Re: Sundial Puzzle Corner

[Richard Langley]

"As far as I know, these days, this
subject isn't taught ANYWHERE in
the UK even in Universities.

Geometry is deemed a useless subject
because 'you don't really need it'."

Might be taught in geomatics programs in the U.K. We certainly teach it in our 
own courses as well as a bespoke course from the Math and Stats Department:
http://www2.unb.ca/gge/Study/Undergraduate/CourseSequence.pdf
http://www.unb.ca/academics/calendar/undergraduate/current/frederictoncourses/mathematics/math-3543.html

-- Richard Langley

-
| Richard B. LangleyE-mail: l...@unb.ca |
| Geodetic Research Laboratory  Web: http://gge.unb.ca/ |
| Dept. of Geodesy and Geomatics EngineeringPhone:+1 506 453-5142   |
| University of New Brunswick   Fax:  +1 506 453-4943   |
| Fredericton, N.B., Canada  E3B 5A3|
|Fredericton?  Where's that?  See: http://www.fredericton.ca/   |
-



> On Oct 31, 2016, at 5:50 AM, Frank King  wrote:
> 
> Dear Karl,
> 
> Your idea is not without merit:
> 
>> Wrap the slate with a reflective strip ...
>> Playing around with a laser should find the
>> focii.
> 
> This is sometimes referred to as the
> "Elliptical Billiard Table Problem"...
> 
> If you aim at a focus, the laser path will
> reflect through the other focus and so on.
> Eventually it will settle down into
> running backwards and forwards along the
> major axis BUT...
> 
> If you aim the laser so that its path
> passes OUTSIDE the line joining the
> two foci, the path traced will, after
> an indefinite number of reflections,
> leave a dead area in the centre which
> is ITSELF an ellipse.
> 
> If you aim the laser so that its path
> passes BETWEEN the two foci, the path
> traced will, after an indefinite number
> of bounces, leave two dead areas around
> each end of the major axis.  The inner
> boundaries of these areas are the turning
> points of a hyperbola.
> 
> The real excitement comes if you aim
> the laser so that you get a return to
> the starting point after a finite
> number of reflections.
> 
> You then get a nice pretty pattern.  I
> have knocked up the attached example
> where there are 46 reflections.
> 
> You can prove all this using
> Projective Geometry.  This is a
> delightful subject which includes
> splendid concepts such as "The
> Circular Points at Infinity".
> 
> In the 1950's, Projective Geometry
> was in the UK A-level Mathematics
> syllabus and taught to 17- and
> 18-year olds.
> 
> As far as I know, these days, this
> subject isn't taught ANYWHERE in
> the UK even in Universities.
> 
> Geometry is deemed a useless subject
> because "you don't really need it".
> 
> End of rant.
> 
> Frank
> 
> ---
> https://lists.uni-koeln.de/mailman/listinfo/sundial
> 

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Re: Sundial Puzzle Corner

[Geoff Thurston]

I like David's solution and I am in awe of his stone-cutting capability.
However, to return to your problem where the ellipse has been returned
without any markings, why not draw an ellipse of the same specified
dimensions on paper with the axes already marked on it? This could then be
matched to the "close-to-perfect" slate ellipse to locate the axes and the
centre. It probably would not be necessary to go to the trouble of drawing
a complete ellipse. A polygonal approximation might work.

On 31 October 2016 at 08:22, David  wrote:

> I cut my own slate ellipses, often up to 3cm thick. In the process of
> marking out the ellipse I will have drawn both the major and minor axes. I
> make a small indentation with a metal scribe at both ends of both axes as
> well as the centre point. These indentations are deep enough to be
> noticeable but shallow enough to be erased easily on the cleaning-up
> operation after the eventual painting/gilding of the finished inscription.
> The indentations can be made more prominent by painting them with a spot of
> light-coloured acrylic paint. After the ellipse has been cut out (which I
> do with a series of short straight cuts from a bench-mounted, water-fed
> circular saw, followed by a hand-held water-fed cylindrical abrasive drum)
> the axes are easily drawn through the still-visible marks.
> If the job of drawing and cutting out has to be done by another person,
> such as at the stonemason's yard, they could be asked to leave similar
> marks to help you with the alignment of the inscription.
> David Brown
> Somerton, Somerset, UK
>
>  On 30/10/2016 21:29, Karl Billeter wrote:
>
>> On Mon, Oct 31, 2016 at 08:24:04AM +1100, Karl Billeter wrote:
>>
>>> On Sun, Oct 30, 2016 at 02:37:04PM +, Frank King wrote:
>>>   ...
>>>
 Almost the first task is to find
 the centre and the axes.  Clearly
 you cannot fold a slate in half
 and the traditional way to proceed
 is to put a large sheet of paper
 over the slate and crease it down
 all round the rim.

 You then cut round the crease and
 attempt to follow your procedure!

>>> Wrap the slate with a reflective strip (smooth, shiny plastic? thin
>>> polished
>>> metal?).  Playing around with a laser should find the focii.
>>>
>> You could also try balancing the slate on rod to find the centre but it's
>> probably too hard to get the required accuracy and it might be a little
>> risky!
>>
>> K
>> ---
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>>
>>
>
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Re: Sundial Puzzle Corner

[Frank King]

Dear Karl,

Your idea is not without merit:

> Wrap the slate with a reflective strip ...
> Playing around with a laser should find the
> focii.

This is sometimes referred to as the
"Elliptical Billiard Table Problem"...

If you aim at a focus, the laser path will
reflect through the other focus and so on.
Eventually it will settle down into
running backwards and forwards along the
major axis BUT...

If you aim the laser so that its path
passes OUTSIDE the line joining the
two foci, the path traced will, after
an indefinite number of reflections,
leave a dead area in the centre which
is ITSELF an ellipse.

If you aim the laser so that its path
passes BETWEEN the two foci, the path
traced will, after an indefinite number
of bounces, leave two dead areas around
each end of the major axis.  The inner
boundaries of these areas are the turning
points of a hyperbola.

The real excitement comes if you aim
the laser so that you get a return to
the starting point after a finite
number of reflections.

You then get a nice pretty pattern.  I
have knocked up the attached example
where there are 46 reflections.

You can prove all this using
Projective Geometry.  This is a
delightful subject which includes
splendid concepts such as "The
Circular Points at Infinity".

In the 1950's, Projective Geometry
was in the UK A-level Mathematics
syllabus and taught to 17- and
18-year olds.

As far as I know, these days, this
subject isn't taught ANYWHERE in
the UK even in Universities.

Geometry is deemed a useless subject
because "you don't really need it".

End of rant.

Frank



Ellipse.pdf
Description: Ellipse.pdf
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Re: Sundial Puzzle Corner

[David]

I cut my own slate ellipses, often up to 3cm thick. In the process of 
marking out the ellipse I will have drawn both the major and minor axes. 
I make a small indentation with a metal scribe at both ends of both axes 
as well as the centre point. These indentations are deep enough to be 
noticeable but shallow enough to be erased easily on the cleaning-up 
operation after the eventual painting/gilding of the finished 
inscription. The indentations can be made more prominent by painting 
them with a spot of light-coloured acrylic paint. After the ellipse has 
been cut out (which I do with a series of short straight cuts from a 
bench-mounted, water-fed circular saw, followed by a hand-held water-fed 
cylindrical abrasive drum) the axes are easily drawn through the 
still-visible marks.
If the job of drawing and cutting out has to be done by another person, 
such as at the stonemason's yard, they could be asked to leave similar 
marks to help you with the alignment of the inscription.

David Brown
Somerton, Somerset, UK

 On 30/10/2016 21:29, Karl Billeter wrote:

On Mon, Oct 31, 2016 at 08:24:04AM +1100, Karl Billeter wrote:

On Sun, Oct 30, 2016 at 02:37:04PM +, Frank King wrote:
  
...

Almost the first task is to find
the centre and the axes.  Clearly
you cannot fold a slate in half
and the traditional way to proceed
is to put a large sheet of paper
over the slate and crease it down
all round the rim.

You then cut round the crease and
attempt to follow your procedure!

Wrap the slate with a reflective strip (smooth, shiny plastic? thin polished
metal?).  Playing around with a laser should find the focii.

You could also try balancing the slate on rod to find the centre but it's
probably too hard to get the required accuracy and it might be a little risky!

K
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Re: Sundial Puzzle Corner

[Karl Billeter]

On Mon, Oct 31, 2016 at 08:24:04AM +1100, Karl Billeter wrote:
> On Sun, Oct 30, 2016 at 02:37:04PM +, Frank King wrote:
>  
> ...
> > Almost the first task is to find
> > the centre and the axes.  Clearly
> > you cannot fold a slate in half
> > and the traditional way to proceed
> > is to put a large sheet of paper
> > over the slate and crease it down
> > all round the rim.
> > 
> > You then cut round the crease and
> > attempt to follow your procedure!
> 
> Wrap the slate with a reflective strip (smooth, shiny plastic? thin polished
> metal?).  Playing around with a laser should find the focii.

You could also try balancing the slate on rod to find the centre but it's
probably too hard to get the required accuracy and it might be a little risky!

K
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Re: Sundial Puzzle Corner

[Karl Billeter]

On Sun, Oct 30, 2016 at 02:37:04PM +, Frank King wrote:
 
...
> Almost the first task is to find
> the centre and the axes.  Clearly
> you cannot fold a slate in half
> and the traditional way to proceed
> is to put a large sheet of paper
> over the slate and crease it down
> all round the rim.
> 
> You then cut round the crease and
> attempt to follow your procedure!

Wrap the slate with a reflective strip (smooth, shiny plastic? thin polished
metal?).  Playing around with a laser should find the focii.

Completely off the top of my head and untried in any way :-)


K
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RE: Sundial Puzzle Corner

[Dave Bell]

Well, my "method" was mostly meant tongue-in-cheek, and based upon the
premise that the ellipse was n a (normal sized) sheet of paper. I'd have to
actually try it, but it seems the approximation in the second fold is no
more critical than the first. Assuming the paper is somewhat translucent,
you should be able to overlay the halves pretty accurately.

I do agree that the geometric method is elegant and gives a much better
result!

Dave

-Original Message-
From: Frank King [mailto:f...@cl.cam.ac.uk] 
Sent: Sunday, October 30, 2016 7:37 AM
To: Dave Bell <db...@thebells.net>; Donald L Snyder <dsny...@wustl.edu>
Cc: 'Frank King' <f...@cl.cam.ac.uk>; sundial@uni-koeln.de
Subject: Re: Sundial Puzzle Corner

Dear Dave and Donald,

This puzzle actually has serious
practical sundial applications
as I shall illustrate.  First:

Good try Dave...

> Personally, I'd fold the paper,
> superimposing the reflected
> half-ellipse, crease it, unfold,
> rotate ~90 degrees and repeat!

Although this is not what I had in
mind, it is undoubtedly a practical
method.  I wonder whether you have
actually tried it?  The clue to the
difficulty lies in the approximation
sign you have wisely written before
the "90"!

A fair proportion of my sundials
are hand-cut into elliptical
slabs of slate.  These days,
some stone-yards will simply
accept the major and minor axes
of the ellipse and, somehow,
(water-jet?) cut a slate which
is a close-to-perfect ellipse.

Almost the first task is to find
the centre and the axes.  Clearly
you cannot fold a slate in half
and the traditional way to proceed
is to put a large sheet of paper
over the slate and crease it down
all round the rim.

You then cut round the crease and
attempt to follow your procedure!

No doubt in practised hands this
can give a good result but I find
that, no matter how carefully I
work, the two axes are invariably
not (quite) at 90 degrees.  

It is quite hopeless to refold the
paper; the folds are too close
together.  You have to start again.
I usually get it to my satisfaction
at second attempt but occasionally
I have had to have three tries.

It is especially hard if the ellipse
is not too far off being a circle,
say 1200mm x 1050mm.

This is a big sheet of paper.  Just
try it for yourself and see the
challenges!

Donald pointed to a link which gives
the answer I was aiming at.  Alas,
although I find the geometry a
delight, the practicalities are
just as challenging as the folding
method!  You can get the axes at
right-angles fairly easily but
you find the "centre" is nearer
one end of the major axis than
the other, ditto the minor axis.

As so often with sundials, the
theory may be elegant and
straightforward but real life has
a way of making implementation a
struggle!

Frank


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Re: Sundial Puzzle Corner

[Frank King]

Dear Andrew,

You are quite right...

> I thought that you were looking for a
> "straight-edge" and compass construction
> on the sheet of paper.

Yes.  I was!

Don Snyder gave us pointers to what I was
hoping for...

>   first find the ellipse center 
>
> http://www.had2know.com/makeit/find-ellipse-center.html
>   
>   then the ellipse axes
>
> http://www.had2know.com/makeit/find-ellipse-axis-focus.html

Unfortunately, I got side-tracked by
the "folding-paper" method.  That is
a practical method but not what I
really wanted!

> One way to draw the ellipse in the
> first place is to adopt the "Trammel
> Method".

Yes.  That is probably almost as well
known as the String and Pins method.
It also gives good results if you are
careful.

> If you start by drawing the major &
> minor axes on a sheet of paper and
> plotting the ellipse from there, you
> have no need to search for the axes
> because they are already drawn on
> the paper template!

Quite so, but my real-life problem is
that I take delivery of an elliptical
piece of slate WITHOUT the axes marked
on so I have to find them!

> Am I missing something?

No.  You stayed with the question and
didn't get side-tracked like the rest
of us :-)

Frank



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RE: Sundial Puzzle Corner

[R. Hooijenga]

Hi Frank,

Yes, in fact the ellipse for my front yard analemmatic dial was constructed
using pegs and rope.
It did not turn out half bad, but I know what you mean.

Now that you have pointed it out to me, the string proof is obvious... :)
Thank you.

Rudolf

-Oorspronkelijk bericht-
Van: Frank King [mailto:f...@cl.cam.ac.uk] 
Onderwerp: Re: Sundial Puzzle Corner

Dear Rudolph,
Glad you enjoyed the puzzle or, at least, the solution...
> Brilliant!

You add...
> I never realized that the distance from the focus to the end of the 
> minor axis equals the half major axis...
Have you never tried drawing an ellipse with a Length of String and Two
Pins?  This is illustrated all over the web, for example here:

[proof]


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RE: Sundial Puzzle Corner

[Andrew Pettit]

Hello Frank

 

I had not responded to your query because I thought that you were looking
for a "straight-edge" and compass construction on the sheet of paper.

 

Looking at your practical problem:

 

One way to draw the ellipse in the first place is to adopt the "Trammel
Method".

 

See:

 

http://bpptg.weebly.com/uploads/1/3/6/8/13685178/ellipses_paper_method.pdf  

 

If you start by drawing the major & minor axes on a sheet of paper and
plotting the ellipse from there, you have no need to search for the axes
because they are already drawn on the paper template!

 

Am I missing something?

 

Regards

 

Andrew Pettit

 

 

-Original Message-
From: sundial [mailto:sundial-boun...@uni-koeln.de] On Behalf Of Frank King
Sent: Sunday, October 30, 2016 2:37 PM
To: Dave Bell; Donald L Snyder
Cc: sundial@uni-koeln.de
Subject: Re: Sundial Puzzle Corner

 

Dear Dave and Donald,

 

This puzzle actually has serious

practical sundial applications

as I shall illustrate.  First:

 

Good try Dave...

 

> Personally, I'd fold the paper,

> superimposing the reflected

> half-ellipse, crease it, unfold,

> rotate ~90 degrees and repeat!

 

Although this is not what I had in

mind, it is undoubtedly a practical

method.  I wonder whether you have

actually tried it?  The clue to the

difficulty lies in the approximation

sign you have wisely written before

the "90"!

 

A fair proportion of my sundials

are hand-cut into elliptical

slabs of slate.  These days,

some stone-yards will simply

accept the major and minor axes

of the ellipse and, somehow,

(water-jet?) cut a slate which

is a close-to-perfect ellipse.

 

Almost the first task is to find

the centre and the axes.  Clearly

you cannot fold a slate in half

and the traditional way to proceed

is to put a large sheet of paper

over the slate and crease it down

all round the rim.

 

You then cut round the crease and

attempt to follow your procedure!

 

No doubt in practised hands this

can give a good result but I find

that, no matter how carefully I

work, the two axes are invariably

not (quite) at 90 degrees.  

 

It is quite hopeless to refold the

paper; the folds are too close

together.  You have to start again.

I usually get it to my satisfaction

at second attempt but occasionally

I have had to have three tries.

 

It is especially hard if the ellipse

is not too far off being a circle,

say 1200mm x 1050mm.

 

This is a big sheet of paper.  Just

try it for yourself and see the

challenges!

 

Donald pointed to a link which gives

the answer I was aiming at.  Alas,

although I find the geometry a

delight, the practicalities are

just as challenging as the folding

method!  You can get the axes at

right-angles fairly easily but

you find the "centre" is nearer

one end of the major axis than

the other, ditto the minor axis.

 

As so often with sundials, the

theory may be elegant and

straightforward but real life has

a way of making implementation a

struggle!

 

Frank

 

 

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Re: Sundial Puzzle Corner

[Frank King]

Dear Rudolph,

Glad you enjoyed the puzzle or, at least,
the solution...

> Brilliant!

You add...

> I never realized that the distance from
> the focus to the end of the minor axis
> equals the half major axis...

Have you never tried drawing an ellipse
with a Length of String and Two Pins?

This is illustrated all over the web,
for example here:

  http://bogan.ca/orbits/kepler.html

This is a very bad way of drawing an
ellipse but the theory is sound!

In this illustration, the two ends of
the string are each tied to one of
the two pins.  The pins are at the
two foci.

Now imagine the pencil point at one
end of the major axis.  It is then
obvious that the total length of
the string equals the major axis.

Then imagine the pencil point at
one end of the minor axis.  It is
then obvious that the total length
of the string equals twice the
distance from a focus to one end
of the minor axis.

If you try this in real life you
will find you draw a potato shape!

[Well that's what happens if I do
it but there is nothing wrong with
the idea!]

Frank

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Re: Sundial Puzzle Corner

[Frank King]

Dear Dave and Donald,

This puzzle actually has serious
practical sundial applications
as I shall illustrate.  First:

Good try Dave...

> Personally, I'd fold the paper,
> superimposing the reflected
> half-ellipse, crease it, unfold,
> rotate ~90 degrees and repeat!

Although this is not what I had in
mind, it is undoubtedly a practical
method.  I wonder whether you have
actually tried it?  The clue to the
difficulty lies in the approximation
sign you have wisely written before
the "90"!

A fair proportion of my sundials
are hand-cut into elliptical
slabs of slate.  These days,
some stone-yards will simply
accept the major and minor axes
of the ellipse and, somehow,
(water-jet?) cut a slate which
is a close-to-perfect ellipse.

Almost the first task is to find
the centre and the axes.  Clearly
you cannot fold a slate in half
and the traditional way to proceed
is to put a large sheet of paper
over the slate and crease it down
all round the rim.

You then cut round the crease and
attempt to follow your procedure!

No doubt in practised hands this
can give a good result but I find
that, no matter how carefully I
work, the two axes are invariably
not (quite) at 90 degrees.  

It is quite hopeless to refold the
paper; the folds are too close
together.  You have to start again.
I usually get it to my satisfaction
at second attempt but occasionally
I have had to have three tries.

It is especially hard if the ellipse
is not too far off being a circle,
say 1200mm x 1050mm.

This is a big sheet of paper.  Just
try it for yourself and see the
challenges!

Donald pointed to a link which gives
the answer I was aiming at.  Alas,
although I find the geometry a
delight, the practicalities are
just as challenging as the folding
method!  You can get the axes at
right-angles fairly easily but
you find the "centre" is nearer
one end of the major axis than
the other, ditto the minor axis.

As so often with sundials, the
theory may be elegant and
straightforward but real life has
a way of making implementation a
struggle!

Frank


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RE: Sundial Puzzle Corner

[R. Hooijenga]

Brilliant!

Also, I never realized that the distance from the focus to the end of the
minor axis equals the half major axis (second link, step 2).
What bliss on this Sunday morning (CET).

Rudolf 

-Oorspronkelijk bericht-
[...] Here are two websites that together give a graphical approach to
determining the axes of the ellipse:
  first find the ellipse center
http://www.had2know.com/makeit/find-ellipse-center.html
  then the ellipse axes
http://www.had2know.com/makeit/find-ellipse-axis-focus.html
   Don Snyder

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Re: Sundial Puzzle Corner

[Donald L Snyder]

That was a neat answer for an ellipse drawn on paper, as Frank posed the 
problem.  But, what if the ellipse is marked out on a patch of grass or 
a walkway?  Here are two websites that together give a graphical 
approach to determining the axes of the ellipse:
 first find the ellipse center 
http://www.had2know.com/makeit/find-ellipse-center.html
 then the ellipse axes 
http://www.had2know.com/makeit/find-ellipse-axis-focus.html

  Don Snyder
P.S.   Come to St. Louis for the 2017 NASS Annual Meeting and a peek 
at the total solar-eclipse that will occur.  C U there (or here since 
I'm already there).





On 10/29/2016 7:56 AM, Frank King wrote:

Dear Geoff,

I had a private list of people I
expected to respond and I have been
waiting for your answer!


I think that these instructions might
work if you ventured to the antarctic
circle during the southern winter and
then trecked to a position such that
your latitude is greater than 90-Dec
(so that the midnight sun is visible)
but less than 90.  The shadow of the
stake would then be longest and point
to true north at true solar midnight.

Well, that is easily the best answer so
far!

I have just two teensy quibbles:

1. "during the southern winter"

You won't get too much sun then.
I suspect you meant southern
summer!

2. "your latitude is greater than 90-Dec"

This is difficult to express.  In the
southern summer the Dec is negative.
Suppose Dec = -20 then you would
require me to be at a latitude
greater than 110 which is a bit of
a challenge!

Taking southern latitudes to be
negative and southern summer
declinations to be negative too,
The condition is:

 Dec < -90 -Lat

So if you are at -70 then the
declination would need to be
less than (i.e. more negative
than) -20.

Alternatively you need:

 Lat + Dec < -90
or:

 Lat < -90-Dec

Almost using your words: you venture to
the antarctic during the southern summer
and choose any day when you have midnight
sun.  The shadow of the stake would then
be longest and point to true north at
true solar midnight.

I feel sure that the writer of the
website didn't mean this!!

There is a practical consideration too...

Over a 24-hour period the shadow traces
out an approximate ellipse.  [It is not
quite closed because the declination
doesn't stay still but that's not the
problem.]  The difficulty is pinpointing
the ends of the major axis.  The solar
altitude is changing very slowly at each
end and the end-points are not sharply
presented.

Ancillary Question:

I hand you a perfect ellipse drawn
on a sheet of paper.  The ellipse is
unornamented.  What is the geometrical
construction required to determine the
axes?

Aren't sundial questions fun?

Frank

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RE: Sundial Puzzle Corner

[Dave Bell]

Personally, I'd fold the paper, superimposing the reflected half-ellipse,
crease it, unfold, rotate ~90 degrees and repeat!

Dave

-Original Message-
From: sundial [mailto:sundial-boun...@uni-koeln.de] On Behalf Of Frank King

Ancillary Question:

I hand you a perfect ellipse drawn
on a sheet of paper.  The ellipse is
unornamented.  What is the geometrical
construction required to determine the
axes?

Aren't sundial questions fun?

Frank

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Re: Sundial Puzzle Corner

[Frank King]

Dear Geoff,

I had a private list of people I
expected to respond and I have been
waiting for your answer!

> I think that these instructions might
> work if you ventured to the antarctic
> circle during the southern winter and
> then trecked to a position such that
> your latitude is greater than 90-Dec
> (so that the midnight sun is visible)
> but less than 90.  The shadow of the
> stake would then be longest and point
> to true north at true solar midnight.

Well, that is easily the best answer so
far!

I have just two teensy quibbles:

1. "during the southern winter"

   You won't get too much sun then.
   I suspect you meant southern
   summer!

2. "your latitude is greater than 90-Dec"

   This is difficult to express.  In the
   southern summer the Dec is negative.
   Suppose Dec = -20 then you would
   require me to be at a latitude
   greater than 110 which is a bit of
   a challenge!

   Taking southern latitudes to be
   negative and southern summer
   declinations to be negative too,
   The condition is:

Dec < -90 -Lat

   So if you are at -70 then the
   declination would need to be
   less than (i.e. more negative
   than) -20.

   Alternatively you need:

Lat + Dec < -90
   or:

Lat < -90-Dec

Almost using your words: you venture to
the antarctic during the southern summer
and choose any day when you have midnight
sun.  The shadow of the stake would then
be longest and point to true north at
true solar midnight.

I feel sure that the writer of the
website didn't mean this!!

There is a practical consideration too...

Over a 24-hour period the shadow traces
out an approximate ellipse.  [It is not
quite closed because the declination
doesn't stay still but that's not the
problem.]  The difficulty is pinpointing
the ends of the major axis.  The solar
altitude is changing very slowly at each
end and the end-points are not sharply
presented.

Ancillary Question:

I hand you a perfect ellipse drawn
on a sheet of paper.  The ellipse is
unornamented.  What is the geometrical
construction required to determine the
axes?

Aren't sundial questions fun?

Frank

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Re: Sundial Puzzle Corner

[Barry Wainwright]

If you stand at the South Pole, any time that the sun is visible, then any 
shadow is pointing North, because all directions are north!


> On 28 Oct 2016, at 07:58, Frank King  > wrote:
> 
> When you have recovered, ponder this puzzle:
> 
>  Where on the planet would you have to
>  be, and at what time of year, for these
>  instructions to give the correct result?

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Re: Sundial Puzzle Corner

[Geoff Thurston]

Frank,

Thanks for the puzzle. I think that these instructions might work if you
ventured to the antarctic circle during the southern winter and then
trecked to a position such that your latitude is greater than 90-Dec (so
that the midnight sun is visible) but less than 90. The shadow of the stake
would then be longest and point to true north at true solar midnight.

Geoff



On 28 October 2016 at 06:58, Frank King  wrote:

> Dear All,
>
> I have been looking at new U.K. educational
> website which has a whole category devoted
> to sundials.  Early on, there is section
> "Finding True North".  See:
>
>   http://wiki.dtonline.org/index.php/Finding_True_North
>
> This is what it asserts:
>
>   The Sun can be used to find True North
>   quite simply by placing a vertical stake
>   in the ground and noting which direction
>   the longest shadow points to.
>
> You may now take a two-minute break while
> you recover from rolling on the floor in
> a state of helpless laughter.
>
> When you have recovered, ponder this puzzle:
>
>   Where on the planet would you have to
>   be, and at what time of year, for these
>   instructions to give the correct result?
>
> Moral:
>
>   It is better to learn from other people's
>   mistakes than from your own.
>
> Frank King
> Cambridge, U.K.
>
> P.S. The home page of the website is:
>
>   http://wiki.dtonline.org/index.php/Category:Sundials
>
> It won't take long before you notice other
> erroneous assertions :-)
>
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Re: Sundial Puzzle Corner

[John Lynes]

Shadow will be longest to the north at midnight, twice a year, at any point
north of the Arctic Circle.
John Lynes

On 28 October 2016 at 07:58, Frank King  wrote:
>
>
>   Where on the planet would you have to
>   be, and at what time of year, for these
>   instructions to give the correct result?
>
>
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Sundial Puzzle Corner

[Frank King]

Dear All,

I have been looking at new U.K. educational
website which has a whole category devoted
to sundials.  Early on, there is section
"Finding True North".  See:

  http://wiki.dtonline.org/index.php/Finding_True_North

This is what it asserts:

  The Sun can be used to find True North
  quite simply by placing a vertical stake
  in the ground and noting which direction
  the longest shadow points to.

You may now take a two-minute break while
you recover from rolling on the floor in
a state of helpless laughter.

When you have recovered, ponder this puzzle:

  Where on the planet would you have to
  be, and at what time of year, for these
  instructions to give the correct result?

Moral:

  It is better to learn from other people's
  mistakes than from your own.

Frank King
Cambridge, U.K.

P.S. The home page of the website is:

  http://wiki.dtonline.org/index.php/Category:Sundials

It won't take long before you notice other
erroneous assertions :-)

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