Re: Golden Ratio and Sundials

[Geoff Thurston]

Rod,

Thanks for the information. They look as if they have been been
attractively produced but in the UK, Issue 1 is available at £1.99 while
Issue 3 onwards costs £8.99.  So I shall not be subscribing £450 for a
complete set even if I had room on my shelves.

Best wishes,

Geoff

On 24 June 2017 at 12:26, rodwall1...@gmail.com 
wrote:

> Hi Geoff and Frank,
>
> My book also shows this.
>
> By The Way, the book is one of a series called the Mathematical World by
> National Geographic. Where in Australia the 1st book. The Golden Ratio. The
> mathematical language of beauty. Was AUD $2. $2 to get you to purchase the
> 1st book. And then you may want to purchase the series of books. Of course
> they will not $2.
>
> Regards,
>
> Roderick Wall.
>
> - Reply message -
> From: "Frank King" 
> To: "Geoff Thurston" 
> Cc: "Michael Ossipoff" , "Sundial Mailing List" <
> sundial@uni-koeln.de>
> Subject: Golden Ratio and Sundials
> Date: Sat, Jun 24, 2017 8:16 PM
>
> Dear Geoff,
>
> Many congratulations on your proof...
>
> When I set the puzzle, I thought three things:
>
>  1. I am really setting this for Geoff to solve.
>
>  2. He will certainly solve it and will probably
> be the first to publish.
>
>  3. His proof will either match mine or be more
> elegant.
>
> I was right on all three counts.  Your proof is
> just what I had in mind.  Once you spot those
> two triangles it is obvious that they are
> similar and the rest comes out in the wash!
>
> Very best wishes
>
> Frank
>
> ---https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
---
https://lists.uni-koeln.de/mailman/listinfo/sundial

Re: Golden Ratio and Sundials

[rodwall1...@gmail.com]

Hi Geoff and Frank,
My book also shows this.

By The Way, the book is one of a series called the Mathematical World by 
National Geographic. Where in Australia the 1st book. The Golden Ratio. The 
mathematical language of beauty. Was AUD $2. $2 to get you to purchase the 1st 
book. And then you may want to purchase the series of books. Of course they 
will not $2.

Regards,

Roderick Wall.

- Reply message -
From: "Frank King" 
To: "Geoff Thurston" 
Cc: "Michael Ossipoff" , "Sundial Mailing List" 

Subject: Golden Ratio and Sundials
Date: Sat, Jun 24, 2017 8:16 PM

Dear Geoff,

Many congratulations on your proof...

When I set the puzzle, I thought three things:

1. I am really setting this for Geoff to solve.

2. He will certainly solve it and will probably
be the first to publish.

3. His proof will either match mine or be more
elegant.

I was right on all three counts.  Your proof is
just what I had in mind.  Once you spot those
two triangles it is obvious that they are
similar and the rest comes out in the wash!

Very best wishes

Frank

---
https://lists.uni-koeln.de/mailman/listinfo/sundial---
https://lists.uni-koeln.de/mailman/listinfo/sundial

Re: Golden Ratio and Sundials

[Frank King]

Dear Geoff,

Many congratulations on your proof...

When I set the puzzle, I thought three things:

 1. I am really setting this for Geoff to solve.

 2. He will certainly solve it and will probably
be the first to publish.

 3. His proof will either match mine or be more
elegant.

I was right on all three counts.  Your proof is
just what I had in mind.  Once you spot those
two triangles it is obvious that they are
similar and the rest comes out in the wash!

Very best wishes

Frank

---
https://lists.uni-koeln.de/mailman/listinfo/sundial

Re: Golden Ratio and Sundials

[Geoff Thurston]

Frank,

I think the most elegant proof that the diagonal to side ratio in a
pentagon equals phi is as shown in the attachment.

Geoff



On 23 June 2017 at 08:08, Frank King  wrote:

> Dear All,
>
> Referring to the Golden Ratio and Sundials, Donald
> Snyder wrote:
>
>   I see nothing obvious except ... trivial
>   possibilities.
>
> Try Googling   Dodecahedral Sundial  and you will
> see many examples.  Here is one chosen at random:
>   http://stretchingtheboundaries.blogspot.co.uk/2012/09/
> dodecahedral-sundial.ht
> ml
>
> The faces are all regular pentagons and the ratio of
> the distance between any two non-adjacent vertices
> and the length of a side is the golden ratio.
>
> Exercise for the reader:
>
>   Come up with a simple proof of this!
>
> In some (slightly contrived) sense, a regular
> pentagon incorporates 25 instances of the
> Golden Ratio, so a Dodecahedron incorporates 300
> such instances.
>
> Frank H. King
> Cambridge, U.K.
>
> ---
> https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
---
https://lists.uni-koeln.de/mailman/listinfo/sundial

Re: Golden Ratio and Sundials

[Peter Mayer]

Hi,

Further to Fred's puzzle solution, here's an illustration from Martin 
Gardner's /More Mathematical Puzzles and Diversions/ (Harmondsworth: 
Penguin Books, Ltd., 1961) p. 74.  The base angles are 72 degrees, the 
apex 36 degrees, so a suitable gnomon for Abany in Western Australia, 
Las Vegas, Tulsa, Gibraltar  Gardner notes this is "an isosceles 
triangle that has sides in golden ratio to its base" (p. 71).


best wishes,

Peter


On 24/06/2017 10:48, Fred Sawyer wrote:

In 1997, I presented the following problem in The Compendium:

Problem:  It is required to know in what Latitude of this terraqueous 
Globe, an Erect South Declining Dial might be fixed to have these 
Properties, viz. the Declination of the Plane, the Distance of the 
Substyle from the Meridian, and the Style’s height [are] all equal.


The problem originated with Edward Hauxley in a challenge issued to 
Charles Leadbetter Feb. 1, 1736/7.  Leadbetter struggled with the 
solution, developing a 4th degree polynomial whose solution gave him a 
value for the declination.  He then fit this value into other 
equations to come up with a slightly different value for the latitude.


The correct solution is that the latitude is 38d 10m 22s and that this 
is also the value of the other angles sought.  The solution involves 
finding that the sine of the required latitude is the reciprocal of 
the golden ratio.


To see the article, download it at: 
https://www.dropbox.com/s/bj2qk6s1hg3a5m2/Pages%20from%20Nass43.pdf?dl=0


Fred Sawyer


On Wed, Jun 21, 2017 at 5:04 PM, rodwall1...@gmail.com 
> wrote:


Hi all,

I have been reading a book on the Golden Ratio which is
1.6180339887. It describes how the Golden Ratio describes how the
spiral of a sea shell is produced. And how nature uses the Golden
Ratio on the size of leaves etc.

Does anyone know if sundials have ever been produced useing the
Golden Ratio. The Golden Ratio goes back in history so one wonders
if it was ever applied to sundials.

The book describes that the short and long sizes of credit cards
are close to being the Golden Ratio.

LongSide/ShortSide = Golden Ratio.

Regards,

Roderick Wall.


---
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--
Peter Mayer
Department of Politics & International Studies (POLIS)
School of Social Sciences
http://www.arts.adelaide.edu.au/polis/
The University of Adelaide, AUSTRALIA 5005
Ph : +61 8 8313 5609
Fax : +61 8 8313 3443
e-mail: peter.ma...@adelaide.edu.au
CRICOS Provider Number 00123M
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Re: Golden Ratio and Sundials

[Fred Sawyer]

In 1997, I presented the following problem in The Compendium:

Problem:  It is required to know in what Latitude of this terraqueous
Globe, an Erect South Declining Dial might be fixed to have these
Properties, viz. the Declination of the Plane, the Distance of the Substyle
from the Meridian, and the Style’s height [are] all equal.

The problem originated with Edward Hauxley in a challenge issued to Charles
Leadbetter Feb. 1, 1736/7.  Leadbetter struggled with the solution,
developing a 4th degree polynomial whose solution gave him a value for the
declination.  He then fit this value into other equations to come up with a
slightly different value for the latitude.

The correct solution is that the latitude is 38d 10m 22s and that this is
also the value of the other angles sought.  The solution involves finding
that the sine of the required latitude is the reciprocal of the golden
ratio.

To see the article, download it at:
https://www.dropbox.com/s/bj2qk6s1hg3a5m2/Pages%20from%20Nass43.pdf?dl=0

Fred Sawyer


On Wed, Jun 21, 2017 at 5:04 PM, rodwall1...@gmail.com <
rodwall1...@gmail.com> wrote:

> Hi all,
>
> I have been reading a book on the Golden Ratio which is 1.6180339887. It
> describes how the Golden Ratio describes how the spiral of a sea shell is
> produced. And how nature uses the Golden Ratio on the size of leaves etc.
>
> Does anyone know if sundials have ever been produced useing the Golden
> Ratio. The Golden Ratio goes back in history so one wonders if it was ever
> applied to sundials.
>
> The book describes that the short and long sizes of credit cards are close
> to being the Golden Ratio.
>
> LongSide/ShortSide = Golden Ratio.
>
> Regards,
>
> Roderick Wall.
>
>
> ---
> https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
>
---
https://lists.uni-koeln.de/mailman/listinfo/sundial

RE: Golden Ratio and Sundials

[John Carmichael]

Rod:

 

Here are photos of actual clocks for sale with spiral faces.  Notice that the 
hour numerals in each turn of the spiral are radially aligned so that the clock 
hands point to the same numerals in all the turns.  A planar equatorial dial 
would be the perfect dial type to use in this design.  See : 
http://www.cafepress.com/+spiral+clocks   The dial you referred to was invented 
by Bill Gottesman and it takes the form of a 3D helix.

 

regards

 

From: rodwall1...@gmail.com [mailto:rodwall1...@gmail.com] 
Sent: Friday, June 23, 2017 2:36 PM
To: John Carmichael; sundial@uni-koeln.de
Subject: Re: Golden Ratio and Sundials

 

Hi all and thanks to everyone who responded to my questions. All very 
interesting.

 

John I have never see a spiral clock face. Very interesting thanks. Learn 
something every day.

 

That had me thinking. I think I have seen somewhere where there is a spiral 
sundial. Where a spot of light marked the time on the numbers on the spiral.

 

Wonder if it was a Golden Ratio spiral.

 

Have fun,

 

Roderick Wall.

 

- Reply message -
From: "John Carmichael" <jlcarmich...@comcast.net>
To: "'rodwall1...@gmail.com'" <rodwall1...@gmail.com>, <sundial@uni-koeln.de>
Subject: Golden Ratio and Sundials
Date: Sat, Jun 24, 2017 2:34 AM





Rod:
 
 
 
Do a goggle image search on “spiral clock face”.  A similar sundial face design 
could be made.   Time marks and numerals could be adjusted to be in the proper 
positions to be a functional sundial.
 
 
 
 
 
 
 
From: sundial [mailto:sundial-boun...@uni-koeln.de] On Behalf Of 
rodwall1...@gmail.com
Sent: Wednesday, June 21, 2017 2:04 PM
To: sundial@uni-koeln.de
Subject: Golden Ratio and Sundials
 
 
 
Hi all,
 
 
 
I have been reading a book on the Golden Ratio which is 1.6180339887. It 
describes how the Golden Ratio describes how the spiral of a sea shell is 
produced. And how nature uses the Golden Ratio on the size of leaves etc.
 
 
 
Does anyone know if sundials have ever been produced useing the Golden Ratio. 
The Golden Ratio goes back in history so one wonders if it was ever applied to 
sundials.
 
 
 
The book describes that the short and long sizes of credit cards are close to 
being the Golden Ratio.
 
 
 
LongSide/ShortSide = Golden Ratio.
 
 
 
Regards,
 
 
 
Roderick Wall.
 
 
 
---
https://lists.uni-koeln.de/mailman/listinfo/sundial

Re: Golden Ratio and Sundials

[rodwall1...@gmail.com]

Hi all and thanks to everyone who responded to my questions. All very 
interesting.
John I have never see a spiral clock face. Very interesting thanks. Learn 
something every day.

That had me thinking. I think I have seen somewhere where there is a spiral 
sundial. Where a spot of light marked the time on the numbers on the spiral.

Wonder if it was a Golden Ratio spiral.

Have fun,

Roderick Wall.

- Reply message -
From: "John Carmichael" 
To: "'rodwall1...@gmail.com'" , 
Subject: Golden Ratio and Sundials
Date: Sat, Jun 24, 2017 2:34 AM

Rod:



Do a goggle image search on “spiral clock face”.  A similar sundial face design 
could be made.   Time marks and numerals could be adjusted to be in the proper 
positions to be a functional sundial.







From: sundial [mailto:sundial-boun...@uni-koeln.de] On Behalf Of 
rodwall1...@gmail.com
Sent: Wednesday, June 21, 2017 2:04 PM
To: sundial@uni-koeln.de
Subject: Golden Ratio and Sundials



Hi all,



I have been reading a book on the Golden Ratio which is 1.6180339887. It 
describes how the Golden Ratio describes how the spiral of a sea shell is 
produced. And how nature uses the Golden Ratio on the size of leaves etc.



Does anyone know if sundials have ever been produced useing the Golden Ratio. 
The Golden Ratio goes back in history so one wonders if it was ever applied to 
sundials.



The book describes that the short and long sizes of credit cards are close to 
being the Golden Ratio.



LongSide/ShortSide = Golden Ratio.



Regards,



Roderick Wall.---
https://lists.uni-koeln.de/mailman/listinfo/sundial

Re: Golden Ratio and Sundials

[Frank King]

Dear All,

Referring to the Golden Ratio and Sundials, Donald
Snyder wrote:

  I see nothing obvious except ... trivial
  possibilities.

Try Googling   Dodecahedral Sundial  and you will
see many examples.  Here is one chosen at random:
  http://stretchingtheboundaries.blogspot.co.uk/2012/09/dodecahedral-sundial.ht
ml

The faces are all regular pentagons and the ratio of
the distance between any two non-adjacent vertices
and the length of a side is the golden ratio.

Exercise for the reader:

  Come up with a simple proof of this!

In some (slightly contrived) sense, a regular
pentagon incorporates 25 instances of the
Golden Ratio, so a Dodecahedron incorporates 300
such instances.

Frank H. King
Cambridge, U.K.

---
https://lists.uni-koeln.de/mailman/listinfo/sundial

Re: Golden Ratio and Sundials

[Fred Sawyer]

Traveling now so I don't have access to it at the moment, but several years
ago I published a quiz in The Compendium that had the golden ratio as the
answer.   It's was an actual historical example and the author back in the
17th? Century wasn't aware that the number he was approximating was phi.
Readers who have the (fantastic) NASS Repository. DVD can do an  easy
search to find the quiz.



On Jun 22, 2017 3:31 PM, "Donald L Snyder"  wrote:

Thanks, Michael, for setting that right.  I would add only that the golden
ratio
equals (sqroot(5) + 1)/2, which is a number approximately equal to 1.61803.
The inverse of the golden ratio is approximately 0.61803.
  The original question posted by Roderick asked if the golden ratio could
ever apply to a sundial.  I see nothing obvious except these trivial
possibilities.  One could certainly construct a sundial on a rectangular
plate having a long side to short side ratio that is golden.  A
second possibility has the golden ratio pretty much hidden.  Since
atan(1.61803) equals 58.28 degrees, a horizontal sundial in a city at
this latitude could have a triangular gnomon with a height to base
ratio that is golden.
   Don Snyder



On 6/21/2017 10:00 PM, Michael Ossipoff wrote:



On Wed, Jun 21, 2017 at 5:27 PM, Brooke Clarke  wrote:

> Hi Roderick:
>
> I also have a book on this number that makes the case that there is no
> such ratio.
>


Your book is mistaken.

If A/B = (A+B)/A, then A/B is the golden ratio.

If a line-segment is divided into two parts related by that ratio, then the
golden ratio is also called the golden section.

If the interval between two numbers is divided into two intervals related
by the golden ratio, then the golden ratio is also called the golden mean.



> For example if you look at a photograph of something where do you put the
> markers to make the measurement?
>

Along two mutually-perpendicular edges, measured from a common corner?  :^)


Michael Ossipoff



> Brooke 
> Clarkehttp://www.PRC68.comhttp://www.end2partygovernment.com/2012Issues.html
>
>  Original Message 
>
> Hi all,
>
> I have been reading a book on the Golden Ratio which is 1.6180339887. It
> describes how the Golden Ratio describes how the spiral of a sea shell is
> produced. And how nature uses the Golden Ratio on the size of leaves etc.
>
> Does anyone know if sundials have ever been produced useing the Golden
> Ratio. The Golden Ratio goes back in history so one wonders if it was ever
> applied to sundials.
>
> The book describes that the short and long sizes of credit cards are close
> to being the Golden Ratio.
>
> LongSide/ShortSide = Golden Ratio.
>
> Regards,
>
> Roderick Wall.
>
>
>
> ---https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
>
> ---
> https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
>


---https://lists.uni-koeln.de/mailman/listinfo/sundial



---
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---
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Re: Golden Ratio and Sundials

[Steve Lelievre]

On Thu, Jun 22, 2017 at 12:31, Donald L Snyder  wrote:

> Since
> atan(1.61803) equals 58.28 degrees, a horizontal sundial in a city at
> this latitude could have a triangular gnomon with a height to base
> ratio that is golden.
>Don Snyder
>
> On 6/21/2017 10:00 PM, Michael Ossipoff wrote:
>
>
>
> On Wed, Jun 21, 2017 at 5:27 PM, Brooke Clarke  wrote:
>
>> Hi Roderick:
>>
>> I also have a book on this number that makes the case that there is no
>> such ratio.
>>
>
>
> Your book is mistaken.
>
> If A/B = (A+B)/A, then A/B is the golden ratio.
>
> If a line-segment is divided into two parts related by that ratio, then
> the golden ratio is also called the golden section.
>
> If the interval between two numbers is divided into two intervals related
> by the golden ratio, then the golden ratio is also called the golden mean.
>
>
>
>> For example if you look at a photograph of something where do you put the
>> markers to make the measurement?
>>
>
> Along two mutually-perpendicular edges, measured from a common corner?  :^)
>
>
> Michael Ossipoff
>
>
>
>> Brooke 
>> Clarkehttp://www.PRC68.comhttp://www.end2partygovernment.com/2012Issues.html
>>
>>  Original Message 
>>
>> Hi all,
>>
>> I have been reading a book on the Golden Ratio which is 1.6180339887. It
>> describes how the Golden Ratio describes how the spiral of a sea shell is
>> produced. And how nature uses the Golden Ratio on the size of leaves etc.
>>
>> Does anyone know if sundials have ever been produced useing the Golden
>> Ratio. The Golden Ratio goes back in history so one wonders if it was ever
>> applied to sundials.
>>
>> The book describes that the short and long sizes of credit cards are
>> close to being the Golden Ratio.
>>
>> LongSide/ShortSide = Golden Ratio.
>>
>> Regards,
>>
>> Roderick Wall.
>>
>>
>>
>> ---https://lists.uni-koeln.de/mailman/listinfo/sundial
>>
>>
>>
>> ---
>> https://lists.uni-koeln.de/mailman/listinfo/sundial
>>
>>
>>
>
>
> ---https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
> ---
> https://lists.uni-koeln.de/mailman/listinfo/sundial
>
> --
Cell +1 778 837 5771
---
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Re: Golden Ratio and Sundials

[Donald L Snyder]

Thanks, Michael, for setting that right.  I would add only that the 
golden ratio

equals (sqroot(5) + 1)/2, which is a number approximately equal to 1.61803.
The inverse of the golden ratio is approximately 0.61803.
  The original question posted by Roderick asked if the golden ratio could
ever apply to a sundial.  I see nothing obvious except these trivial
possibilities.  One could certainly construct a sundial on a rectangular
plate having a long side to short side ratio that is golden.  A
second possibility has the golden ratio pretty much hidden.  Since
atan(1.61803) equals 58.28 degrees, a horizontal sundial in a city at
this latitude could have a triangular gnomon with a height to base
ratio that is golden.
   Don Snyder


On 6/21/2017 10:00 PM, Michael Ossipoff wrote:



On Wed, Jun 21, 2017 at 5:27 PM, Brooke Clarke > wrote:


Hi Roderick:

I also have a book on this number that makes the case that there
is no such ratio.



Your book is mistaken.

If A/B = (A+B)/A, then A/B is the golden ratio.

If a line-segment is divided into two parts related by that ratio, 
then the golden ratio is also called the golden section.


If the interval between two numbers is divided into two intervals 
related by the golden ratio, then the golden ratio is also called the 
golden mean.


For example if you look at a photograph of something where do you
put the markers to make the measurement?


Along two mutually-perpendicular edges, measured from a common 
corner?  :^)



Michael Ossipoff

Brooke Clarke
http://www.PRC68.com
http://www.end2partygovernment.com/2012Issues.html


 Original Message 

Hi all,

I have been reading a book on the Golden Ratio which is
1.6180339887. It describes how the Golden Ratio describes how the
spiral of a sea shell is produced. And how nature uses the Golden
Ratio on the size of leaves etc.

Does anyone know if sundials have ever been produced useing the
Golden Ratio. The Golden Ratio goes back in history so one
wonders if it was ever applied to sundials.

The book describes that the short and long sizes of credit cards
are close to being the Golden Ratio.

LongSide/ShortSide = Golden Ratio.

Regards,

Roderick Wall.



---
https://lists.uni-koeln.de/mailman/listinfo/sundial





---
https://lists.uni-koeln.de/mailman/listinfo/sundial






---
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---
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Re: Golden Ratio and Sundials

[Michael Ossipoff]

On Wed, Jun 21, 2017 at 5:27 PM, Brooke Clarke  wrote:

> Hi Roderick:
>
> I also have a book on this number that makes the case that there is no
> such ratio.
>


Your book is mistaken.

If A/B = (A+B)/A, then A/B is the golden ratio.

If a line-segment is divided into two parts related by that ratio, then the
golden ratio is also called the golden section.

If the interval between two numbers is divided into two intervals related
by the golden ratio, then the golden ratio is also called the golden mean.



> For example if you look at a photograph of something where do you put the
> markers to make the measurement?
>

Along two mutually-perpendicular edges, measured from a common corner?  :^)


Michael Ossipoff



> Brooke 
> Clarkehttp://www.PRC68.comhttp://www.end2partygovernment.com/2012Issues.html
>
>  Original Message 
>
> Hi all,
>
> I have been reading a book on the Golden Ratio which is 1.6180339887. It
> describes how the Golden Ratio describes how the spiral of a sea shell is
> produced. And how nature uses the Golden Ratio on the size of leaves etc.
>
> Does anyone know if sundials have ever been produced useing the Golden
> Ratio. The Golden Ratio goes back in history so one wonders if it was ever
> applied to sundials.
>
> The book describes that the short and long sizes of credit cards are close
> to being the Golden Ratio.
>
> LongSide/ShortSide = Golden Ratio.
>
> Regards,
>
> Roderick Wall.
>
>
>
> ---https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
>
> ---
> https://lists.uni-koeln.de/mailman/listinfo/sundial
>
>
>
---
https://lists.uni-koeln.de/mailman/listinfo/sundial

Re: Golden Ratio and Sundials

[rodwall1...@gmail.com]

Hi Brooke,
Thanks, I can see that your information is from Da Book.

Roderick.

- Reply message -
From: "Brooke Clarke" 
To: "'Sundial Mailing List'" 
Subject: Golden Ratio and Sundials
Date: Thu, Jun 22, 2017 7:27 AM

Hi Roderick:

I also have a book on this number that makes the case that there is no such 
ratio.  For example if you look at a 
photograph of something where do you put the markers to make the measurement?

It's interesting that 4x5, 8x10 film cameras have aspect ratios of 1.25.  35mm 
film cameras (36 x 24mm) have a ratio of 
1.5 the same as most DSLR cameras, yet the most common printer paper aspect 
ratios are still 1.25 (4x5 or 8x10).  But 
you can get 4x6 or 8x12" paper for an aspect ratio of 1.5.  Note all of these 
are smaller than 1.618.

-- 
Have Fun,

Brooke Clarke
http://www.PRC68.com
http://www.end2partygovernment.com/2012Issues.html

 Original Message 
> Hi all,
>
> I have been reading a book on the Golden Ratio which is 1.6180339887. It 
> describes how the Golden Ratio describes how 
> the spiral of a sea shell is produced. And how nature uses the Golden Ratio 
> on the size of leaves etc.
>
> Does anyone know if sundials have ever been produced useing the Golden Ratio. 
> The Golden Ratio goes back in history so 
> one wonders if it was ever applied to sundials.
>
> The book describes that the short and long sizes of credit cards are close to 
> being the Golden Ratio.
>
> LongSide/ShortSide = Golden Ratio.
>
> Regards,
>
> Roderick Wall.
>
>
>
> ---
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Re: Golden Ratio and Sundials

[Brooke Clarke]

Hi Roderick:

I also have a book on this number that makes the case that there is no such ratio.  For example if you look at a 
photograph of something where do you put the markers to make the measurement?


It's interesting that 4x5, 8x10 film cameras have aspect ratios of 1.25.  35mm film cameras (36 x 24mm) have a ratio of 
1.5 the same as most DSLR cameras, yet the most common printer paper aspect ratios are still 1.25 (4x5 or 8x10).  But 
you can get 4x6 or 8x12" paper for an aspect ratio of 1.5.  Note all of these are smaller than 1.618.


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Have Fun,

Brooke Clarke
http://www.PRC68.com
http://www.end2partygovernment.com/2012Issues.html

 Original Message 

Hi all,

I have been reading a book on the Golden Ratio which is 1.6180339887. It describes how the Golden Ratio describes how 
the spiral of a sea shell is produced. And how nature uses the Golden Ratio on the size of leaves etc.


Does anyone know if sundials have ever been produced useing the Golden Ratio. The Golden Ratio goes back in history so 
one wonders if it was ever applied to sundials.


The book describes that the short and long sizes of credit cards are close to 
being the Golden Ratio.

LongSide/ShortSide = Golden Ratio.

Regards,

Roderick Wall.



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