Thread: Why angularity not regarded a dimension.

As with the ds/dt topic this is confusing again.

To me angle is a natural part of any dimensional system : x-y system two axis on themselves is not a defined euclidean system. Only when a ninety degree angle is defined the axis become a euclidean system. That adds another dimension to it ; literally and figuratively speaking as far as I,m concerned.

In official math and fysics cirkles this is different. 2 pi for a cycle as a salto can,t be expressed then (related to time). A real axis of a clock or machine can,t rotate then but only move in cirkles.

Maybe someone knows the background for this ? Has this always been or is it a new rule or something ?

I never knew this just read it today after a search on internet.

Pi is a ratio between diameter (a length) and circumference (also a length) and therefore has no units. Any time its used to measure anything other than angles, it'll be multiplied by something with units.

Angles don't have dimensions either. Consider what happens when you scale an object up. Lengths increase linearly with scale. Surface area increases with the square of the scale. Volume increases with the cube. Angles don't change.

Angles are important to the different kinds of geometry, but defining a 90 degree angle doesn't make something Euclidean. Almost any geometry will have 90 degree angles.

Offcourse pi is the ratio between circumference and diameter for instance for a round plate. Units for length to length ratio is m/m it,s a different ratio then masses to masses (M/M) newtons to newtons (N/N) etc. There is not a unit for the ratio but for the parts the ratio relates to there are units. But a rotating line has no cirkleshaped circumference it rotates with an anglespeed.

pi/sec pi does not stand for just a number pi (as n=pi) but for pi radials in such a case. (at least to me it does which would make me unconventional but that,s not necessarily a problem to me)

It is because it,s not regarded to be a unit that it is common to write : 3,14/sec or pi/sec in such a case. If the convention was different you would write : 3,14 rad/sec. Or use other units for angle like degrees.

You can't explain/proof a convention for math and notation with math and notation that follows that convention.....that,s impossible.

Allthough the ratio between circumference and diameter for a plate (or a round shaped something in general) is a ratio between lengths ; cm/cm it follows from radials for instance when you would draw a cirkle round a point (0,0) in an euclidean (x,y) system the full trajektory for the angle r makes to both axis is 2 pi rad.

It just seems more logic to me and there is nothing I can think of - apart from not being conventional - against regarding angle as a dimension working with such systems and using rads or degrees as units. I can think of arguments for it :

(repeat ) with only the axis given
without a particular angle (1/2 pi rad)
me or no-one could ever make-up an (x,y) axis system on paper you need that angle for making such a system....to me that makes it an impliccable as a dimension to it.

And also everything projekted in it. A coffeepot or constructive element for a machine, only with the position (x,y) the koffee pot could have all kinds of angular orientations....not defined it can,t be drawn related to the axes.

Is a Arc a dimension or just a half-piece of a circle's dimension?

Originally Posted by FuturePasTimeCE

Is a Arc a dimension or just a half-piece of a circle's dimension?

It depends on what you are doing. You may need to study some mathematics to understand what "dimension" means.

Do not let Ghrasp confuse you. Please ignore him. He has no idea what he is talking about -- he never does.

As I understand it, dimensions are the most basic parameters required to describe the position of a point relative to that of another. A single angle is largely arbitrary and is automatically oriented in relation to other parameters, like the primary axes.

The angle is just a natural requirement of choosing the most basic coordinate system, as any line can be described in relation to them. If any other angle was chosen, then you would not get correct calculations of area or volume.

@Ghrasp, two things. One, N/N and m/m and s/s are all dimensionless and can be multiplied with other units without changing them.

Second, I didn't say you didn't need angles to make sense of geometry. I said having a 90 degree angle doesn't uniquely describe much of anything (beyond perpendicular lines). All continuous geometries have 90 degree angles, whether it's spherical, hyperbolic, Euclidean or something else.

Originally Posted by KALSTER

As I understand it, dimensions are the most basic parameters required to describe the position of a point relative to that of another. A single angle is largely arbitrary and is automatically oriented in relation to other parameters, like the primary axes.

The angle is just a natural requirement of choosing the most basic coordinate system, as any line can be described in relation to them. If any other angle was chosen, then you would not get correct calculations of area or volume.

no

Dimension is different from calculation of distance, volume etc.

To get into this requires som real mathematics that you would do well to learn from a text.

Suffice it to say that not all coordinate systems are orthogonal, and the number of parameers required is the dimension.

Do not let Ghrasp confuse you. Please ignore him. He has no idea what he is talking about -- he never does.

Ignorance ; "always afraid to be ignored, always there ready to tell who and what to ignore."

Originally Posted by KALSTER

As I understand it, dimensions are the most basic parameters required to describe the position of a point relative to that of another. A single angle is largely arbitrary and is automatically oriented in relation to other parameters, like the primary axes.

The angle is just a natural requirement of choosing the most basic coordinate system, as any line can be described in relation to them. If any other angle was chosen, then you would not get correct calculations of area or volume.

What is "just a natural recquirement" ? It,s required then and with other angles then perpendicular for other systems then these are also recquired for these systems I suppose only different angles but angularity stays an intrinsic part to distinguish these systems from other systems.
Most cases the paper you construkt such a system on will allready have rectangular sides making it seem more natural maybe as you can draw the axis nicely parallel to the sides. But in essence you (Kalster) are not arguing against the possibillity to regard angularity as dimension. You also confirm that a 1/2 pi / 90 degree / perpendicular angle is essential and intrinsic to such dimensional systems before you can even think of putting points or figures or things to it.
To make a rectangular paper like a page of a book the length of the sides only is also not enough.

It,s intrinsic part of position, orientation also when it,s no longer just about mathematical points but about things.

I,m not arguing that it,s possible to regard angularity not to be a dimension. I,m sure it will be. I,m just bring on the question what would be against it ? And ..possibly .... could be for it ?

The meaning for anglespeed would change because of it. Now anglespeed is m/s then it would be (rad*m/sec) put between brackets behind the outcome for anglespeed for instance. Rad /2pi /sec is frecquency hence anglespeed would become 2pi * f * r for a point at distance r from the rotational axis. (As a matter of fact this is exactly the math I learned to use programming a Lathe machine).

Putting units not regarded as dimensions behind brackets is common also. for example (Nm) meter is dimension but Newton is not. Degrees, radials are a unicque exception in this.

Another example is a cirkelsegment with an angle 1 rad.

C for circumference C=2pi/2pi*r= r. But r is only the length not telling anything anymore about the arc and how the line arcs (or bends).

For this expression (2pi/2pi*r) it is clear that 2 pi can,t mean the same both times it occurs in the formula and is notated. then I could just as well calculate with any random quotient. It would make no sense to produce first with 2 pi then annihilate immediately with adding it as a quotient to itself. To me it says C=2pi(rad) /2pi(n)*r (m)....C = C (rad*m)
That,s an arc to me then distinguishable from a straight line or different arc.
Only r it can arc in all possible ways, be pulled straight or rolled out to a straight line parallel to one of the axis (a projektion to one of the axis, the length becomes different) it,s no longer an arc in all these cases ; only it,s length the angularity as a dimension lacks.

MA straight line of the same length rads would become zero. It would result in a produkt : 0*r/2pi....hence length become zero. For an arc this is right it has zero length then. For a straight line

arc for anglespeed is change of arc offcourse to a line or axis not a static arc.

Then with a calliper on the paper d(A) = 0 with A for angle...meaning no arc and no length.

Originally Posted by DrRocket

Originally Posted by FuturePasTimeCE

Is a Arc a dimension or just a half-piece of a circle's dimension?

It depends on what you are doing. You may need to study some mathematics to understand what "dimension" means.

Do not let Ghrasp confuse you. Please ignore him. He has no idea what he is talking about -- he never does.

LOL. Neither do I. :-) just a mathematical skeptic.

Originally Posted by FuturePasTimeCE

I. :-) just a mathematical skeptic.

Mathematics is the one subject where there is no need for skepticism. Mathematics is based on rigorous logic and proof. That is proof, not just evidence.

Physics is based on evidence.

Mathematics is the one subject where there is no need for skepticism. Mathematics is based on rigorous logic and proof. That is proof, not just evidence.

Haha, that will help against scepticism. But this is not the issue here.

The thing is you can,t construkt pi on a euclidean space using a calliper while I can (only not with a calliper) it,s quite simple.

It goes like this :

A round piece of coloured painters chalk or charcoal is made with diameter = length = 1 (meter or cm whatever unit used for the spatial system, it can be dimensioned to it).

Laying the piece of chalk on paper on one of the axis (or anywhere parallel to one of the axis) there is a line where the chalk touches the paper.
Mathematical the paper defines a tangent surface for the piece of chalk.

Make a mark on the piece of chalk anywhere and roll it 2pi rad,s, a full cycle.

The plane construkted has a surface : pi D *D= Pi d^2.

and a length pi.

Originally Posted by Ghrasp

As with the ds/dt topic this is confusing again.

To me angle is a natural part of any dimensional system : x-y system two axis on themselves is not a defined euclidean system. Only when a ninety degree angle is defined the axis become a euclidean system. That adds another dimension to it ; literally and figuratively speaking as far as I,m concerned.

In official math and fysics cirkles this is different. 2 pi for a cycle as a salto can,t be expressed then (related to time). A real axis of a clock or machine can,t rotate then but only move in cirkles.

Maybe someone knows the background for this ? Has this always been or is it a new rule or something ?

I never knew this just read it today after a search on internet.

Then it's presumeably infinitive dimensions if you apply geometry.

We have to limit things and boil it down to the simplest terms, else we waste too much time with too many things that doesn't matter.

You know how many dimensions there are ? ......Always one more then you can think of...So maybe better forget thinking about it.

Originally Posted by Ghrasp

You know how many dimensions there are ? ......Always one more then you can think of...:-)

No, I don't buy this Superstring, M-Theory and Super Symmentry stuff about all these exotic dimensions, and this "proof" with time dialation isn't any proof, welll ..if so then ordinary water waves and heat also distorts time, just look at the "The Search for Longitude" project.

It was more like a humorous hint to these theories where each time a new dimension is added.....so obviously there is always one more.

For instance you can add a dimension to a point (not as applying a force or anything, just see a moving point)... if it moves the point has a direction. If angle is a dimension then that direction on itself becomes a dimension for the point. Hence a point - dynamically - has one dimension then. Only static, staring at it, it hasn,t.

Interesting site about dimensions (To me, dr rocket will sure be able to explain in his words and as usual without real arguments that it's rubbish, crackpottery etc) is the tgd site of Matti pitkanen.

http://tgd.wippiespace.com/public_html/

TGD is for topological-geometro-dynamics.

From a logically and logistically point of view, isn't it counterproductive theories?

Originally Posted by Ghrasp

It was more like a humorous hint to these theories where each time a new dimension is added.....so obviously there is always one more.

For instance you can add a dimension to a point... if it moves the point has a direction. If angle is a dimension then that direction on itself becomes a dimension for the point. Hence a point - dynamically - has one dimension then. Only static, staring at it, it hasn,t.

Would this mean a surface is 2+1 dimensional then or maybe 2*2=4.
And a fysical body would it be only 1+1+1+1 or 2^2 + 2^2=8 ?
Maybe it depends how you construkt it in real geometry or imaginary. I can imagine a surface as a cut out from paper (allthough the paper has a weight and thickness) but also as a trajektory for a moving line perpendicular direction to it,s length. This is a way of construkting. Printing a surface on paper or draw it with a piece of charcoal with it,s full length touching the paper. If the tangent moves perpendicular to the length over the paper you also get a surface but with a direction and as the paper a material existence hence 2^2*2= 2*4 dimensional ?

Four (for four dimensions) comes back in this but as 2^2 to the fourth and the thickness of the layer multiplies it with two.

Statical as a secquency of pictures a dimension only would add one up but dynamically ?

Interesting site about dimensions (To me, dr rocket will sure be able to explain in his words and as usual without real arguments that it's rubbish, crackpottery etc) is the tgd site of Matti pitkanen.

http://tgd.wippiespace.com/public_html/

TGD stands for topological-geometro-dynamics.

Even though I can,t understand even half of it it is at least intruiging to me (but string - and m-theory also at least intrigue me)

A point is 0-dimensional. An angle is 0-dimensional. If the point is moving, it's velocity vector can be any dimension greater than 0, but the point is still 0-dimensional.

Also, perpendiculars aren't what's fundamental in geometry. Parallels are. There's nothing about a 90 degree angle that's particularly special. 90 degrees is 90 degrees in any geometry, but parallel lines are different in different geometries.

What,s parallel ?..... It,s 0 pi or one pi angle so you do regard angle as dimension here then ? You just do it naturally and think of it like that maybe...it,s just natural.
That,s sloppy thinking. One could just as well say "length is just natural", "everything is just natural" and we can stop doing math at all (and culture in general, "everything is just natural").

A point is 0-dimensional. An angle is 0-dimensional. If the point is moving, it's velocity vector can be any dimension greater than 0, but the point is still 0-dimensional.

A point having a vector there is no line (yet or anymore) if it moves in vacuum. Vectoriality is only indicated by a stripe. But the vectoriality is really part of the point in relation to other points (and lines, surfaces etc) you can,t seperate "vectoriality" and point like that. Maybe you can but I would not advice it.
direction/ angularity is part of a vector and the point if only determined by an observer (other line with an angle to the vectoriality of the point otherwise you can,t see the vectoriality, a point doesn,t get bigger or smaller dependant by distance as a dot does.

A line without vectorial (activ, moving) direktion still is not symmetric ; it connects one point to another. Connecting A and B and A is not B or vice versa they exist in a larger context if only because being observed. It,s length also gives it angularity to me and to my surroundings.


You,re explanation (magimaster) you mix things up ( or try to ?) between what is and what is regarded to be. Explaining to me what I allready new and learnt long time ago. : a point is regarded 0-dim an angle is regarded not to be a dimension. Only difference is you leave the "regarded" away. This is all simply on some agreement (or give me real/mathematical reasons). Explaining it this way is patronizing and I,m not asking to be patronized. Ideas and conclusions are all made within math as a system with these agreements working as axioms...Offcourse you come to conclusions then but these are not arguments then for these axioms : "point is 0-dimensional", "angularity (vectorial or position) is not a dimension".


I can,t proof or disproof a point is 1 dim (dr Rocket made that argument allready and it,s not about that). I can,t proof or disproof these things with math because they funktion as axioms within the math. But I also don,t have to for that same reason. The only possible approach in such cases is change the axioms, use other axioms and see what happens....Then you can compare between two familiar games and at least come to know the meaning and influence of the existing axioms better.

Comparing this way with an open mind when problems arise it,s not just testing other axioms but also existing axioms. Learning another language can help learning about you,re own language...the other language is not English then maybe but still a language.

Ghrasp, this is math. There are only definitions and things provable from those definitions.

The definition of a dimension, in this sense, is the number of different numbers that need to be specified to describe what you're trying to describe. A point just is, it doesn't need any dimensions to specify it. A direction needs some specification, but exactly how many degrees of freedom it has depends. A direction on a piece of string just needs one number, so a string is 1-dimensional. A piece of paper needs 2. Space needs 3. Spacetime needs 4.

Angles, in that sense, are 1-dimensional. See the circle. However, they are still a dimensionless quantity, which is something serarate.

If you want to redefine point as something else, that's fine, but don't pretend that your definition is somehow better or more fundamental than the standard one.

And the reason everyone uses the standard definitions is that for centuries people have been doing useful stuff with those definitions.

Earlier in this topic, Dr Rocket mentioned that it depends what you do, wether angle is a dimension or not and warned to not get confused by me, that,s always good thing to warn for.

But the confusing part to me (as I see it) is that pi is sometimes an angle sometimes a ratio-number. When it,s an angle it,s allowed, possible to regard it as unit/dimension and as all units it can be exchanged with a different unit then.
That has to do with 2 pi radials being 1 cycle or cirkle , pi radials halve a cycle or cirkle. As a ratio pi can,t be exchanged with 180 degreesand can,t be seen as a dimension because it,s not an angle it,s a - universal - ratio.

So it seems a good thing in general using degrees now and then to distinct/notice when pi is for radials or for ratio.
Because I suppose it is clear that you can,t divide a ratio and a unit away to each other as "just numbers". That certainly is a guarantee for getting yourself confused.

Also 1 cycle is two full changes of angle. Halve a cycle something turns round (a line) completely for direktion.

Mathematical f=n/s n stands for an amount of full cycles 2 pi, 4 pi, 6 pi but geometrical to dimensional axes if you turn something with one side painted black the other white the frecquency at which black and white turn is 2n/sec.

I think that,s why the 2 comes in with 2pi*f*r. It can be seen as 2/s * pi.

In practice such type of "double frecquency" is used often and also using f for it : a trafic light jumping red-green-red -green for instance. The term frecquency can be used for how often the light changes colour not a full cycle from one colour to the same colour again as f for a wheel or a lathemachine from one angular position to the same again .