖

The Universe is multi-dimensional. ✓

A structure is multi-dimensional if it has 2 or more dimensions. A property of some figures.

Codimension is a term used in a number of algebraic and geometric contexts to indicate the difference between the dimension of certain objects and the dimension of a smaller object contained in it.

Codimension is a relative concept: it is only defined for one object inside another. Like dimension and dimensionality, we also distinguish codimension and codimensionality. For example with respect to a 4-dimensional figure of dimensions (x, y, z, w), the 3-dimensional figure of dimensions (x, y, z) has a codimensionality of 1, and a codimension w.

3-Dimensional "Solid"

There are two ways to name a 3-dimensional figures: solid or space. However, neither of these are really good names.

Solid implies that the structure must be firm with a fixed shape as opposed to fluids. However, fluids are also 3-dimensional.
Space creates even more confusion:
- It implies emptiness. For example, everyone knows a square is a type of plane, but nobody calls a cube "a type of space" because cubes are objects that exists in space, but not space itself.
- It can also imply that it must be made up of spatial dimensions. For example, 3 dimensions of time is still a mathematical space even though it has no spatial dimension.

Considering both, I would still prefer naming 3-dimensional structures solids. This is because space is a very ambiguous word which people can actually use for any number of dimensions.

Emptiness
In 3 dimensions more so than any other, there is emphasis on the emptiness of the 3-dimensional figure. For instance, most consider both a sphere and a ball to be 3-dimensional, but with one difference. A sphere is empty, making it intuitively more akin to space. Meanwhile, a ball is the solid enclosed by a sphere, making it more akin to solid. Ideally, there should be a word to generalize both space and solid into one term, similar to how I generalized (mathematical) space and objects into one term called figure at the start of this entire post. Unfortunately, there is no such word in the language, which is why we have two competing terms in the first place.

Emptiness

In 3 dimensions more so than any other, there is emphasis on the emptiness of the 3-dimensional figure.

For instance, most consider both a sphere and a ball to be 3-dimensional, but with one difference. A sphere is empty, making it intuitively more akin to space. Meanwhile, a ball is the solid enclosed by a sphere, making it more akin to solid.

Ideally, there should be a word to generalize both space and solid into one term, similar to how I generalized (mathematical) space and objects into one term called figure at the start of this entire post. Unfortunately, there is no such word in the language, which is why we have two competing terms in the first place.

Space
In common language, the word space is often just a short for more specific phrases. Short for Empty Space. Most common usage, for example "to make space" means to de-occupy an region Short for Outer Space. Refers to the region beyond the atmosphere. A type of dimension in physics. For example, "across space and time". Short for 3-space. For example, "point, line, plane, followed by space". Refers to 3-dimensional figures in general. The first two points are usually apparent when used, the confusion usually comes with the third, fourth point, and the true definition of space. A name for 3-dimensional figures in general. The 3-dimensional equivalent to point, line and surface. Short for 3-space. A type of dimension. Contrast with the dimension of time. For example, a figure with both spatial dimension and temporal dimension is called spacetime. An empty figure of any number (and any type) of dimension. I name them with separate words to prevent unnecessary confusion: solid (1), p-space (2; stands for physical space) and m-space (3; stands for mathematical space).

Space

In common language, the word space is often just a short for more specific phrases.

Short for Empty Space. Most common usage, for example "to make space" means to de-occupy an region
Short for Outer Space. Refers to the region beyond the atmosphere.
A type of dimension in physics. For example, "across space and time".
Short for 3-space. For example, "point, line, plane, followed by space". Refers to 3-dimensional figures in general.

The first two points are usually apparent when used, the confusion usually comes with the third, fourth point, and the true definition of space.

A name for 3-dimensional figures in general. The 3-dimensional equivalent to point, line and surface. Short for 3-space.
A type of dimension. Contrast with the dimension of time. For example, a figure with both spatial dimension and temporal dimension is called spacetime.
An empty figure of any number (and any type) of dimension.

I name them with separate words to prevent unnecessary confusion: solid (1), p-space (2; stands for physical space) and m-space (3; stands for mathematical space).

4-Dimensional "???"

4-dimensional figures are called Spacetime. ✖

Spacetime is a 4-dimensional figure. ✓

4-dimensional Spacetime is probably the most well-known figure for anything 4-dimensional. However, that is only one specific type of 4-dimensional figure, with 3 dimensions of space and 1 dimension of time.

Unlike dimensions 1 to 3, there is no general term for 4-dimensional figures, other than simply "4-space". Although there is no general term for 4-space, 4-cube in particular is known as tesseract, and 5-cube is known as a penteract.

Time
Like space, time has more than one definition/usage: A dimension, similar to "length", "width" and "height". A type of dimension, similar to "space". For example, dimensions of time are called temporal dimensions, just like how dimensions of space are called spatial dimensions. A direction, similar to "up", "down", "left", "right", "forward", "backward". Unlike the space counterparts, time is a word for both the type of dimension and the dimension itself. For example, length is a dimension of space, while time is a dimension of ... well, time. This lack of distinction between definition 1 and definition 2 is never properly addressed as even the most cutting-edge theories in physics mostly involves only one dimension of time. In the rare models with two (or more) dimensions of time, there are mostly differentiated with words like: "real time" and "imaginary time" "time", "time squared", "time cubed" etc. Apart from that, there are also efforts to distinguish definition 1 from definition 3. For instance, one could call "forward in time" as time, and "backward in time" as anti-time. While time becomes a direction, the name of the dimension is instead duration. However, that does not align with its definition. Duration is the separation of time (Definition 2). It applies to any temporal dimensions in general, independent of any dimension in specific.

Time

Like space, time has more than one definition/usage:

A dimension, similar to "length", "width" and "height".
A type of dimension, similar to "space". For example, dimensions of time are called temporal dimensions, just like how dimensions of space are called spatial dimensions.
A direction, similar to "up", "down", "left", "right", "forward", "backward".

Unlike the space counterparts, time is a word for both the type of dimension and the dimension itself. For example, length is a dimension of space, while time is a dimension of ... well, time. This lack of distinction between definition 1 and definition 2 is never properly addressed as even the most cutting-edge theories in physics mostly involves only one dimension of time. In the rare models with two (or more) dimensions of time, there are mostly differentiated with words like:

"real time" and "imaginary time"
"time", "time squared", "time cubed" etc.

Apart from that, there are also efforts to distinguish definition 1 from definition 3. For instance, one could call "forward in time" as time, and "backward in time" as anti-time. While time becomes a direction, the name of the dimension is instead duration.

However, that does not align with its definition. Duration is the separation of time (Definition 2). It applies to any temporal dimensions in general, independent of any dimension in specific.

5-Dimensional "???"

While it is debatable if the Universe is 5-dimensional, many multiverse theories, especially in fiction, rely on a 5th dimension to create a range of infinite universes.

Extra-Dimensionality
In physics, extra dimensions are the proposed additional dimensions beyond the 3 dimensions of space and 1 dimension of time. Extra-dimensional figures can be further classified into Hyperspace, Hypertime and others. Hyperspace is an extra-dimensional space, in other words a figure with more than 3 dimensions of space. Hypertime are figures with more than one dimension of time. Have dimensions that are neither spatial or temporal. Even if the figure has less spatial and/or temporal dimensions, it is still considered to be extra-dimensional. Note that there are no such things as "higher" dimensions, only extra dimensions, due to the Homogenity of Dimensions (no order). The fifth dimension might as well be the first dimension, and there is no difference. A hyperplane is a subspace of one dimension less than the dimension of the hyperspace.

Extra-Dimensionality

In physics, extra dimensions are the proposed additional dimensions beyond the 3 dimensions of space and 1 dimension of time. Extra-dimensional figures can be further classified into Hyperspace, Hypertime and others.

Hyperspace is an extra-dimensional space, in other words a figure with more than 3 dimensions of space.
Hypertime are figures with more than one dimension of time.
Have dimensions that are neither spatial or temporal. Even if the figure has less spatial and/or temporal dimensions, it is still considered to be extra-dimensional.

Note that there are no such things as "higher" dimensions, only extra dimensions, due to the Homogenity of Dimensions (no order). The fifth dimension might as well be the first dimension, and there is no difference.

A hyperplane is a subspace of one dimension less than the dimension of the hyperspace.

∞-Dimensional "???"

In mathematics, it is not uncommon to have a infinite-dimensional or transfinite-dimensional space, such as Hilbert Space.

Alternate Dimensions
The popular usage of this term is incorrect; "other dimensions" are not locations, but the means by which you get to them... Free movement in the fifth dimension, or "time squared", is usually seen as jumping sideways from branch to branch within the tree of choices and alternate events that make up the multiverse- so called alternate universes. [TV Tropes]

Alternate Dimensions

The popular usage of this term is incorrect; "other dimensions" are not locations, but the means by which you get to them... Free movement in the fifth dimension, or "time squared", is usually seen as jumping sideways from branch to branch within the tree of choices and alternate events that make up the multiverse- so called alternate universes. [TV Tropes]

-1-Dimensional "???"

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