The dimensions of numbers

[DevilSolution]

I'm curious as to the nature of numbers and have come up with a little concept.

 

I've recently found a new love for maths so ive been going over some geometry, trig, algebra, calc etc and have a few basic questions.

 

What dimension are numbers? Based on the fact logic works in binary i concluded they are 2D but have no idea.

 

The euclidean geometry is 2D? can all shapes be expressed in these terms? i know there is 3D geometry but it too is express in mathematical terms which seems to impose its only a 2D manifistation of 3D reality by the use of logic.

 

In my mind right now i see numbers as 2D, shapes as 3D and reality as 4D.

 

Each heirarchy can be expressed in the lower dimension but not in absolute terms. Much in the same way you can create a 3D world on a computer screen that exists in a 2D array of colour and whose numbers are being crunched by a 2D process(or).

 

1D Vertex.

2D Logic. (of vertex's / vertices)

3D imposes a physical limit to geometry.

4D all the above + time.

 

What dimension are numbers???

[John]

I'm assuming you're talking about the real numbers and Euclidean geometry.

 

A real number by itself has no spatial extent, thus we might call it dimensionless or zero-dimensional, i.e. it is simply a point.

 

If we take the entire set of real numbers, then we have an infinitely long line on which every point is associated with a real number, and this line is one-dimensional.

 

We can then take all ordered pairs (x, y) where x and y are real numbers (this is the Cartesian product [math]\mathbb{R} \times \mathbb{R}[/math]), and the result is the familiar Euclidean plane, which is two-dimensional.

 

If we then add a third real coordinate z, and take all the resulting ordered triples (x, y, z) (which, similarly to before, is the Cartesian product [math]\mathbb{R} \times \mathbb{R} \times \mathbb{R}[/math]), then we have Euclidean space, which is three-dimensional.

 

We can go even higher than three, though, by adding more coordinates, taking the set of all n-tuples (a₁, a₂, ..., a_n) for any natural number n, resulting in Euclidean n-space, which is n-dimensional. We often denote this by [math]\mathbb{R}^{n}[/math], e.g. the plane is [math]\mathbb{R}^{2}[/math] and the number line is just [math]\mathbb{R}[/math]. Sometimes [math]\mathbb{E}^{n}[/math] is used to emphasize that we're dealing with a Euclidean space.

Given all this, we might define the dimensionality of a space as the number of coordinates required to specify a point in that space.

 

I'm not sure what you mean when you say shapes are 3D. For instance, polygons are shapes, but they are two-dimensional. Polyhedra are also shapes, and they are indeed three-dimensional. We also have higher-dimensional shapes, e.g. the tesseract in four dimensions. The tesseract is just one example of a class of n-dimensional objects known as hypercubes, which, as the name implies, are n-dimensional analogues of the familiar square or cube.

 

As for reality being 4D, I guess with the introduction of Minkowski spacetime that's valid, with the understanding that the time dimension isn't the same as the three space dimensions. The physicists could provide more details about that.

 

This post is a bit ramblish, as I need sleep, but hopefully it's helped some.

Edited November 13, 2013 by John

[DevilSolution]

I understood your post to the most extent, ive done some matrix functions with trig to manipulate triangles, wasnt aware that it created the Cartesian plane, i thought that was the product of 2D geometry.

 

I still dont understand how numbers can be dimensionless entities, they exist in some form to represent some other.

[imatfaal]

I understood your post to the most extent, ive done some matrix functions with trig to manipulate triangles, wasnt aware that it created the Cartesian plane, i thought that was the product of 2D geometry...

I am not sure that Matrix functions do - and I don't think that's what John was getting at; the cartesian plane is the simplest representation of two number sets whose numbers lines/axes are orthogonal ie two dimensions. The bold capital \mathbb{R} denotes the set of Real Numbers.

 

[latex] \mathbb{R} - Real

\mathbb{Z} - Integers

\mathbb{Q} -Rationals[/latex]

 

etc

 

Dimensions - at least in my conception - must be mutually distinct and by that I mean that no amount of one can ever change the value of another. Geometrically and with vectors this is represented by orthogonality - they are at right angles to each other; if you start at x=0 y=1 then no matter how much you travel in the x direction (+ve or -ve) you will remain on the line of y=1

 

...

I still dont understand how numbers can be dimensionless entities, they exist in some form to represent some other.

Numbers exist? Really? I am not sure they have any form of existence - they are the product of a logical argument. As John mentioned a group of numbers sits on a line that has extension - but one number has no extension either physically or logically in a non-concrete representation.

[ADreamIveDreamt]

I wonder what happens if you put Numbers on my Line

 

 

 

 

 

 

 

"They used Numbers to make a Point."

 

Also, if you Understand 4D, you also Understand 22D.

[studiot]

 

I still dont understand how numbers can be dimensionless entities, they exist in some form to represent some other.

This difficulty occurs because what is meant by numbers and dimensions is a profound question.

 

For many purposes the physicist's view suffices and Imatfaal's comment sums it up nicely.

 

 

Dimensions - at least in my conception - must be mutually distinct and by that I mean that no amount of one can ever change the value of another.

 

Mathematicians go much further so neither the plain word number nor the equally plain word dimension are sufficient by themselves.

 

So mathematicians use qualifiers for particular well defined situations.

 

Cardinal number

Ordinal number

Complex number

 

and so on, there are many more.

 

Equally there are types of mathematical dimension

 

Hausdorf dimension

Topological dimension

Euclidian dimension

 

again there are more.

 

Each of these have different properties, applicable to the situations in which they are used or found.

Edited November 13, 2013 by studiot

[DevilSolution]

So these:

 

[latex] \mathbb{R} - Real
\mathbb{Z} - Integers
\mathbb{Q} -Rationals[/latex]

etc

 

Define the logic for all of the different 'forms' of number such as:

 

Cardinal number

Ordinal number

Complex number

 

And these forms of logic notaton regarding the definitions of numbers relates to the dimensional space in which they are used? such as:

 

Hausdorf dimension

Topological dimension

Euclidian dimension

 

Which would mean all numbers are dimensionless unless applied to some dimensional space at which point they cease to become abstract and are tangible to the space they are represented in?

[studiot]

 

And these forms of logic notaton regarding the definitions of numbers relates to the dimensional space in which they are used? such as:

 

No the idea was to give you some (useful) terms you could investigate further (look up).

 

I said nothing about the word 'space' which is another term that needs qualifiers and has particular defined meanings in physics and in mathematics.

 

Numbers, dimensions and space are different technical terms that are independent of each other, just like apples, oranges and bananas are independent different fruit.

 

And equally number, dimension and space can be used in combination to build up more complicated concepts, justs as the fruits can be used combined to build up a more complicated food preparation.

Edited November 14, 2013 by studiot

[DevilSolution]

No the idea was to give you some (useful) terms you could investigate further (look up).

 

I said nothing about the word 'space' which is another term that needs qualifiers and has particular defined meanings in physics and in mathematics.

 

Thanks you for your input, I will certainly go ahead and read about them. (I've done some topology design and worked in the cartesian plane at uni).

Im just struggling to abstract each concept from each other, since i was a kid there has never been any attempt on my part or academic to do such a thing. To seperate this 'form' of number from this 'form' never made much sense as they were always interwined and served some physical purpose.

 

This does lead to my next question though, what dimension(al space) do circles belong? in 2D they dont seem to fit and as im currently seeing it, the 3D is just 2D with limits.

 

Here's some philosophical concept which has got me confused equally as much: The computer monitor is an array of 2D LED's (or cathode rays etc), then when we apply sets of logic to this 2D space, such as our current laws of geometry and trigonometry in the form of logic circuits, we can create a highly sophisticated illusion of the 3D in the 2D array. Now if we were self aware being (or beings with the abillity to comprehend and perceive) and we were placed inside this 2D array; Would we perceive it as 3D reality? I.E do the laws super imposed by the logic allow us to perceive 3D reality as a form of the logic applied to the 2D space??

 

Regards.

 

Just as a little extra thought: Would not all numbers have to physically exist for them to have been conceived? For lack of a greater example i cannot comprehend of a number (or logical system) that could exist if it does not fit into some model of reality. I.E (oyf3498t45t45f8548t89489gfj;;;;><><><>< dgfdgfdgdfgfg) <- this is a definition of a logical system, yet its conception would only arise should i have purpose to model it on, such that it represents how the washingup liquid relates the changing of seasonal temperatures (totally totally abstract but it shows how abstract the logical system can be).

 

However certain logical laws may be more related than others, atleast from our perspective and specifically for our use.

Edited November 14, 2013 by DevilSolution

[John]

A circle is two-dimensional. In the Euclidean plane, it is the set of all (x, y) such that x² + y² = r², where r is the radius of the circle.

As for the 2D reality being perceived as 3D concept, you might enjoy reading about the holographic principle. That article may be a bit too technical, but I'm sure Googling around would yield some more accessible introductions to the concept. It's not precisely what you're talking about, but it's similar enough in spirit.

Edited November 14, 2013 by John

[DevilSolution]

A circle is two-dimensional. In the Euclidean plane, it is the set of all (x, y) such that x² + y² = r², where r is the radius of the circle.

 

As for the 2D reality being perceived as 3D concept, you might enjoy reading about the holographic principle. That article may be a bit too technical, but I'm sure Googling around would yield some more accessible introductions to the concept. It's not precisely what you're talking about, but it's similar enough in spirit.

 

The principle is very similar indeed, ive not really tried comprehending how energies would relate though i would imagine they are also invoked from the 2D using a particular set of logic. In these terms it would seem like physics and mathematics is the process of reverse engineering the logic invoked on the 2D to the 3D and 4D.

 

We have finely deducted certain principles / axioms / laws / postulates et cetera by which were being processed except a few major gaps.

 

The main one being time, which i personally think relates to circles.

The other core gaps are the links between fundamental forces, which is waiting to be logically solved.

 

And then ofcourse all the paradoxes.

Edited November 14, 2013 by DevilSolution

[imatfaal]

So these:

 

 

Define the logic for all of the different 'forms' of number such as:

 

 

And these forms of logic notaton regarding the definitions of numbers relates to the dimensional space in which they are used? such as:

 

 

Which would mean all numbers are dimensionless unless applied to some dimensional space at which point they cease to become abstract and are tangible to the space they are represented in?

 

You seem intent on shoehorning one concept into another before you truly understand either The logic of a number is a very difficult idea - what you must first accomplish is understand the relationships between the Reals, the Complexes, the Rationals/Irrationals, the Algebraic/Transcendental etc. ie before searching for deeper meaning, a more fundamental reason, you must become completely at ease with what we actually already know. The sorts of numbers groups listed do not define spaces nor dimensions- perhaps one could talk about them subdividing or being a fraction of the complex number plane (ie the reals form a line in the complex number plane).

 

 

 

Thanks you for your input, I will certainly go ahead and read about them. (I've done some topology design and worked in the cartesian plane at uni).

Im just struggling to abstract each concept from each other, since i was a kid there has never been any attempt on my part or academic to do such a thing. To seperate this 'form' of number from this 'form' never made much sense as they were always interwined and served some physical purpose.

 

This does lead to my next question though, what dimension(al space) do circles belong? in 2D they dont seem to fit and as im currently seeing it, the 3D is just 2D with limits.

 

Here's some philosophical concept which has got me confused equally as much: The computer monitor is an array of 2D LED's (or cathode rays etc), then when we apply sets of logic to this 2D space, such as our current laws of geometry and trigonometry in the form of logic circuits, we can create a highly sophisticated illusion of the 3D in the 2D array. Now if we were self aware being (or beings with the abillity to comprehend and perceive) and we were placed inside this 2D array; Would we perceive it as 3D reality? I.E do the laws super imposed by the logic allow us to perceive 3D reality as a form of the logic applied to the 2D space??

 

Regards.

 

Just as a little extra thought: Would not all numbers have to physically exist for them to have been conceived? For lack of a greater example i cannot comprehend of a number (or logical system) that could exist if it does not fit into some model of reality. I.E (oyf3498t45t45f8548t89489gfj;;;;><><><>< dgfdgfdgdfgfg) <- this is a definition of a logical system, yet its conception would only arise should i have purpose to model it on, such that it represents how the washingup liquid relates the changing of seasonal temperatures (totally totally abstract but it shows how abstract the logical system can be).

 

However certain logical laws may be more related than others, atleast from our perspective and specifically for our use.

 

Disagreeing slightly with John I might almost claim that the circle is one-dimensional embedded in the 2 d plane. You only need one parameter to uniquely define a circle - the radius; and only one parameter changes when you move along it (theta). On the question of beings in a 2d reality that we perceive to be a simulacrum of 3d - I believe it would be perceived as 2d but with a great deal of sophistication they might realise that viewed from outside the plane of their existence an optical illusion would occur - how you can have sophisticated 2d beings Darwin only knows.

 

 

The principle is very similar indeed, ive not really tried comprehending how energies would relate though i would imagine they are also invoked from the 2D using a particular set of logic. In these terms it would seem like physics and mathematics is the process of reverse engineering the logic invoked on the 2D to the 3D and 4D.

 

We have finely deducted certain principles / axioms / laws / postulates et cetera by which were being processed except a few major gaps.

 

The main one being time, which i personally think relates to circles.

The other core gaps are the links between fundamental forces, which is waiting to be logically solved.

 

And then ofcourse all the paradoxes.

 

Just as a note axioms are not deduced - they are postulated and affirmed, they form an agreed basis for your argument from which you move forward. Laws of science are empirically evidenced. The logic of the 2d, 3d, 4d is all the same - mathematicians have been thinking about fewer and extra dimensions for a long time now. What we cannot do is move physical laws from 3d to something else without being very careful - for instance the inverse square law.

 

And there are no paradoxes - just badly stated postulates

[DevilSolution]

 

You seem intent on shoehorning one concept into another before you truly understand either The logic of a number is a very difficult idea - what you must first accomplish is understand the relationships between the Reals, the Complexes, the Rationals/Irrationals, the Algebraic/Transcendental etc. ie before searching for deeper meaning, a more fundamental reason, you must become completely at ease with what we actually already know. The sorts of numbers groups listed do not define spaces nor dimensions- perhaps one could talk about them subdividing or being a fraction of the complex number plane (ie the reals form a line in the complex number plane).

 

I've studies logic quite indepth, sort of I've covered the basics for combinatorial, propositional and sequential logic but logic in these terms seems like a very 2D concept, they are the relationship between points, or ateast they could be. If we deduct that all math is only a form of logic regardless of the "type" of number being used, its still only a 2D representation of logic.

 

A radius is a line between 2 points making it 2D, but again a perfect circle doesnt really exist, not in the 2D or 3D plane anyway.

 

I personally think circles are 4D, thats why they cant be expressed in rational terms in the dimensions below (when i say expressed i mean comprehended correctly, the visual representation and the math defining it are 3D and 2D).

[studiot]

Let us say you are on a circle. By that I mean you are confined to travel a circular path as for instance if you were on the circle line on the London Underground.

 

You only need to know one number to reach any other part of the circle.

 

Now this number may increase steadily and evenly eg 1 km along the line, 2, km, 5 km and so on, or it may change in jumps eg 1 stop, 2 stops, 5 stops along the line. In this second case the distance along the circle represented are uneven.

 

From this point of view the circle is 1 dimensional.

 

But now examine the situation from the point of view of the cleaning contractors for the stations, who, suprisingly, are not allowed to travel on the trains, but must arrive by road.

 

They need two numbers to specify the position of each station, easting and northing coordinates on their map.

 

So they regard the circle as 2 dimensional.

 

Yet again look at it from the point of view of the train authority, wanting to plan an extra line connecting two of the stations across the circle.

 

Yes, they need the easting and northings of both stations, but that is not enough. They need a third number for each station to make sure the new line comes in at the right level.

 

So they think of the situation and their circle as existing in three dimensions.

 

This point of view of the dimension being how many numbers you need to specifiy something works well in ordinary geometry, but recent advances in mathematics has rather upset this. The discovery of fractal geometry to be precise.

 

This is why I urged you to look up Euclidian, Hausdorf and topological dimensions.

Mathematically there may be more than one dimension of an object or situation, depending upon the point of view.

 

But a number for instance 1 or pi is the same from whatever point of view you look.

Edited November 16, 2013 by studiot

[imatfaal]

 

I've studies logic quite indepth, sort of I've covered the basics for combinatorial, propositional and sequential logic but logic in these terms seems like a very 2D concept, they are the relationship between points, or ateast they could be. If we deduct that all math is only a form of logic regardless of the "type" of number being used, its still only a 2D representation of logic.

 

No - really no. You are comparing incommensurable ideas

 

 

A radius is a line between 2 points making it 2D, but again a perfect circle doesnt really exist, not in the 2D or 3D plane anyway.

 

Again no. If you use words you really have to use them in the same manner that everyone else does. What is your idea of definition that allows you to say the above. And the circle in geometry as the nexus of all the points equidistant from a single reference point "exists" as much ass any mathematical construct.

 

 

I personally think circles are 4D, thats why they cant be expressed in rational terms in the dimensions below (when i say expressed i mean comprehended correctly, the visual representation and the math defining it are 3D and 2D).

 

I disagree with Studiot's extension to the third dimension and a little bit the second - using that logic removes the utility of the concept of dimension if the circumstances change the nature of the beast. A point doesn't become 4d because we embed it in spacetime; to locate it we need 4 values but it has no extension in those dimensions. That said I don't have studiot's advanced knowledge of the more abstruse maths. Howver your argument is not correct - physical representation is not important and the definition is as I give it above and relates to one variable.

[studiot]

 

I disagree with Studiot's extension to the third dimension and a little bit the second

 

Crafting a mundane common or garden example to describe a concept is never as perfect as a carefully worked out formal definition scheme.

[DevilSolution]

No - really no. You are comparing incommensurable ideas

 

I'm not alone in my thinking, others like russell have attempted to prove indefinitely that maths is infact only an extension of logic. Calculators and more specifically processors somewhat prove this. They are accurate relative to the amount of binary digits (tracks) we can use. Perhaps saying logic is 2D may be more ambiguous because im not thoroughly defining what 2 dimensions are. My definition in these terms simply refers to the connecting of any 2 points, a single vertices is suffice for a 2D definition, though in our reality this is almost boundless.

 

 

Again no. If you use words you really have to use them in the same manner that everyone else does. What is your idea of definition that allows you to say the above. And the circle in geometry as the nexus of all the points equidistant from a single reference point "exists" as much ass any mathematical construct.

 

This requires a level of articulation i dont posess, however i do have a visual representation of how i understand it, which should help give some indication for my belief.

 

The 2D

 

The 3D

 

As you can see, the 2D circle has 90" missing but still accounts for x,y and z from the source. If the source is the point of origin and the x, y and z have no bounds, we have a boundless 3D space. (obviously mine are bound by the size of the paper and circle but in reality they aren't)

 

This where my comprehension of the cirlce gets fuzzy, because were missing the 90" and we've made 3D using the 270" i equate that this remainder 90 somehow accounts for time, though time is a non-visual entity so it cant be shown. I suggest it creates a looping process much in the same way the 2D visual representation of a circle loops infinitely. However i only believe this because there is no such thing in our form of mathematics and reality as a perfect circle. The perfect circle is the looping of 4D. I have no idea how. But it does.

 

A final note, spheres and all other shapes can exist within the second image, Its strange to conceptualise but imagine your floating somewhere in that 3D, you would exist identically to how you do now except you exist in 3/4 of a circle AND without time. There is nothing inside that 3/4 of a circle to create time, it requires the last 1/4, using some non-visual process, it loops, like the sine wave of the EM spectrum, it loops.

 

Its also a recursive process if you define a point as a circle. Everything exist's as an amalgamation of that point.

Edited November 16, 2013 by DevilSolution

[studiot]

 

However i only believe this because there is no such thing in our form of mathematics and reality as a perfect circle. The perfect circle is the looping of 4D.

 

Of course there is, both in maths and reality.

 

Neither you, nor I, nor anyone else can redefine the circle in mathematics, which is what you are trying to do, and imatfaal has already commented on.

 

Mathematically a circle is defined as a particular locus.

 

In the real world we can actually trace out this locus, so the 'real' circle has real existence. It does not matter that there is not perfectly circular real world object. A (circular) cut through an object will do if you want a concrete noun. If you want abstract nouns then consider a range-range navigation system. The position lines of a rover in this system are perfectly circular. And yes the position line exist as much as other abstract nouns such as the colour red or the emotion joy.

Edited November 16, 2013 by studiot

[DevilSolution]

 

Of course there is, both in maths and reality.

 

Neither you, nor I, nor anyone else can redefine the circle in mathematics, which is what you are trying to do, and imatfaal has already commented on.

 

Mathematically a circle is defined as a particular locus.

 

In the real world we can actually trace out this locus, so the 'real' circle has real existence. It does not matter that there is not perfectly circular real world object. A (circular) cut through an object will do if you want a concrete noun. If you want abstract nouns then consider a range-range navigation system. The position lines of a rover in this system are perfectly circular. And yes the position line exist as much as other abstract nouns such as the colour red or the emotion joy.

 

Well there isn't in reality and maths is mapped to reality. The 2D representation of the circle gives a good approximation, but its not perfect either because its attempting to use 2D maths to explain a 4D object. Thats why pi or 2rads never ends.

 

How about the example of 3/4 of a circle, Is there anything wrong in using that as a representation of 3D space? If 3D + time is 4D then its obvious where i got my conclusion that the last 1/4 creates time.

Edited November 16, 2013 by DevilSolution

[studiot]

I think you said that English is not your first language.

 

Well it is certainly mine, so I recommend you take some notice and respond to points made about it.

 

Not all languages have the concept of abstract and concrete nouns, so do you understand this?

 

It is vitally important to an english word like circle which is an abstract noun.

[DevilSolution]

Mathematically a circle is defined as a particular locus.

 

I had already defined it in these terms. Your only iterating my point.

 

Its also a recursive process if you define a point as a circle. Everything exist's as an amalgamation of that point.

 

You refute my hypothesis and conclusion but have expressed no reason why, not logically, not even in english.

 

If a circle is an abstract noun, what is time?

 

Neither you, nor I, nor anyone else can redefine the circle in mathematics, which is what you are trying to do, and imatfaal has already commented on.

 

I'm not, im explaining the relationship of time to a circle, shedding light on its irrational nature.

[studiot]

 

You refute my hypothesis and conclusion but have expressed no reason why, not logically, not even in english.

 

The only statement of yours that I have refuted is the claim you made several times (without proof or justification) that the circle does not exist in either mathematics or reality.

 

I offered real world examples of the existence of a circles, as evidence for my statemnt that they do indeed exist in reality.

 

If it does not exist in mathematics why do you say this

 

 

I had already defined it in these terms. Your only iterating my point.

 

 

In fact we are apparently agreed on a definition.

 

 

Its also a recursive process if you define a point as a circle.

 

Yes and I have pointed you several times at the fact this this notion of a circle, although different from the locus definition, has implementation in both mathematics and English.

 

In English we talk of a circular walk, a circular argument etc

 

In maths the idea is reflected in the topological notion of a closed loop

 

So you are seeing an adversarial stance where none exists.

 

However I think that talking in circles has taken this thread away from the original point which was about the relastionship between dimension and number.

 

Talking about time (there is no need to shout) would diverge even further, which is why I have avoided it so far.

 

I'm not actually sure whether time is an abstract or concrete noun, or falls somewhere in between in the 'unclassified' basket.

 

In any event, you need to be careful slapping the word time against an axis and calling the result 4D. The forms of physics that create 4D spaces use ct for that axis to preserve compatibility with the other spatial axes.

Edited November 16, 2013 by studiot

[DevilSolution]

However I think that talking in circles has taken this thread away from the original point which was about the relastionship between dimension and number.

 

I see the pun, i think its the natural direction though.

 

The dimensions and circles, atleast in mind, are entwined. I've tried to demonstrate it but its very difficult to illistrate it without repeating myself inadvertently. In all honesty i've no idea how to express going from the 3D to the 4D, that looping process, its hard to explain it. I could be totally wrong.

I said an EM sine wave is a good example but i think they use both sine and cosine, one for the electro wave and one for the magnetic wave, working together but at opposed angles.

 

I just wanted to know if theres any flaws in my concept really.

[studiot]

I think you are still pulling technical terms out of a hat, guessing at what you think or would like their accepted definition to be and then building your castle upon your own notions.

 

Unfortunately that impedes communication with others.

 

Why not simply use accepted definition of terms and if you introduce a new notion then use a new term?

 

The most famous system of logic/maths in history is Euclid's 'Elements'

 

It is based on 23 definitions, 5 axioms (he called them propositions) and 5 'common notions', all stated without proof.

 

The very first definition is

 

1) A point is that which has no part. (This is taken to mean a point is indivisible into parts.)

 

By definition 15 we find out that

 

15) A circle is a plane figure contained by one line such that all straight lines falling upon it from one point, among those lying within the figure, are equal to one and other.

 

To recap.

 

Numbers and points do not have dimensions.

 

There are multiple definitions of the terms number, point and dimension already in use.

 

If you wish to pursue your 3/4 idea perhaps you should compare with Mobius strips and Peano curves? They have allied interesting properties.

[studiot]

With regard to your model, it is an interesting idea.

 

You can certainly map an octant of 3D space to 3 orthogonal planes, formed as you have shown from folded paper.

And as you say all points in each plane can be reached by a radius vector from the origin and 0< an angle <90, which makes 270 in all.

 

This is similar to the process often used called normalisation for a graph or the formation of nomograms.

The normalisation process uses the fact that there are as many points between 0 and 1 as there are between 0 and infinity.

 

Note that your map only covers 1/8 of 3D space, although if you introduce signs you can cover 2/8.

 

Since you like models have a look at

 

Mathematical Models by

 

Cundy and Rollet

 

They show how to make some fascinating paper, string, rod, glass and other models in mechanics, geometry, non euclidian geometry (Mobius strips Klein bottles etc), tesselations, knots, fancy curves, fractals, logic and more fields of maths.

Edited November 17, 2013 by studiot