What Is The Energy of a Point In Space?

[abrogard]

Please, moderator, just delete this question if it is too silly.

I am wondering what is happening in anyone point in the SpaceTime field. That's what it is called, I think?

Because my simple understanding is that we see colours/light because of propagated electromagnetic radiation at certain frequencies.

And certain frequencies all have certain energies.

So if I see green light, say, then electromagnetic radiation of a certain energy was passing through a point in space directly before my eyes and then entered my eye. Right?

Now that point before my eye can be observed from an infinite number of places - i.e. anywhere on the surface of a sphere around that point.

And it were then the colours, i.e. the energies, perceived would be different more or less for every single observation.

So the energy you'd ascribe to that point would appear to be different for each observation post.

Could be dramatically so. From one direction - white light. From 180 degrees away it could be black.

What is going on in these points in spacetime? How are we to think of it?

 

[Dale]

From your question it sounds like you are interested in the energy of the electromagnetic field at a point rather than the energy of spacetime at a point. Assuming this is so, then the energy and momentum density and fluxes are described by the electromagnetic stress energy tensor.

https://en.m.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor

The stress energy tensor captures the idea you mentioned about how the same field looks from different directions and at different speeds.

 

[osilmag]

The energy of a photon is E=hf where h is Planck's constant and f is the frequency. So, that would be the energy of the green light you mention in your post.

 

[jorgenspinozasalad]

Please, moderator, just delete this question if it is too silly.

I am wondering what is happening in anyone point in the SpaceTime field. That's what it is called, I think?

Because my simple understanding is that we see colours/light because of propagated electromagnetic radiation at certain frequencies.

And certain frequencies all have certain energies.

So if I see green light, say, then electromagnetic radiation of a certain energy was passing through a point in space directly before my eyes and then entered my eye. Right?

Now that point before my eye can be observed from an infinite number of places - i.e. anywhere on the surface of a sphere around that point.

And it were then the colours, i.e. the energies, perceived would be different more or less for every single observation.

So the energy you'd ascribe to that point would appear to be different for each observation post.

Could be dramatically so. From one direction - white light. From 180 degrees away it could be black.

What is going on in these points in spacetime? How are we to think of it?

What You are asking is really undefinable, a single point is really undefined in terms of energy or curvature, like asking about the volume of a 1d line, you have to add up infinite points to make up anything that can be described physicly in any way, you can however look at neigboring points at any scale to determine energy or curvature. Doing it without a framework for assigning meaning to the values is meaningless. What people mean when they talk about zero point energy is really the intrincic wobble or energy diffrances that remain when you remove all the knows «forces» from the past, aka cancalation of momentum/velocity for exsample. What remains is the energy precent when all known imputs have been canceled out, in the case of temprature(momentum), or defining position to be constant over time you might get really a meassurment of how to enviorment fluctuates relative to what you think is a well defined possition. In the danger of sounding like langham i leave out maths here because QM is really only well defined in regimes where this flux in position(t) roughly cancel out to 0. And thus using the framework of superposition won't work of nonstatisticly sound definitive position that exsist within the plank scales. Anyway considering the actuall complexity of the vacuume if you believe in a causal structure of the uncertainy principles of qm, then you have to admit that complexity grows along with emergent structure. As an exsample imagine these kind of fluctuations on a makroscopic scale(spacial and time scales) the Earth might wobble like crazy and in a matter of fractions of seconds mass distobutions would be shuffled like a snowball hitting a baloon, both waveform and masses would be distorted and complexity would rise exsponentially, this being less probable in the real world than thomas the tank engine randomly coming into exsistance to help you pass theology. Its probably safe to say that the vacuum has to have some complex form of entropy. This turned cooko quite fast, the conjecture is probably true in some sense. Points however are completely dimensionless, just abstract tools in some sence, you can use them but don't think of them as real objects. I claim no absolute or definitive knowlage of vacuum complexity but its really just an undifined way of saying what you see is causally connected in some way to the cause whether it is zero point energy or uncertainty in general.

IMO: i think the enegry absolute value of a section of space is infinite in the sense that all values that cancel completely doesn't effect anything at all our universe could ever meassure, and in a larger landscape of minima/maxima of effective vacuum energies require it to be consistant for all values and geometries of causality possible in all definable regions of space. This however doesn't really affect the mathematical structure of any 1 set of laws describing some relative reality that is aproximatly measurable. This absolute fact leads to the notion that avsolute facts arnt provable within any system, just because they contain all truth just 1 time for each absolute truth. In essense the death of philosophy in science is real if you don't have truly ininite compatence, information and capacity to understand, that in itself is not selfconsistance, summed up we cannot ever know everything, the relation between knowlage and the physical analog is itself complexity and can never be fully defined, thos is true for math as well, tall to gödel.

 

[jerromyjon]

And it were then the colours, i.e. the energies, perceived would be different more or less for every single observation.

That is explained by Doppler effect for the most part. Relative velocity though spacetime is what changes the observed energy of waves, which is what electromagnetic radiation solely consists of.

From one direction - white light. From 180 degrees away it could be black.

Visibly "white" light contains many colors of the rainbow, moving towards the source would "blueshift" the light towards the ultraviolet, x-ray, gamma ray invisible range and moving away from the source would "redshift" the light into the infrared, microwave, radio wave end of the spectrum. There is no such thing as "black" light, only that which cannot be seen by our eyes.

 

[abrogard]

From your question it sounds like you are interested in the energy of the electromagnetic field at a point rather than the energy of spacetime at a point. Assuming this is so, then the energy and momentum density and fluxes are described by the electromagnetic stress energy tensor.

Unfortunately I am so ignorant I don't know which of the two I am interested in.

Let me rephrase it:

How is it possible that there are different energies existing at the same time at one point?

Why do I think there are?

First: I think energies exist because I think an electromagnetic wave is energetic light we perceive is such a wave hence at that place in our eye where the wave excites a cell in order to initiate 'seeing' there's energy. That's in the eye. But, obviously that same packet of energy ( a photon? ) a moment ago was in a position just before the eye, outside the eye.

Second: At that position 'just outside the eye' one could imagine an observer - an eye - looking through that position from any of an infinite number of places.

Third: At each of those places they'd see something different. That signifies 'seeing', or 'absorbing' or 'interacting with' different energies.
Different even to the extent of seeing white in one direction: the full sum of light energies, and black in another: apparently zero light energy.

Fourth: Hence at that position we must conclude all these different energies exist at once. In the extreme, black and white, none and all.I just thought I would make an attempt to clarify what I am talking about, if it makes any difference.

You have probably provided the definitive answer but unfortunately I'm unable to comprehend it. I followed the link and it is way beyond me.

Jorgenspinozasalad seems to be saying something interesting but I don't understand it, either. Unfortunately. Lots of 'unfortunately's ' here aren't there? :)

 

[Dale]

First: I think energies exist because I think an electromagnetic wave is energetic light we perceive is such a wave hence at that place in our eye where the wave excites a cell in order to initiate 'seeing' there's energy. That's in the eye. But, obviously that same packet of energy ( a photon? ) a moment ago was in a position just before the eye, outside the eye

I agree. This is the energy of an electromagnetic field (which is well defined), not the energy of spacetime.

Second: At that position 'just outside the eye' one could imagine an observer - an eye - looking through that position from any of an infinite number of places.

Third: At each of those places they'd see something different. That signifies 'seeing', or 'absorbing' or 'interacting with' different energies.
Different even to the extent of seeing white in one direction: the full sum of light energies, and black in another: apparently zero light energy.

It doesn’t matter where you are located the energy is the same. You may not absorb it yourself if your detector is poorly positioned, but conservation of energy requires that it be there anyway. So, no, looking in different directions does not change the energy.

However, the energy will be different in different reference frames. So if you observe it from different states of motion.

Fourth: Hence at that position we must conclude all these different energies exist at once. In the extreme, black and white, none and all.

At that position you would indeed conclude that all of the different energies exist at once. Not based on different directions, but based on different reference frames.

Energy is conserved, but it is not invariant.

 

[abrogard]

However, the energy will be different in different reference frames. So if you observe it from different states of motion.

Thanks for the reply.

This seems to me to suggest we are talking about a static detached thing which can be perceived from difference 'reference frames'.

'reference frame' just meaning 'different places' or 'different angles' to the layman, right?

But the essence of what I'm querying is that the location in question is wholly a part of a process with its own definition. i.e. a wavetrain of blue light energy. From one direction. But from another a wavetrain of red light energy. And so on.

Forget the observer. Not really what I'm asking. Just my way of trying to clarify what I'm asking.

Just consider that location. What is happening there? What IS there?

Can I impose on your patience just a little more and try one more illustration?

How about a railroad crossing for the location?
And a train of railroad cars - timber wagons - going through the crossing.
At anyone time there's a timber wagon on the crossing.

Now the essence of this puzzle (to me) is that at anyone time there's an infinite number of different trains going through that crossing.

There are flatbeds and refrigerated cars and passenger trains and every variety in between all simultaneously going through that crossing.

It is not me perceiving the fabled spaceship moving at the speed of light somewhere over there, etc, etc... with different appearances according to my reference frame.

It is not me looking at the train head on in one instance and then looking at it depart in the other.

It is not really me looking at it at all.

It is what is happening. We're told. Aren't we? Wavetrains are proceeding through that point (any point whatever, anywhere in the universe) from an infinite number of directions. Each wavetrain subject to clear definition down to measurement of energy. I think. Right?

It may be that Planck lengths and such prevent infinite division. That's fine. I don't mind. In that case talk about what's happening at the smallest length. The smallest part of that wavetrain. It gets a definite energy ascribed to it. And then we can say then, as though it occupied the whole of that area, the the energy there is that.

But then we have to consider there's an infinite number of trains passing through that place. Likewise 'Wholly' occupying that area.

What's the upshot of all that?

 

[Dale]

reference frame' just meaning 'different places' or 'different angles' to the layman, right?

No, meaning different speeds or directions of motion. Not different places or different angles. The energy is the same under rotations or translations, but it changes under boosts.

Just consider that location. What is happening there? What IS there?

The EM field. It is a tensor field with one given tensor value at that location.

Now the essence of this puzzle (to me) is that at anyone time there's an infinite number of different trains going through that crossing.

This is incorrect. There is one train (EM field). It can be described many ways (reference frames), but there is only one train.

It is what is happening. We're told. Aren't we?

No, you have misunderstood somewhere

 

[abrogard]

No, you have misunderstood somewhere

I have misunderstood? There are not 'wavetrains' of electromagnetic energy propagating through space?

 

[Dale]

I have misunderstood? There are not 'wavetrains' of electromagnetic energy propagating through space?

At a given point in space the EM field has one definite (tensor) value. That tensor value may be due to a propagating EM wave from one or more sources located elsewhere, but there is still only one (tensor) value of the field at that point.

i.e. a wavetrain of blue light energy. From one direction. But from another a wavetrain of red light energy. And so on.

I missed this earlier. This does not happen. If a wave of light is blue in a given reference frame then it is blue regardless of the orientation. It is only if you boost to a different reference frame that the color may Doppler shift in the other frame.

 

[abrogard]

This does not happen.

It does happen, what I'm talking about. So there's the problem between us. You're not understanding what I'm thinking of. Doubtless because my thinking is so unmathematical and so untutored it'd be unreal in your world, I guess. Causing me to ask questions that are too 'wrong' to be answered.

Let me try this:

I'm talking about what happens at a point when we shine a myriad of different coloured lights through it from a myriad of different locations all around it - a sphere.

Which is what is happening everywhere in the real world right now.

Let me ask this: does the energy of a point reflect the energy of the EM wave passing through it?

 

[Dale]

I'm talking about what happens at a point when we shine a myriad of different coloured lights through it from a myriad of different locations all around it - a sphere.

OK, it would have been nice if you had led with that. I thought you were talking about one emitter with detectors looking at the same light from different angles.

Which is what is happening everywhere in the real world right now.

Not everywhere, but I get your meaning.

Let me ask this: does the energy of a point reflect the energy of the EM wave passing through it?

Yes. Again, even with your multicolor spherical emitter, there is still only one EM field value at each point. It does not matter how many sources with how many different colors converge on that point, the field still has one unique (tensor) value, as does the energy.

 

[abrogard]

Yes. Again, even with your multicolor spherical emitter, there is still only one EM field value at each point. It does not matter how many sources with how many different colors converge on that point, the field still has one unique (tensor) value, as does the energy.

I don't have any multicolor spherical emitter. That's the nature of reality is it not? There is no point in reality that is not completely surrounded by more reality, is there? And all of reality emits radiation, right? Hence every point can be considered as at the centre of sphere being 'illuminated' by radiation.

And: back to my question.. I am asking what happens at that (ubiquitous) point?

Because you look at it from here and it appears black. No energy, let us say. You look at it from there and it appears white. Mucho energy, let us say.

What energy does it actually have?

Put it another way: How is it possible for every point to transmit radiation at all frequencies from all directions simulltaneously?

Put it another way: How can an infinite number of photons coexist at every point in the universe?

Put it another way: When a packet of energy meets an atom it causes an electron perhaps to rise to a higher energy level. According to the energy in that arriving packet. What energy arrives from these points? If white light then the energy of white light, yes, we know. If blue light then the energy from blue light, yes, we know. The point is why? That blue light was present in a point location along with much other radiation from all around and that total package - the EM field at that point? - delivers its total energy to the atom, surely, not merely the portion due to the blue light component?

The question is: What is happening in anyone point... ?

 

[davenn]

I don't have any multicolor spherical emitter.

well as an example, you said you did ...

I'm talking about what happens at a point when we shine a myriad of different coloured lights through it from a myriad of different locations all around it - a sphere.

And: back to my question.. I am asking what happens at that (ubiquitous) point?

Because you look at it from here and it appears black. No energy, let us say. You look at it from there and it appears white. Mucho energy, let us say.

What energy does it actually have?

Dale has already answered that several times ... how many times need it be repeated ...

there is still only one EM field value at each point. It does not matter how many sources with how many different colors converge on that point, the field still has one unique (tensor) value, as does the energy.

Put it another way: How is it possible for every point to transmit radiation at all frequencies from all directions simulltaneously?

Put it another way: How can an infinite number of photons coexist at every point in the universe?

photons are not little balls/bullets shooting through space ... they don't even exist till detected by something and absorbed eg...
you eye, a radio antenna etc

As you , Dale and I sit at our computers typing all this stuff, we are "swimming" in electromagnetic radiation coming from all directions
cell phones, TV signals, AM and FM radio signals, IR Visible and UV light and radio signals from the sun and other galactic sources to name just a few.
These EM waves are passing through you all the time from all directions

Put it another way: When a packet of energy meets an atom it causes an electron perhaps to rise to a higher energy level. According to the energy in that arriving packet. What energy arrives from these points?

If it has enough energy to do so

electromagnetic energy

If white light then the energy of white light, yes, we know. If blue light then the energy from blue light, yes, we know.

you don't seem to understand or maybe you don't know/realise that light is just a part of the EM spectrum that the eye happens to be sensitive to

The point is why?

why what ??

That blue light was present in a point location along with much other radiation from all around and that total package - the EM field at that point? - delivers its total energy to the atom, surely, not merely the portion due to the blue light component?

forget about calling it blue/green/red or any other colour light ... colour is only assigned by our eye and brain. And what your eye/brain discriminate as say blue, could be totally different to what my eye/brain interprets the colour as, if I was, say, colour blind. What you see as blue may be red for me.
Before your eye detects a certain wavelength of EM and interprets it as a colour ... colour DOESNT EXIST
I repeat just to let it sink in for you
Before your eye detects a certain wavelength of EM and interprets it as a colour ... colour DOESNT EXIST

The question is: What is happening in anyone point... ?

Honestly ... you really are not going to ask that again are you ?? Dave

 

[PeroK]

@abrogard it's not that different from what happens with water waves, in a busy swimming pool, say. There are waves propagating in all directions, and each point on the surface of the water has a displacement equal to the sum of all the waves passing through it at that time.

Also, if you imagine a light wave traveling through space in front of you. If you are not in the path of that radiation and not detecting it in any way, doesn't mean that you can conclude that it isn't there and that region of space has no energy.

If there is another light wave crossing that path, which you do detect, then you are detecting the wave from its source. And there is no contradiction with what anyone else sees.

You could google for wave "interference" for more info.

 

[abrogard]

displacement equal to the sum of all the waves passing through it at that time.

Thank you for your gentlemanly answer, Perok. It is much appreciated.

Yes: '...the sum of all the waves.. '

Are you saying that is what happens at each point in space? That is what I would expect.

In which case I'd expect an energy sum greater than that merely of the 'main' source. You can understand, I feel, if I refer to that 'main source' as a ray of coloured light for the sake of illustration (no pun intended)?

 

[Dale]

And: back to my question.. I am asking what happens at that (ubiquitous) point?

Because you look at it from here and it appears black. No energy, let us say. You look at it from there and it appears white. Mucho energy, let us say.

What energy does it actually have?

I have answered this several times now. It is getting frustrating repeating the answer. Once again: Regardless of how many sources are converging on a single point, the EM field at that point adds up to a single (tensor) value at that point. That single value of the field determines a single value of the energy at that point. It does not matter what direction you look at it.

Put it another way: How is it possible for every point to transmit radiation at all frequencies from all directions simulltaneously?

Maxwells equations are linear.

Put it another way: How can an infinite number of photons coexist at every point in the universe?

Photons are bosons

the EM field at that point? - delivers its total energy to the atom, surely, not merely the portion due to the blue light component?

Not typically, no.

The question is: What is happening in anyone point... ?

And I have answered it multiple times. Please stop repeating the question and instead respond to the answer: At anyone point the EM field has a single (tensor) value. That single field value corresponds to a single value for the stress energy also.

 

[abrogard]

Please stop repeating the question

Please stop 'replying' to my questions. Your responses do not help me. You are increasingly irascible about my postings. The interaction is simply not helpful.

I post a question in good faith and I exercise politeness, restraint and refrain from any personal intimations, slights, aspersions, untoward inferences or rudenesses.

I ask no one in particular to involve themselves in interaction with me.

I repeat the question because it is not answered.

There is a single value is no answer to a question 'what happens' wherever, whenever the question. What happens when a speeding car meets a pedestrian? There is a single value. A non sequitur.

I am awaiting a response from Perok, is all.

 

[Dale]

I post a question in good faith

And I also posted the answer in good faith. Repeatedly! Presumably your goal in asking the question was to obtain the answer, and so I provided it.

I exercise politeness,

Ignoring answers and telling people to stop replying is not terribly polite. It is unfortunately inherent to the medium of communication that it is easy to become unintentionally offensive. I think that has happened to both of us here. I have become offended by your dismissal of my answer, and you have become offended by my expressed irritation. Let’s both accept the offense as unintended and concentrate on the answer.

There is a single value is no answer to a question 'what happens' wherever, whenever the question.

Ok, with this response we can make progress.

So the question is about what happens to the energy at a point, the idea being that the energy perhaps has multiple values at that point depending on the direction. The energy depends on the field, and the field has a unique value at that point. Therefore the energy has a unique value also, and does not depend on the direction.

Edit: the stress energy tensor, which depends only on the field tensor, includes information not just about the energy at each point, but also about the energy flux. Detectors oriented in a particular direction would only receive energy flux in that direction. So the stress energy tensor both describes the energy at the point as well as the directional energy flux that you are confusing with the energy.

 

[Khashishi]

A single point is not enough to have electromagnetic waves. The physics of electromagnetic waves are described by differential equations, with both space and time derivatives. The wave behavior at a point, such as the propagation directions, doesn't just depend on the value of the field at the point, but the values in a small neighborhood around the point. You cannot simply treat space as a set of independent points.

 

[Khashishi]

Because you look at it from here and it appears black. No energy, let us say. You look at it from there and it appears white. Mucho energy, let us say.

Hmm, you seem to be confusing what you see with what is actually there. You just see the light that enters your eyes. If an object is emitting more light in one direction than another, then it will look brighter from one direction than another. The object itself is losing energy equal to the total amount of light it gives out, not just the part you see.

 

[jerromyjon]

Now that point before my eye can be observed from an infinite number of places - i.e. anywhere on the surface of a sphere around that point.

So you think that there are infinite "degrees of freedom" around any such point you choose to specify, that have a unique point of view and can make unique observations of said point? While there are infinite values that can be observed, think about the simplistic view that every wave has a unique source and you are either looking and moving towards it or looking away and moving with it, which never happens.
In my simplified view of the situation there are only 2 relevant opposing points of view and an infinite spectrum of possibilities.

 

[abrogard]

A single point is not enough to have electromagnetic waves.

I surmised this might be the case. See post #8 I think it was.

you seem to be confusing what you see with what is actually there

No, I am not confused about that. I try to talk about a point in space. Any point in space in the centre of a sphere from which come radiations. Probably from all directions but perhaps not. Perhaps there's 'empty' areas of the universe that seem to emit no radiation. Okay. Doesn't matter to me. The point is that the point location receives radiation from a myriad of sources all around, all the time.

So what's happening at that point?

Please see post #1. I have reread it numerous times and thought how to make it clearer, simpler but finally I believe it is clear and simple enough.

If it cannot be understood in this forum then it is obviously in the wrong place. I'd be happy to move it. That'd be affected by having the moderator delete all posts and I'll raise the question elsewhere. Fine with me.

:)

 

[Dale]

The energy flux at that point depends on the direction, the energy does not.

Instead of a point, consider a small cubic volume with faces on the positive and negative x, y, and z directions. Energy can flow across the faces, that is energy flux. If more flows in than flows out then the energy inside increases. We can do the same with momentum. Now, take the limit as the size of the cube goes to 0 and we have the stress energy tensor at a point.

 

[Khashishi]

Please see post #1. I have reread it numerous times and thought how to make it clearer, simpler but finally I believe it is clear and simple enough.

Well, I think you are confused about multiple things and therefore it's hard to pin down what's bothering you. Perhaps you aren't able to accept the fact that there can be more than one thing in the same place at once. The energy at that point is just the total of all the kinds of energy at that point.

You are probably aware of the relationship between frequency and energy of a photon: $E = h \nu$, but you probably have some misunderstanding of what it means. This is not the energy of the electromagnetic field at a point, but the energy of a single photon. A single blue photon has more energy than a single red photon, but you can have red light with more energy than blue light. That's because a beam of light can have many overlapping photons. The energy of the electromagnetic field in a region can be any positive value. If you don't understand classical electromagnetism, then don't worry about what a photon is. That is a more advanced topic.

You can overlap light beams with no problem. Just take two flashlights/torches and point the beams across each other. The electromagnetic field is just the sum (aka superposition) of the contributions from each beam. Since you are talking about single points in space, you should be talking about energy density, rather than just energy. The energy density in the place where two beams cross is on average the sum of the energy densities if you just had each one beam separately.

If you can understand how one beam works, then you can understand how two beams work, since it's the same thing. If you don't understand how two beams work, then you actually don't understand how one beam works. When you ask, "what is happening at the point?" that's such a vague and open ended question. What kind of answer are you looking for? Do you want the equations that describe how to compute what is happening? (Look up Maxwell's equations)

 

[jorgenspinozasalad]

Any point in space in the centre of a sphere from which come radiations.

Well you are asking about a point, when you have to ask about a volume for any of the theories of how space, light or even causality makes sense, if you want to know about a volume of space infinatesally small, much smaller than the planc scale( approaching a point sort of) you would need a theory describing the materiality of the vacuum, and that doesn't exist yet. You can think in terms of wast numbers of low energy particles or somthing like that all moving in all directions everywhere and cancel out mostly, in terms of meassured wavelenghts of elecrtomagnetic radiation you won't ever have enough photons so that they would do anything else than slip past each other or converge with a density that creates a black hole and get stuck long before you reach a single point, true for any energies if they where actually converging at a point, this black hole would radiate away very quickly.

 

[abrogard]

when you have to ask about a volume

Yes, I earlier referred to that aspect, referring back to my post #8.

So what happens in that volume?

No, I don't want to think of an infinitesimal space smaller than the Planck length. I don't intend anything revolutionary or extraordinary nor even abstruse or sophisticated.

I simply use the words 'a point' as labels for the smallest sensible volume to consider.

I have read that when you try to resolve space down to the Planck length just about all physics breaks down. Well consider that I'm querying at the smallest you can get before that happens.

 

[Dale]

Well consider that I'm querying at the smallest you can get before that happens.

See post 25

 

[abrogard]

Yes, I saw it, Dale.

 

[gary350]

How far is it from point A to point B?

 

[abrogard]

How far is it from point A to point B?

I've often thought about that. Seems to me you cannot have a 'point A and a point B' alone. Because as points there's no way to measure their separation.

A dimensionless point has no dimensions in itself. In a surroundings lacking dimensions there's nothing we can say in terms of dimension.

Given a third point we can start measuring. That third point could be part of the first point if it is given dimensions, thickness, height, width...

Then the distance from A to B becomes in terms of A the number of times you can divide the width of A into the interval.

Hence the need for three. Three is the smallest number.

 

[fresh_42]

I think it's time to close this thread as the contributions become more and more meaningless and tend towards unsubstantial speculations.

Thread closed.