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JAIMENJONATHAN (Guest)
guest
Jan-30-01, 02:49 PM (EST)
 
"Answer to "Inside a spherical mirror""
 
   If i seem illogical, forgive me. I'm only 10, and I'm still in 5th grade.

Anyway I think, assuming that the mirror is ONLY the inside surface, you would see yourself and a beam of light reflecting into some glass.


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  Subject     Author     Message Date     ID  
  RE: Answer to "Inside a spherical m... alexb Jan-30-01 1
     RE: Answer to "Inside a spherical m... JAIMENJONATHAN (Guest) Jan-30-01 2
         RE: Answer to "Inside a spherical m... alexb Jan-30-01 3
  RE: Answer to "Inside a spherical m... ktyson Mar-16-01 4
     RE: Answer to "Inside a spherical m... alexb Mar-16-01 5
         RE: Answer to "Inside a spherical m... ktyson Mar-16-01 6
             RE: Answer to "Inside a spherical m... alexb Mar-16-01 7

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alexb
Charter Member
672 posts
Jan-30-01, 03:02 PM (EST)
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1. "RE: Answer to "Inside a spherical mirror""
In response to message #0
 
   Never mind the age. As long as you are really curious and sincere, any thing you say is all right with me.

My question to you is, How did the applet help you arrive to your conclusion?

All the best,
Alexander Bogomolny


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JAIMENJONATHAN (Guest)
guest
Jan-30-01, 03:16 PM (EST)
 
2. "RE: Answer to "Inside a spherical mirror""
In response to message #1
 
   Well, if you mean how I got my answer, I used what I know:


  1. Light from a mirror bounces back into glass and leaves.
  2. Light from a mirror will bounces back and forth between two mirrors INFINITELY.

If you mean how I got the logic for my answer, I just formed a picture in my head.


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alexb
Charter Member
672 posts
Jan-30-01, 03:17 PM (EST)
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3. "RE: Answer to "Inside a spherical mirror""
In response to message #2
 
   You are very young and have plenty of time to form the right habits of mind that will foster your creativity, imagination and the innate gifts you may have.

One of this may be to be able to critically investigate your own opinions. So you formed an idea in your head. Have you tried to somehow verify that idea?

Another good habit may be to look for a second solution. You may not find anything, or may find support for your first solution - one never knows. But it always is worth a try.

I think the applet there may be very helpful.


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ktyson
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2 posts
Mar-16-01, 09:19 AM (EST)
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4. "RE: Answer to "Inside a spherical mirror""
In response to message #0
 
   Hello! I have questions concerning the spherical mirror model. I am posting this to Jaime's thread to keep this subject in one place.

In reference to the spherical mirror applet, I notice that by positioning the two nodes (the origin node at the bottom of the circle and the circumferential node at various locations around the circle), one can inscribe by reflection regular polyhedra inside the circle, as well as many related dense and complex geometric patterns. If we were to associate the patterns with real numbers, such that a triangle would correspond to 1/3, and a square to 1/4 and so on, we could represent points on a real number line between zero and one as the associated patterns in the circle. We can notice that as we move the nodes onto and off of the most primitive patterns, the surrounding patterns are quite dense. This might correspond to the progression of, say, 0.3329..., 0.3333..., 0.3334..., as we approach 1/3 on either side. My hypothesis from observation is that a slow but smooth motion of the circumferential node passes through the simple real number pattern (the triangle) and pushes the density of the pattern towards infinity on either side.

If this scenario is a valid interpretation (is it?), I have a few questions: 1. Will the patterns generated by the java applet be limited artificially by the processor of the digital machine on which the model is run, and is this a general problem affecting digital machines? 2. The actual system described, a small beam of light inside a sherical mirror, is an analog system. If such an apparatus was actually constructed and used, would the analog instrument surpass the power of a digital processor to resolve these patterns? 3. Could such an analog system be used as a part of a super-sliderule, to achieve numerical results by analog means?


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alexb
Charter Member
672 posts
Mar-16-01, 09:37 AM (EST)
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5. "RE: Answer to "Inside a spherical mirror""
In response to message #4
 
   >In reference to the spherical mirror
>applet, I notice that by
>positioning the two nodes (the
>origin node at the bottom
>of the circle and the
>circumferential node at various locations
>around the circle),

The applet has two control points:


  1. The one that moves along the vertical diameter of the circle represents the viewer's eyes.
  2. The one that is restricted to the circumference represents the point at which the viewer's eyes are focused.

>one can
>inscribe by reflection regular polyhedra
>inside the circle, as well
>as many related dense and
>complex geometric patterns. If
>we were to associate the
>patterns with real numbers, such
>that a triangle would correspond
>to 1/3, and a square
>to 1/4 and so on,
>we could represent points on
>a real number line between
>zero and one as the
>associated patterns in the circle.
> We can notice that
>as we move the nodes
>onto and off of the
>most primitive patterns, the surrounding
>patterns are quite dense.
>This might correspond to the
>progression of, say, 0.3329..., 0.3333...,

It does.

>0.3334..., as we approach 1/3
>on either side. My
>hypothesis from observation is that
>a slow but smooth motion
>of the circumferential node passes
>through the simple real number
>pattern (the triangle) and pushes
>the density of the pattern
>towards infinity on either side.

In the "real" world it would not matter whether your move your eyes slowly or fast. The speed of the motion only matters because your computer is not as fast as your eyes or brain.

>If this scenario is a valid
>interpretation (is it?),

(Yes.)

>I have
>a few questions: 1.
> Will the patterns generated
>by the java applet be
>limited artificially by the processor

They are limited not only by the processor but also by the program, i.e. up front by me - the programmer.

>of the digital machine on
>which the model is run,
>and is this a general
>problem affecting digital machines?

Yes of course. Naturally, computers store information in digital form which nowadays implies finite precision, unless special programs are used. The applet makes use of standard facilities available to Java, meaning that all the numbers obtained by calculations have finite precision. Therefore, the totality of all possible numbers that might be produced by such calculations is finite.

>2. The actual system
>described, a small beam of
>light inside a sherical mirror,
>is an analog system.
>If such an apparatus was
>actually constructed and used, would
>the analog instrument surpass the
>power of a digital processor
>to resolve these patterns?

You use a very loose language. I am not sure of what power you are talking about. But probably yes, an analogue system would be potentially capable of more patterns than the digital one. Human ability to differentiate between the patterns would still limit the number of perceived patterns to a large but a finite quantity.

>3. Could such an
>analog system be used as
>a part of a super-sliderule,
>to achieve numerical results by
>analog means?

Be the inventor.

All the best,
Alexander Bogomolny


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ktyson
Charter Member
2 posts
Mar-16-01, 02:58 PM (EST)
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6. "RE: Answer to "Inside a spherical mirror""
In response to message #5
 
   Thank you very much for your reply. I would like to make a few additional comments if I may.

Although I am not a trained mathematician, this problem set has been on my mind for 10 years. I regularly search the internet for "mirrored torus" because I first conceived of light reflecting and revolving around a torus. The original CTK problem asked about either a torus or a sphere, and it is the first and most specific hit I have found on the internet applicable to my question. Now that I have found CTK, please accept my high praise and gratitude for your work here.

You illustrated a mirrored sphere (actually a 2-D circle) rather than the more complex torus, but that was a great help for me, since it presented a much simplified digital-analog model to study.

My language is very loose because it is no more than a hunch. The hunch is that by using light reflected on the inside of curved volumes one might gain insights into numbers in a manner distinct from using computers, and that there might be advantages to such an approach, primarily for the reason of digital rounding.

I would like to mention the following tie-ins, which may be common knowledge to you. The association of reals between 0 and 1 proposed for the mirrored sphere can be similarly mapped to the Mandelbrot set. In fact, if you overlay such a circle on an image of the M-set you can see that the features on the boundary of the main cardioid correspond to what I called primitive reals (1/3, 1/4, 1/5, etc.). I also believe there is a connection in the patterns leading up to the primitives (in the dense region surrounding them) to what I have read about as the 'Devil's staircase' pattern.

I want very much to pursue actually building a working model of the apparatus, perhaps simplified. I would greatly appreciate any assistance or direction you feel ready to give. But regardless, I am very thankful for the encouragement above.

Sincerely,
Karl Tyson


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alexb
Charter Member
672 posts
Mar-16-01, 03:16 PM (EST)
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7. "RE: Answer to "Inside a spherical mirror""
In response to message #6
 
   >My language is very loose because
>it is no more than
>a hunch. The hunch
>is that by using light
>reflected on the inside of
>curved volumes one might gain
>insights into numbers in a
>manner distinct from using computers,

You lost me here. I do not need computers
to get insights into numbers.

>and that there might be
>advantages to such an approach,

The more, the better. Any variety is good.

>primarily for the reason of
>digital rounding.

Do not understand this. But the rounding is
OK as it is, believe me. To use spherical
mirrors for this purpose - if possible -
is like shooting sparrows with a canon.

>I would like to mention the
>following tie-ins, which may be
>common knowledge to you.
>The association of reals between
>0 and 1 proposed for
>the mirrored sphere can be
>similarly mapped to the Mandelbrot
>set.

Are you sure you know what you are
talking about?

> In fact, if
>you overlay such a circle

Overlay? You may use a conformal mapping too.

>on an image of the
>M-set you can see that
>the features on the boundary
>of the main cardioid correspond
>to what I called primitive
>reals (1/3, 1/4, 1/5, etc.).

Nah, have a look at

https://math.bu.edu/DYSYS/FRACGEOM2/FRACGEOM2.html

to imput some meaning to your hunches.

>I want very much to pursue
>actually building a working model
>of the apparatus, perhaps simplified.

By all means.

> I would greatly appreciate
>any assistance or direction you
>feel ready to give.

Nah, I am too busy with my own idiosyncrasies.

>But regardless, I am very
>thankful for the encouragement above.

I did not mean to encourage you. You are on your own.

Best,
Alexander Bogomolny


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