[Tom Bishop]

The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch. There may be varying explanations for why they touch.

It might have something to do with resolution. It might also be more than that. For instance, if we shine a laser beam at the point on railroad tracks where they appear to touch in the distance, the beam will widen and touch both of the tracks at the same time.

From http://wiki.tfes.org/Magnification_of_the_Sun_at_Sunset --

Beam Divergence

This phenomenon of enlarging rays is also seen in lasers. Supposedly "straight" rays of light will spread out when shining over long distances.



From the Wikipedia entry on Beam Divergence we read:

    "The beam divergence of an electromagnetic beam is an angular measure of
    the increase in beam diameter or radius with distance from the optical
    aperture or antenna aperture from which the electromagnetic beam emerges."

The light is broadcasted towards the small scene  in the distance and widens appropriately to cover that area it sees. Under a perfect universe the laser beam should only be able to touch only one of the tracks at a time when it reaches the destination. However, the beam is seen to widen significantly, easily covering both tracks and an area of landscape. It seems to suggest that, if the small laser beam diameter can cover a large area, the squishing of the tracks to a single point is more than a resolution limitation of the eye.

[thatsnice]

Well first of all, Greek math states that they DO appear to touch and that they don't physically touch, not the other way around.

No, they don't. We never learn that in Geometry class. The greek's geometry math assumes that we live in a continuous universe where resolution is infinite and where perfect circles could exist. The model says that two parallel lines should never touch in such a perfect universe.

That's because by definition, parallel lines will never intersect on any given plane. If we can show that over any finite distance two parallel lines will not touch, why wouldn't we assume that over longer distances approaching infinity that they would continue to not touch? They only seem to touch because we do not have eyes that can show extremely small distances, i.e infinite resolution.

Also, hey Totes, how you doin?

[thatsnice]

The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch. There may be varying explanations for why they touch.

It might have something to do with resolution. It might also be more than that. For instance, if we shine a laser beam at the point on railroad tracks where they appear to touch in the distance, the beam will widen and touch both of the tracks at the same time.

From http://wiki.tfes.org/Magnification_of_the_Sun_at_Sunset --

Beam Divergence

This phenomenon of enlarging rays is also seen in lasers. Supposedly "straight" rays of light will spread out when shining over long distances.



From the Wikipedia entry on Beam Divergence we read:

    "The beam divergence of an electromagnetic beam is an angular measure of
    the increase in beam diameter or radius with distance from the optical
    aperture or antenna aperture from which the electromagnetic beam emerges."

The light is broadcasted towards the small scene  in the distance and widens appropriately to cover that area it sees. Under a perfect universe the laser beam should only be able to touch only one of the tracks at a time when it reaches the destination. However, the beam is seen to widen significantly, easily covering both tracks and an area of landscape. It seems to suggest that the squishing of the tracks to a single point is more than a resolution limitation of the eye.

Beam divergence occurs because light shines in every direction from a light source. Because it's confined to emit from a one sided light, the "laser beam" will actually be a very narrow cone that spreads out because all of the light in this cone is moving in a straight line. It's not as if the light will start bending midway through transmission. Say instead that I took a single photon and fired it in a direction, never will that photon change direction unless it is reflected or refracted by other outside influence.

[TotesNotReptilian]

The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch. There may be varying explanations for why they touch.

It might have something to do with resolution. It might also be more than that. For instance, if we shine a laser beam at the point on railroad tracks where they appear to touch in the distance, the beam will widen and touch both of the tracks at the same time.

Yes, there are multiple reasons why they can appear to touch. Insufficient resolving power of the eye/camera (as has been stated multiple times) is one of them. Decreased clarity due to atmospheric scattering is another. But this is all irrelevant. The basic geometry deals with a perfect world. There is no such thing as perfectly parallel lines, or a perfectly clear atmosphere. But that doesn't mean the entire theory is useless. We can still make VERY accurate approximations with the theory. This is why it is important to be able to estimate how much error is possible in your calculation. How is the divergence of a laser beam going to cause significant errors in how we percieve parallel lines?

Going by your logic, we should never use math for anything, since nothing can be calculated 100% accurately. Is this what you are arguing for?

From http://wiki.tfes.org/Magnification_of_the_Sun_at_Sunset --

Holy Toledo, that page is so full of misinformation... please don't tell me you actually think light sources appear larger the farther you are away from them.

Also, hey Totes, how you doin?

Just chillin' yo

[rabinoz]

The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch. There may be varying explanations for why they touch.

It might have something to do with resolution. It might also be more than that. For instance, if we shine a laser beam at the point on railroad tracks where they appear to touch in the distance, the beam will widen and touch both of the tracks at the same time.

Are you really persisting with this? The simple fact is that the lines do not touch, they only appear to touch!

You claim "There may be varying explanations for why they touch." NO, they DO NOT TOUCH, they appear to touch!

The diverging laser beam means nothing at all. All the laser beam, a spotlight or the sun does is illuminate the railway tracks.

We are NOT looking at the laser beam and how much its beam diverges has absolutely nothing to do with the case.

There are cases when we are looking at the source of the light, such as:

• looking directly at the laser pointer (with appropriate protective eyewear)
• looking directly at the sun (again with appropriate protective eyewear)
• looking directly at stars at light.

In all these cases the apparent size of the object (measured as the angle subtended at the eye) is simple the (size of the object)/(distance to the object) so for these cases the apparent sizes would be at a "guess" (taking the laser pointer at 3 miles - where I claimed the railway lines converged):

• Guess 1 mm (0.025") diameter - apparent size (0.039")/(3 x 12 x 5280") = 2.1x10^-7 rad or 0.043" of arc, far too small to resolve.
• (Globe) Sun's diameter = 900,000 miles, Sun's distance = 93,000,000 miles, so apparent size = 900,000/93,000,000 = 0.01 rad or 0.55° - obviously visible.
• Say Alpha Centauri Estimated diameter = 1,100,000 miles, estimated distance = 4.37 ly or 2.57x10¹³ miles, so apparent size = 0.009" of arc, far too small to resolve.

I am using Globe figures as the FE supporters seem to have no idea of how big the stars might be.

As I posted before a simple test on whether lines meet is to simply go there and check it out, as with the TGV analogy in my previous post.
Of course this is only practical for achievable distances.

But the bottom line is that your diverging laser beam means absolutely nothing.
We all know that a laser beam has a finite (though very small) divergence angle - but so what, other light sources have much bigger divergent angles.
The sun's light shines over a very large angle (a full sphere), but this has not the slightest effect on the apparent size of objects we see!

[Tom Bishop]

Beam divergence occurs because light shines in every direction from a light source. Because it's confined to emit from a one sided light, the "laser beam" will actually be a very narrow cone that spreads out because all of the light in this cone is moving in a straight line. It's not as if the light will start bending midway through transmission. Say instead that I took a single photon and fired it in a direction, never will that photon change direction unless it is reflected or refracted by other outside influence.

The phenomena of beam divergence is certainly a curiosity, particularly because a laser beam is supposed to be straight due to photons between a series of mirrors and a glass amplifier to produce an extremely bright and straight beam of light. It may be argued that some of the photons are not straight, but then the divergence should have a central hot spot as the beam diverges.

[Tom Bishop]

Yes, there are multiple reasons why they can appear to touch. Insufficient resolving power of the eye/camera (as has been stated multiple times) is one of them. Decreased clarity due to atmospheric scattering is another. But this is all irrelevant. The basic geometry deals with a perfect world. There is no such thing as perfectly parallel lines, or a perfectly clear atmosphere. But that doesn't mean the entire theory is useless.

If their theory doesn't match observations it means the theory is wrong and must be modified or discarded.

Going by your logic, we should never use math for anything, since nothing can be calculated 100% accurately. Is this what you are arguing for?

If the math can't calculate things far away accurately, what reason is there to use that math for things that are far away?

From http://wiki.tfes.org/Magnification_of_the_Sun_at_Sunset --

Holy Toledo, that page is so full of misinformation... please don't tell me you actually think light sources appear larger the farther you are away from them.

Have you never seen headlights in fog?

[Tom Bishop]

The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch. There may be varying explanations for why they touch.

It might have something to do with resolution. It might also be more than that. For instance, if we shine a laser beam at the point on railroad tracks where they appear to touch in the distance, the beam will widen and touch both of the tracks at the same time.

Are you really persisting with this? The simple fact is that the lines do not touch, they only appear to touch!

You claim "There may be varying explanations for why they touch." NO, they DO NOT TOUCH, they appear to touch!

Maybe the photons coming from the tracks are hitting your eye in a way that the photons are touching (or getting as close as they physically can to each other). It has not been demonstrated that the effect is due to "lack of resolution". The human eye is incredibly sensitive. Tests have been done where the human eye can detect a single photon in a dark room.

A photon is the physical manifestation of that object at distance, after all. This society seeks to ask and explore such vexing questions, not mindlessly scream that "NO, they DO NOT TOUCH".

[CableDawg]

An experiment for Mr. Bishop, which is absolutely within the realm of possibility.

Items needed:

1.  Yourself and at least one trusted companion
2.  Stretch of train track of sufficient length (3 to 5 miles) in a straight line (no sweeps left or right)
3.  A means of communication between participants (cell phone, radio, semaphore, etc.)
4.  Markers of agreed upon type or method
5.  Two measurement devices agreed upon by you and your trusted companion as being accurate
6.  Two notebooks
7.  At least two pencils, possibly more

This is really quite simple.

To begin, both of you start at the same point in the middle (between the rails) of the track.  At this point both of you make note of the width of the track.  Have your trusted companion move forward along the track, keeping notes of the width of the track at regular intervals and placing a marker at each point measured, to the point where the rails appear, to you, to touch.  Communicate via your chosen method for your trusted companion to stop at this point and make note of the width of the track.

Now, you make your way forward, stopping to measure the width of the track at each point marked by your trusted companion, until you reach the point where you had your trusted companion stop.  At this point make a final measurement of the width of the track.

Once you have made your final measurement, compare your own measurements across the entire length you've traveled to see if your measurements have arrived at zero (as they would if the rails touched).  Once you have done this, compare your measurements to those of your trusted companion to see if their measurements have arrived at zero (as they would if the rails touched).

I'm only suggesting this because it seems to me that you appear capable of performing such an experiment.  Of course my perspective may be skewed.

[rabinoz]

Yes, there are multiple reasons why they can appear to touch. Insufficient resolving power of the eye/camera (as has been stated multiple times) is one of them. Decreased clarity due to atmospheric scattering is another. But this is all irrelevant. The basic geometry deals with a perfect world. There is no such thing as perfectly parallel lines, or a perfectly clear atmosphere. But that doesn't mean the entire theory is useless.

If their theory doesn't match observations it means the theory is wrong and must be modified or discarded.

Going by your logic, we should never use math for anything, since nothing can be calculated 100% accurately. Is this what you are arguing for?

If the math can't calculate things far away accurately, what reason is there to use that math for things that are far away?

From http://wiki.tfes.org/Magnification_of_the_Sun_at_Sunset --

Holy Toledo, that page is so full of misinformation... please don't tell me you actually think light sources appear larger the farther you are away from them.

Have you never seen headlights in fog?

You say "If their theory doesn't match observations it means the theory is wrong and must be modified or discarded." But you have never shown that their theory does not match observations!

You earlier said "The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch" and here YOU are simply mistaken! As I have tried to get across with parallel lines the simple fact is that the lines DO NOT touch, they do APPEAR to TOUCH.

In addition you blame poor Euclid and those other Greeks for everything. But our modern ideas are based on much more than that.
Take a look at: Geometry: A History from Practice to Abstraction, where we learn of:

• Ancient and Classical Geometries
  Aristotle (384-322) BCE, Euclid of Alexandria (325-265) BCE (maybe tacking on Playfair's Axiom), Abul Wafa al-Buzjani (940-998), Omar Khayyam (1048-1131) - yes even him, Nasir al-Din al-Tusi (1201-1274)
• Renaissance and Early Modern Developments
  The Painters' Perspective (Al-Haytham (965-1039)), Leone Battista Alberti (1404-1472), Piero della Francesca (1412-1492) etc, etc.
• Modern Geometries
  Girolamo Saccheri (1667-1733), Johan Heinrich Lambert (1728-1777), Adrien-Marie Legendre (1752-1833), Carl Friedrich Gauss (1777-1855),
  Nikolai Ivanovich Lobachevski (1792-1856) (finally getting into non-Euclidean stuff), János Bolyai (1802-1860), Eugenio Beltrami (1835-1900), etc, etc.

So, the Greeks Philosophers just laid a foundation.

Again as I said before you can either actually travel to where they appeared to touch, and of course you find that they did not touch, or at least do a "thought experiment". If you try to claim that they DO touch you need a good physical reason to make that claim.

Now, of course I do accept that we live in slightly non-Euclidean space as postulated in Einstein's GR, but the actual curvature in our region is smaller that minute. The sun's huge gravitational field only deflects light by about 1.8 seconds of arc, and half that is due to time dilation.

And again "Have you never seen headlights in fog?" Of course I have seen headlights in a fog, but have you seen the sun setting in a perfectly clear sky and appearing exactly the same size (maybe slightly larger - the Ponzo illusion) as it did at midday?

There is no way "atmospheric magnification" can keep the sun exactly the same size all day. Apart from anything else is happens every day. It can appear fuzzy and larger through cloud, but that is obvious when it happens and can happen in any part of the sky.

[Woody]

Tom at what distance does math no longer work?  1, 10, 100, 1000, 10000 or further?

Just your estimate where math fails to be able to give good estimates or right answers?

I can use that math to estimate ranges when I sail and had accurate results.

I used it recently to determine how much fuel a rounded tank on my boat could hold and the answer I got was verified to be correct when I put fuel in it.

I used it for celestial navigation and determined my position accurately with noon and star sightings.

Throughout my life I have used it in my career and personal life and consistently had it verified since it gave me the right answers that were verified correct when applied to projects, the amount of material needed, how much liquid something could hold, etc.

So when does math no longer work?  I ask because you seem assured you are right and must know at what distances it is no longer accurate.  My experience is that it does return the right answers. An experience I highly doubt that the majority of people would say the opposite.

Did you have evidence of being correct or is the only evidence you offer is if the Greeks were right you are wrong about the shape of the Earth?

[Ecthelion]

The phenomena of beam divergence is certainly a curiosity, particularly because a laser beam is supposed to be straight due to photons between a series of mirrors and a glass amplifier to produce an extremely bright and straight beam of light.

According to whom is a laser beam "straight"? How would mirrors an amplifiers result in parallel photons?

It may be argued that some of the photons are not straight, but then the divergence should have a central hot spot as the beam diverges.

Why? Do you refer to a focal point?

If their theory doesn't match observations it means the theory is wrong and must be modified or discarded.

According to Kant, Euclidean Geometry isn't a theory in the traditional sense. It's based on an a-priori understanding of space, not on observation.

If the math can't calculate things far away accurately, what reason is there to use that math for things that are far away?

Define " accurately".

Maybe the photons coming from the tracks are hitting your eye in a way that the photons are touching (or getting as close as they physically can to each other). It has not been demonstrated that the effect is due to "lack of resolution". The human eye is incredibly sensitive. Tests have been done where the human eye can detect a single photon in a dark room.

A photon is the physical manifestation of that object at distance, after all. This society seeks to ask and explore such vexing questions, not mindlessly scream that "NO, they DO NOT TOUCH".

Even if you want to call the photons emitted a "physical manifestation", the image they produce in you brain isn't.

[thatsnice]

Beam divergence occurs because light shines in every direction from a light source. Because it's confined to emit from a one sided light, the "laser beam" will actually be a very narrow cone that spreads out because all of the light in this cone is moving in a straight line. It's not as if the light will start bending midway through transmission. Say instead that I took a single photon and fired it in a direction, never will that photon change direction unless it is reflected or refracted by other outside influence.

The phenomena of beam divergence is certainly a curiosity, particularly because a laser beam is supposed to be straight due to photons between a series of mirrors and a glass amplifier to produce an extremely bright and straight beam of light. It may be argued that some of the photons are not straight, but then the divergence should have a central hot spot as the beam diverges.

It's not a curiosity, it's actually quite the opposite. It just appears that you don't fundamentally understand how light is emitted from a source. Much like sound, light, when produced from a source, moves in all possible directions from its origin point. The casing of a laser pointer isn't exactly dead focal, meaning that the "laser beam" is a uniform and extremely narrow cone. Because it is a narrow cone, it spreads out as it moves farther away from its point of origin. You cannot attribute this to light bending or not moving in a straight line, it's an invalid argument.

[Tom Bishop]

You earlier said "The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch" and here YOU are simply mistaken! As I have tried to get across with parallel lines the simple fact is that the lines DO NOT touch, they do APPEAR to TOUCH.

If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.

According to the mathematical model of the Ancient Greeks, they should never touch.

And again "Have you never seen headlights in fog?" Of course I have seen headlights in a fog, but have you seen the sun setting in a perfectly clear sky and appearing exactly the same size (maybe slightly larger - the Ponzo illusion) as it did at midday?

There is no way "atmospheric magnification" can keep the sun exactly the same size all day. Apart from anything else is happens every day. It can appear fuzzy and larger through cloud, but that is obvious when it happens and can happen in any part of the sky.

Several examples were given in the link showing that the enlargement is proportional to distance, causing the body to seem the same size.

[Tom Bishop]

Tom at what distance does math no longer work?  1, 10, 100, 1000, 10000 or further?

Just your estimate where math fails to be able to give good estimates or right answers?

I can use that math to estimate ranges when I sail and had accurate results.

I used it recently to determine how much fuel a rounded tank on my boat could hold and the answer I got was verified to be correct when I put fuel in it.

I used it for celestial navigation and determined my position accurately with noon and star sightings.

Throughout my life I have used it in my career and personal life and consistently had it verified since it gave me the right answers that were verified correct when applied to projects, the amount of material needed, how much liquid something could hold, etc.

So when does math no longer work?  I ask because you seem assured you are right and must know at what distances it is no longer accurate.  My experience is that it does return the right answers. An experience I highly doubt that the majority of people would say the opposite.

Did you have evidence of being correct or is the only evidence you offer is if the Greeks were right you are wrong about the shape of the Earth?

It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.

[Ecthelion]

It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.

Only if you cherry pick some observations and ignore others. Euclidean geometry is perfectly in accordance with observation since observation is based on euclidean geometry.

[Tom Bishop]

It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.

Only if you cherry pick some observations and ignore others. Euclidean geometry is perfectly in accordance with observation since observation is based on euclidean geometry.

Where have we observed perfect circles?

[rabinoz]

Tom at what distance does math no longer work?  1, 10, 100, 1000, 10000 or further?

Just your estimate where math fails to be able to give good estimates or right answers?

I can use that math to estimate ranges when I sail and had accurate results.

I used it recently to determine how much fuel a rounded tank on my boat could hold and the answer I got was verified to be correct when I put fuel in it.

I used it for celestial navigation and determined my position accurately with noon and star sightings.

Throughout my life I have used it in my career and personal life and consistently had it verified since it gave me the right answers that were verified correct when applied to projects, the amount of material needed, how much liquid something could hold, etc.

So when does math no longer work?  I ask because you seem assured you are right and must know at what distances it is no longer accurate.  My experience is that it does return the right answers. An experience I highly doubt that the majority of people would say the opposite.

Did you have evidence of being correct or is the only evidence you offer is if the Greeks were right you are wrong about the shape of the Earth?

It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.

It certainly does work with the railway tracks. They do not touch, they only appear to touch. How many times do we have to say the same thing?

PROOF:
Imagine the lines in question are railway tracks. They would appear to touch in about 3 miles (at a guess), but quite importantly they clearly do not touch, or that TGV flying past us at 200 mph is going to be in BIG BIG BIG TROUBLE in a bit under one minute! 

Go and have a look here if want to see what might happen Tgv crash, not the same cause.

Yes, I know I posted it before, but sometimes reality takes while to sink in.

Clearly railway tracks DO NOT ACTUALLY MEET, THEY ONLY APPEAR to MEET!

As I posted earlier in many cases we can test whether lines meet, by simply travelling to where they appear to meet.

Of course in some cases we may not be certain that the "lines" are truly parallel. That is no reflection on the geometry.

I'm afraid you must live in a different world to the rest of us, one quite divorced from reality!

[Ecthelion]

Where have we observed perfect circles?

Nowhere. Which is why I said observation is based on euclidean geometry, not the other way round. We don't observe perfect circles or parallel lines. But euclidean geometry does also account for non-perfect circles and almost-parallel lines, so that observation doesn't conflict with the geometry.

In any case, you have shifted the question. The original point was that converging parallel lines only conflict with euclidean geometry if a.) you fail to account for the fact that the lines aren't actually perfectly parallel and b.) you fail to account for the fact that moving the observer makes the lines appear parallel once more.

[Tom Bishop]

Tom at what distance does math no longer work?  1, 10, 100, 1000, 10000 or further?

Just your estimate where math fails to be able to give good estimates or right answers?

I can use that math to estimate ranges when I sail and had accurate results.

I used it recently to determine how much fuel a rounded tank on my boat could hold and the answer I got was verified to be correct when I put fuel in it.

I used it for celestial navigation and determined my position accurately with noon and star sightings.

Throughout my life I have used it in my career and personal life and consistently had it verified since it gave me the right answers that were verified correct when applied to projects, the amount of material needed, how much liquid something could hold, etc.

So when does math no longer work?  I ask because you seem assured you are right and must know at what distances it is no longer accurate.  My experience is that it does return the right answers. An experience I highly doubt that the majority of people would say the opposite.

Did you have evidence of being correct or is the only evidence you offer is if the Greeks were right you are wrong about the shape of the Earth?

It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.

It certainly does work with the railway tracks. They do not touch, they only appear to touch. How many times do we have to say the same thing?

PROOF:
Imagine the lines in question are railway tracks. They would appear to touch in about 3 miles (at a guess), but quite importantly they clearly do not touch, or that TGV flying past us at 200 mph is going to be in BIG BIG BIG TROUBLE in a bit under one minute! 

Go and have a look here if want to see what might happen Tgv crash, not the same cause.

Yes, I know I posted it before, but sometimes reality takes while to sink in.

Clearly railway tracks DO NOT ACTUALLY MEET, THEY ONLY APPEAR to MEET!

As I posted earlier in many cases we can test whether lines meet, by simply travelling to where they appear to meet.

Of course in some cases we may not be certain that the "lines" are truly parallel. That is no reflection on the geometry.

I'm afraid you must live in a different world to the rest of us, one quite divorced from reality!

If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.

According to the mathematical model of the Ancient Greeks, they should never touch.