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
>
>
---
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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
>
>
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