Understanding Time Squared: Exploring Acceleration

[marioisaac]

Hello everyone,

What does mathematically a second squared means?

Let us consider we have a Car that is traveling through space from point A to point B.

At point A it has 0 velocity; and at point B it reaches a velocity of 1m/s.
The acceleration therefore is 1m/s squared.

For example:

Let us have 2 points in space: A and B, and they are 1 meter away from each other.
A---------B ; if I square this distance in space I get the following:

D---------C
! !
! !
A---------B

With this operation I have added a new dimension to space, and it is easy to understand.

Now let us have 2 points in the time domain: A and B, and they are 1 second away from each other.

A---------B ; if I square this distance in time I get the following:

D---------C
! !
! !
A---------B

Now I get in trouble; if I think of time as flowing from A to B, what does time squared actually mean?

Notice that I understand the concept of Acceleration as the rate of change of velocity:

Acceleration = Velocity (meters per second) per Time (second); but mathematically I don't get to understand Time (s) squared;

I appreciate your answers.

Thank you,

 

[symbolipoint]

Numbers in use to count or measure must have the units for the quantity being counted or measured. A time rate will have, usually, a time quantity in its denominator. If this rate is itself a rate with time again as the unit for the denominator, then this makes the simplified ratio of rates have time multiplied by time. This is how a change in travel rate compared to a change in time passage becomes acceleration. Dimensions are DISTANCE/TIME^2

 

[jack action]

Well it is my opinion that it physically means nothing more than $\frac{m/s}{s}$. But if you ever get another answer, maybe someone could try to tell me what a temperature to the 4th power means, like in the Stefan-Boltzmann constant (= 5.670367(13)×10⁻⁸ W/m²/K⁴)?

 

[symbolipoint]

Numbers in use to count or measure must have the units for the quantity being counted or measured. A time rate will have, usually, a time quantity in its denominator. If this rate is itself a rate with time again as the unit for the denominator, then this makes the simplified ratio of rates have time multiplied by time. This is how a change in travel rate compared to a change in time passage becomes acceleration. Dimensions are DISTANCE/TIME^2

Well it is my opinion that it physically means nothing more than $\frac{m/s}{s}$. But if you ever get another answer, maybe someone could try to tell me what a temperature to the 4th power means, like in the Stefan-Boltzmann constant (= 5.670367(13)×10⁻⁸ W/m²/K⁴)?

My discussion is not according to opinion. It was a discussion of numbers and the dimensions or units associated with them. Hopefully someone else can explain this better.

 

[Stephen Tashi]

Hello everyone,

What does mathematically a second squared means?

There is no "mathematical" definition for the interpretation of physical units.

When a mathematical equation is applied to specific physical problem people may find a specific interpretation for the units that are used in that equation.

If a unit like "meters/sec" appears in many physical applications with the same interpretation (velocity) then people are tempted to say that "meters/sec" must mean velocity. However, this is not the case.

For example, suppose you have a machine where the operator pushes the machine y meters. When the pushing stops, the machine runs for x seconds. (You can think of the effect of the pushing as winding up a spring or something like that.) If the behavior of the the machine is given by the equation y/x = 5 meters/sec then interpretation of "meters/sec" isn't "velocity". However the use of the unit "meters/sec" is still useful when you need to convert units. For example, if some one asks "How many miles would you need to push machine in order for it to run 3 minutes ?" then the usual conversion of units can be used to get the answer.

 

[marioisaac]

First of all I want to thank everyone (symbolipoint, Jack, and Stephen) for joining this conversation.

I have to say Stephen that your answer is quite enlightening, I understood what you meant, and it has invited me to think in new ways towards this subject. Still, my question remains and have not being answered.

I want to clarify that I am being "purely mathematical" in my inquiry, and if I use any physical expression or physical magnitude in my words, they should first be considered as metaphorical devices used to refer to the abstract "mathematical realm".

Let us simplify our example:

An horizontal line (let's call it x Axis) depicted in the human mental theatre, expresses the property of extension in one dimension. There is only one degree of possible movement in this hypothetical world.
if we add a vertical line (let's call it y Axis) in this world, it will express the property of extension in another dimension. Now we have 2 dimensions, and 2 possible degrees of movement in this world. We can also notice that a new property have arise in this world, and that is the angular relationship between the x and y axis.
if we add a depth line to this world (let's call it the z Axis), it will express the property of extension in a new dimension. Now we have 3 dimensions, and 3 possible degrees of movement in this world.
We could probably keep adding planes, thus creating zillion of dimensions but that would be hard to imagine, so for the moment let's confine ourselves to this 3 dimensions.

Notice that although there could be zillion of dimensions in this world, they would all still express the same quality or property (extension) but in different planes, thus allowing the movement of this quality (extension) in different directions.

Please take into consideration that this is an abstract world and in this case extension does not mean "Physical Length"; it is purely a mathematical magnitude, devoid of any physical reality.

Now, this abstract world seems to be composed of two properties;
1) Extension.
2) Angular relationship.

And this 2 properties differ from one another, in that one is dimensional (extension), and the other one is dimensionless (angular relationship).

Having said this, let us assign a physical magnitude to one of these lines through which extension is manifested. We will choose the x Axis, and we will assign it the physical unit that represents length, and it is known as meter (m).

As of now, our mathematical representation of the physical realm is constrained to one dimension along the x axis.
Then, if we square m, a new plane comes into play (let's choose the y axis), and a new dimension is thus added to our mathematical representation of the physical realm. We have 2 dimensions now. Notice that when we elevate anything to the power of 2, we refer to it as "squaring something". So in a geometrical sense it seems that whenever you square a magnitude, a new dimension of that magnitude arises.

We could have instead, elevated the meter to the power of 3, or cube the meter, and we would have ended with 3 dimensions of the physical realm, in our mathematical representation of it.

And here is my question, why if I square or cube time (s), new dimensions are not created? or are we actually creating this dimensions, and time is moving in different planes at the same time?

I would appreciate any hardcore physicist or mathematician, or anyone, to answer this questions with mathematical certainty.

 

[jack action]

I think everyone understood your multi-dimensions analysis, which is right in many situations. But there are situations where it doesn't apply. For example, $E = \frac{1}{2}mv^2$. The squared velocity does not represent two velocities, each in its own plane; It is exactly the same velocity - in one plane - multiplied by itself. Where does it come from? Well, the reality is that it's not really twice the velocity:
$$E = \int_{u= 0}^v mudu$$
So it is the summation of the multiplications of the velocity by the infinitesimal change in velocity at any instant between $u = 0$ and $u = v$. It just so happens that the answer to this problem ends up with the final velocity squared.

Similarly, for acceleration, $a = \frac{d}{dt}\left(\frac{dx}{dt}\right)$. Again, you can see that we are not dealing with sizes in different planes (like an area where $A = xy$) but rather infinitesimal changes in sizes. Both $dt$ are exactly the same value (in the same plane).

 

[jim mcnamara]

Hmm - another try.

@marioisaac

I think you are really trying hard to understand. Others have taken very mathematical
approaches. Sometimes that may not help, so - Let's not do that.

You have bad (wrong) assumptions. Because someone somewhere emphasized SI units to you and what
they mean and how you express them. Then I think you went off in the wrong direction. SI
units are a convention - that is a set of definitions everyone agrees on. Nothing more.
That is what @Stephen Tashi explained very well.

The point is that we could define distance per [unit of something] as goldfish or traffic
jams. As long as we are clear how we define it everyone can see what it means. How it
translates. It may not be useful - clearly true in this absurd example.

You are assigning an extra added meaning to "time" that does not exist. At least when
you do Physics problems and expect others to understand your result.

First off. Time is not a universal human concept. In fact, Athpascan languages (Apache,
Navajo, Tlingit) -- the traditional languages anyway -- did not have "time" at all the way
we do in English. Or any Western Language. It got added later to allow Athapascan speakers
to communicate with Western language-based folks. If you speak another language you
already know this: Some concepts do not translate at all well. Does that mean physical laws
did not exist when people spoke to each other in Tlingit in 1750? No! -- In a way you are
doing what I just deliberately did. You are taking a concept, and ascribing extra meaning
to it. More than it has. And falling into a rabbit hole. BTW -- say 'Hi' to Alice if you
see her. She and I are old friends...

Time has a defined meaning for Science. Period. It is not a mathematical operation. It
is a well-defined label. An identifier. Sure, you can perform a set of mathematical
operations to get time units. Or goldfish units for that matter. But only because we
defined it that way beforehand.

Above all, have fun with this stuff because it is fun and interesting.

 

[marioisaac]

Numbers in use to count or measure must have the units for the quantity being counted or measured. A time rate will have, usually, a time quantity in its denominator. If this rate is itself a rate with time again as the unit for the denominator, then this makes the simplified ratio of rates have time multiplied by time. This is how a change in travel rate compared to a change in time passage becomes acceleration. Dimensions are DISTANCE/TIME^2

The word usually leaves space for something else seldom ocurring: are there any physical examples of time quantities being put on the numerator against another physical units on the denominator. e.g.
To measure how many seconds flow per meter (s/m) or how much time is contained in one meter?
The time unit second (s) assumes that time flows in one direction, or along a single plane?
According to Wikipedia a second, is defined by measuring the electronic transition frequency of caesium atoms.

Notice the words transition and frequency:
Transition denotes movement in space.
and frequency is a time rate; a repetition of something; a movement through at least one spatial dimension at a given time.

So it is interesting that within the very definition of the time unit second (s), a spatial dimension is to be found.

Now if you analyze all derived and base units of SI, and dissect them, you will end up with time and spatial physical magnitudes; and if you dissect time in this system (SI) you will find a space magnitude contained within it. So the only real or at least quantifiable property of this physical reality seems to be lenght; time seems to be dependent on space; so it does not remain clear what actually it is.

And this does not seem to contradict the mathematical world I described in my previous post, in which the only dimension found was extension.
Of course extension does not need physical reality to exist; the universe could explode; physical reality could dissapear; extension will remain.

Extension has the quality of being quantifiable and measurable.
Time does not seem to be well defined or understood by physics. And it is not clear wether time can or can not coexist along different dimensions, or is it actually a dimensionless property of the universe, or if it actually does not exist at all.

 

[marioisaac]

There is no "mathematical" definition for the interpretation of physical units.

When a mathematical equation is applied to specific physical problem people may find a specific interpretation for the units that are used in that equation.

By these, are you implying that e.g. by squaring a meter, it is totally arbitrary (and not mathematically certain) for one to decide at whim wether a square meter it is contained within one dimension, or in two dimensions?
So is it totally possible to have a squared or cubed meter contained within a single dimension?
I am concerned why within the realm of mathematics this matters are decided at whim; or is there a mathematical definition why sometimes squaring a unit can derive another dimension, and sometimes not, so that we can get rid of the whiminnes in this matter?

 

[jack action]

By these, are you implying that e.g. by squaring a meter, it is totally arbitrary (and not mathematically certain) for one to decide at whim wether a square meter it is contained within one dimension, or in two dimensions?

There are no arbitrary decisions. The definition of the mathematical relationship defines the number of dimensions. For example an area is defined as (from Wikipedia):

the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.

There, the two dimensions are defined. Now for something in space, where all dimensions have the unit of length, the area unit will be LENGTH². But an area could have other units too. For example, the net work performed by a thermodynamic cycle on a Pressure-Volume diagram is represented by the area enclosed by the PV curve in the diagram. In this case, the units for each dimension of the area are PRESSURE and VOLUME which leads to the «area» unit of ENERGY.

The units don't define the mathematical relationship; The mathematical relationship defines the units.

Here is another definition for a LENGTH², not related to an area: The angular momentum is the linear momentum ($mv$) of a particle times the length of the moment arm ($r$). But the linear speed of the particle ($v$) is also the product of the angular velocity ($\omega$) and the length of the same moment arm ($r$). Hence, an angular momentum contains a double moment that leads to $rmr\omega$ or $r^2m\omega$. (from Wikipedia)

As you can see, $r^2$ is not an area, it is the exact same vector ^[1] multiply by itself, therefore from only one dimension.

Please take into consideration that this is an abstract world and in this case extension does not mean "Physical Length"; it is purely a mathematical magnitude, devoid of any physical reality.

Now, this abstract world seems to be composed of two properties;
1) Extension.
2) Angular relationship.

And this 2 properties differ from one another, in that one is dimensional (extension), and the other one is dimensionless (angular relationship).

I think an angular relationship can only be a property between 2 dimensions if both represents vectors. It won't work if one of them is a scalar dimension. This is why you can't create a plane between a time and a length or between a time and another time.

[1] The general angular momentum definition is $\overrightarrow{r} \times (m\ (\overrightarrow{\omega} \times \overrightarrow{r}))$.

 

[marioisaac]

I think everyone understood your multi-dimensions analysis, which is right in many situations. But there are situations where it doesn't apply. For example, $E = \frac{1}{2}mv^2$. The squared velocity does not represent two velocities, each in its own plane; It is exactly the same velocity - in one plane - multiplied by itself. Where does it come from? Well, the reality is that it's not really twice the velocity:
$$E = \int_{u= 0}^v mudu$$
So it is the summation of the multiplications of the velocity by the infinitesimal change in velocity at any instant between $u = 0$ and $u = v$. It just so happens that the answer to this problem ends up with the final velocity squared.

I decided to explain my question better because the answers I was receiving seemed to be considering mathematics as a tool for measuring physical units, and not as a realm in itself. Yes mathematics can serve physical measurement purposes, but limiting the scope of mathematics to physical utilitarian purposes is a fail to comprehend it's very nature, and the laws that govern this realm.
Similarly Geometry is a very bad name for what it actually is. This name implies that it's essence is to measure earth;this is due to historical reasons which I won't get into detail here. The point is that if you approach Geometry only as a measurement tool, you will limit your comprehension of what it actually is; and so I want first to comprehend the very nature of what are the possible implications of squaring a quantity along a single dimension solely in the mathematical realm in absolute abstraction. Once this is clear, then we can proceed to apply these to physical units, and examine why sometimes new physical dimensions arise while performing certain mathematical operations.

For example, giving deep thought to these conversations we are having here, I see that the only physical unit that seems susceptible to acquire a new dimension when being elevated to a power seems to be the length dimension. (Please correct me if I'm wrong here).
E.g if you square or cube any base SI Unit corresponding to any of the following dimensions: time, mass, electric current, temperature, luminous intensity, amount of substance, or any other derived based SI Unit, they will all remain within the same dimension.

Please take into consideration that I am analyzing first the possibilities of single units to acquire new dimensions; I don't want to get into combinations if the components of this combinations are not well understood. E.g. I want to first understand the implications and possibilites of m2, s2, in themselves; not a combination of units such as m/s2.

Given this, our problem seems to get simplified, and the question seems to be:
Under what conditions a length physical unit while being elevated to the power of something remains in a single dimension? And, what conditions should be met to add a new dimension while performing this same operation?
If this is well understood, there should be rational mathematical questions to this answers. And example here and there does not really answer the question. A simple answer will; an example for facilitating comprehension is a plus.

 

[marioisaac]

Hmm - another try.

@marioisaac
The point is that we could define distance per [unit of something] as goldfish or traffic
jams. As long as we are clear how we define it everyone can see what it means. How it
translates. It may not be useful - clearly true in this absurd example.

Goldfishes and Traffic Jams come in many sizes, so they are not good mathematical examples; standard size cans of Coke or Red Bull are better ones. And no, I don't work for the Coke or Red Bull Marketing Department in case you are wondering :)

First off. Time is not a universal human concept. In fact, Athpascan languages (Apache,
Navajo, Tlingit) -- the traditional languages anyway -- did not have "time" at all the way
we do in English. Or any Western Language. It got added later to allow Athapascan speakers
to communicate with Western language-based folks. If you speak another language you
already know this: Some concepts do not translate at all well. Does that mean physical laws
did not exist when people spoke to each other in Tlingit in 1750? No! -- In a way you are
doing what I just deliberately did. You are taking a concept, and ascribing extra meaning
to it. More than it has. And falling into a rabbit hole. BTW -- say 'Hi' to Alice if you
see her. She and I are old friends...

Who implied that Time was a human concept?
You seem to be wrongly assuming I am assuming things.
I agree that physical laws existed before we ever thought of them; and also that Mathematical laws existed before physical laws were born.
2+2=4; apples are not needed for this mathematical reality to exist.
The physical world seems to be written in mathematical language.
Humans need physics to exist; physics need mathematics to exist; mathematics exist in itself.

I think you went out too rapidly from the rabbit hole thinking you had grasp the very nature of it; I invite you to come back; this time I suggest you bring a microscope along; you might be surprise what on closer inspection these grounds may reveal. BTW, besides Alice and her friends, if you go deeper you will find Liebniz and Newton hanging around with their microscopes; they are open and fun to engage in conversation, and seem very interested in this matters. Alice is still fun I should say, just in a different way :)

Time has a defined meaning for Science. Period. It is not a mathematical operation. It
is a well-defined label. An identifier. Sure, you can performa set of mathematical
operations to get time units. Or goldfish units for that matter. But only because we
defined it that way beforehand. [QUOTE/]

Period? Not so fast.
For practical purposes it might be well defined, but not conceptually.
Time (conceptually) is definitely not well defined. The Time SI second, is a circular definition; it's very definition contains itself (see my previous posts).

 

[jack action]

and examine why sometimes new physical dimensions arise while performing certain mathematical operations.

Physical dimensions do not arise while performing mathematical operations; Mathematical operations are performed on system where physical dimensions are already well defined.

I see that the only physical unit that seems susceptible to acquire a new dimension when being elevated to a power seems to be the length dimension.

That is because space is probably the only coordinate system that we use with more than one dimension that have the same unit.

E.g if you square or cube any base SI Unit corresponding to any of the following dimensions: time, mass, electric current, temperature, luminous intensity, amount of substance, or any other derived based SI Unit, they will all remain within the same dimension.

These are all represented by a scalar. They have a size, but no directions. Geometry can't be applied.

Under what conditions a length physical unit while being elevated to the power of something remains in a single dimension?

1. If the physical quantities are defined as scalars. In that case, I wouldn't say they are within a dimension because they are physical quantities independent of any coordinate system (Ex.: the mass is either big or small). Note that a length can be seen as a scalar too, but it can also be defined with position vectors;
2. If you do a product of two colinear vectors (or multiply the vector by itself).

And, what conditions should be met to add a new dimension while performing this same operation?

1. When you do a product of two non-colinear vectors.

But, again, new dimensions don't appear after a mathematical operation, the mathematical operation is done on dimensions already defined.

 

[marioisaac]

and examine why sometimes new physical dimensions arise while performing certain mathematical operations.
Physical dimensions do not arise while performing mathematical operations; Mathematical operations are performed on system where physical dimensions are already well defined.

By this definition then (3m)³ will only equal to 9m? 9m³ is out of the game because it will "add" 2 extra dimensions; or will it actually equal to 9m³ but we will necessarily consider it to be only in one dimension because we had not previously agreed that were going to consider m³ in 3 dimensions. And only if we had previously agreed that m³ consists of 3 dimensions then we will consider it to be so.
So L³ primarily does not bring about the volume of a cube, it primarily has to be considered as a straight line enclosed in one dimension that has increased by elevating itself to the power of 3.
Then it seems that the naming the power of 2 "square", and the power of 3 "cube" is wrong and misleading, because one at first instance should not consider the figure of a square or a cube when performing these operations; in order for these forms to arise after performing such said operations, we previously have to agreed upon this, and it is not something that mathematically naturally occurs. or actually in the mathematical realm it does naturally occurs at first instance, but in the physical realm it does not occur unless previously agreed upon?

I see that the only physical unit that seems susceptible to acquire a new dimension when being elevated to a power seems to be the length dimension.
That is because space is probably the only coordinate system that we use with more than one dimension that have the same unit.

And why it is not possible for other magnitudes to coexist simultaneously on different dimensions? On what basis can it be affirm that they are constrained to exist in one dimension?

E.g if you square or cube any base SI Unit corresponding to any of the following dimensions: time, mass, electric current, temperature, luminous intensity, amount of substance, or any other derived based SI Unit, they will all remain within the same dimension.
These are all represented by a scalar. They have a size, but no directions. Geometry can't be applied.

Length is a scalar and it is susceptible of being represented in more than one dimension.

Under what conditions a length physical unit while being elevated to the power of something remains in a single dimension?

1. If the physical quantities are defined as scalars. In that case, I wouldn't say they are within a dimension because they are physical quantities independent of any coordinate system (Ex.: the mass is either big or small). Note that a length can be seen as a scalar too, but it can also be defined with position vectors;
2. If you do a product of two colinear vectors (or multiply the vector by itself).

1. If you are very sure scalars don't have dimension, then you can correct this Wikipedia or present your case against SI or ISQ to prove them the contrary, because there radians and steradians are the only units defined to be dimensionless. Everything else has a Dimension.
https://en.wikipedia.org/wiki/International_System_of_Quantities
https://en.wikipedia.org/wiki/SI_derived_unit

So all cases in which length is not considered a scalar, nor a collinear vector, it should be considered that by elevating it to the power of x, the number of dimensions should equal to x. This is the only case were we should consider that a physical magnitude can be represented in more than one dimension.

Is this the answer?

And, what conditions should be met to add a new dimension while performing this same operation?

1. When you do a product of two non-colinear vectors.

Is length the only physical dimension that can be represented as two non-collinear vectors?

But, again, new dimensions don't appear after a mathematical operation, the mathematical operation is done on dimensions already defined.

I guess you are referring to physical dimensions. Does this holds true for mathematical dimensions as well?By the way, through investigation I found out there is a subject named Dimensional Analysis, and deals with this matters. Fourier, Maxwell, and others have been concerned with this matters.
https://en.wikipedia.org/wiki/Dimensional_analysis

 

[Stephen Tashi]

By these, are you implying that e.g. by squaring a meter, it is totally arbitrary (and not mathematically certain) for one to decide at whim wether a square meter it is contained within one dimension, or in two dimensions?
So is it totally possible to have a squared or cubed meter contained within a single dimension?
I am concerned why within the realm of mathematics this matters are decided at whim; or is there a mathematical definition why sometimes squaring a unit can derive another dimension, and sometimes not, so that we can get rid of the whiminnes in this matter?

In mathematics, the term "dimension" is only defined in certain contexts. For example, "dimension" is defined in the context of vector spaces. "Dimension" has a different definition in the context of "dimensional analysis". (There are many mathematical terms that have different definitions in different contexts. For example "factor" means one thing in algebra and something different in the context of statistical "factor analysis".)

You are speaking of "dimension" without establishing any mathematical context for it.

You apparently have some private notion of what "dimension" is and you are trying to relate your concept to physical units. If you are seeking a mathematical answer, you'll have to specify a mathematical context. You won't get a mathematical answer unless you ask a mathematical question.

By the way, through investigation I found out there is a subject named Dimensional Analysis, and deals with this matters. Fourier, Maxwell, and others have been concerned with this matters.
https://en.wikipedia.org/wiki/Dimensional_analysis

That's a very important field of study in physics. It's provides the basis for using scale models to study larger objects. If you study "dimension" in the context of dimensional analysis, you won't find any material about $L^3$ signifying "a straight line enclosing a cube" or thoughts like that.

If you are actually interested in dimensional analysis, I suggest you look at the book "The Physical Basis Of Dimensional Analysis" by Ain A. Sonin (available at http://web.mit.edu/2.25/www/pdf/DA_unified.pdf). This book deals with both the mathematics of dimensional analysis and the interpretation of that mathematics when applied to physics.

 

[jack action]

I think that you are mixing two different concepts. There are dimensions within a vector space and there are measurements defined by a magnitude and a unit. Measurements can also be called dimensions. In my previous post, when I was referring to measurement, I was using (maybe incorrectly?) the term quantity or scalar dimension, when I was using dimension in any other context, I was referring to the vector space kind.

In a vector space, each of the basis vectors can be scaled based on a measurement (i.e. with a unit). But a measurement is not necessarily a vector.

When solving a problem, you first have to define:

1. The vector space and the vectors it contains (if any);
2. The scalar measurements (if any);
3. The mathematical relationship between all of those variables;

Then the notion of what the power of the different units means makes perfect sense.

 

[marioisaac]

In mathematics, the term "dimension" is only defined in certain contexts. For example, "dimension" is defined in the context of vector spaces. "Dimension" has a different definition in the context of "dimensional analysis". (There are many mathematical terms that have different definitions in different contexts. For example "factor" means one thing in algebra and something different in the context of statistical "factor analysis".)

You are speaking of "dimension" without establishing any mathematical context for it.

You apparently have some private notion of what "dimension" is and you are trying to relate your concept to physical units. If you are seeking a mathematical answer, you'll have to specify a mathematical context. You won't get a mathematical answer unless you ask a mathematical question.

Well, in my second post I provided a simple mathematical context; I did not provide a name for it, but it was very simple, and it did not contradict in any way the Euclidean Space, which is adequate (as far as I am concerned) and usually the one employed to deal with simple mathematical figures such as squares, and cubes, and also with basic combinations of physical magnitudes such as velocity and acceleration.
https://en.wikipedia.org/wiki/Euclidean_space#Applications

That's a very important field of study in physics. It's provides the basis for using scale models to study larger objects. If you study "dimension" in the context of dimensional analysis, you won't find any material about $L^3$ signifying "a straight line enclosing a cube" or thoughts like that.

Probably in the context of dimensional analysis the answer to L³ is not a cube. But there have been at least a mathematical definition as to what the dimensional implications of exponentiation upon a physical magnitude are.

I quote Fourier's thoughts on this matters:

"The original meaning of the word dimension, in Fourier's Theorie de la Chaleur, was the numerical value of the exponents of the base units. For example, acceleration had the dimension 1 with respect to the unit of length, and the dimension -2 with respect to the unit of time.[15] This was slightly changed by Maxwell, who said the dimensions of acceleration are LT−2, instead of just the exponents.[16]"
https://en.wikipedia.org/wiki/Dimensional_analysis#History

I don't know on what basis Fourier express this, but it was my main concern that exponentiation upon a physical magnitude should have at least dimensional consequences. So it seems that for Fourier T and T² are 2 different dimensions. They might not be coexisting at the same time, but it seems he is suggesting they are different dimensions, and we don't need to agree beforehand that they are going to be different dimensions; the exponentiation itself will inform us the dimension of the physical magnitude. And here I've been told quite the contrary, that it is required from us to agree beforehand what dimensions are going to be considered and that they will not result from exponentiation. We all should be glad I kept inquiring.

If you are actually interested in dimensional analysis, I suggest you look at the book "The Physical Basis Of Dimensional Analysis" by Ain A. Sonin (available at http://web.mit.edu/2.25/www/pdf/DA_unified.pdf). This book deals with both the mathematics of dimensional analysis and the interpretation of that mathematics when applied to physics.

Thank you, I appreciate your input in this thread, and the book suggestion.

 

[marioisaac]

By the way,

If exponentiating a number or unit to the power or 2 or 3, does not necessarily render (because it is dependent on the geometrical space, or whatever else) a 2-dimensional space, or a 3-dimensional space, why should we keep naming this precise operations as "squaring", or "cubing" something.
Squares and cubes are one of the many possible outcomes that can result from exponentiating a unit.

For the sake of mathematical sanity I think we should stop naming this operations as squaring and cubing, because they suggest something they are not.