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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