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PLAYING WITH ABOK #525 and Figure-of-9  
( this possibility was evoked without detail in a post on Igkt forum, Dan_Lehman it must
have been if memory serves )

Being given this tip about these two knots will you succeed in making one of them and then
only using Reidemeister's moves transmogrify it into the other one?  Doing thus you will be
pragmatically establishing, if not proving in a formalized mathematical way, that the one is
'equivalent' to the other

Here are the first stages and the last stages of the metamorphosis that can be made in a
quicker manner but that will be less easy to read for a first contact with the notion.

This is the resume of the moves I used.
Another summary may be of help to clarify.

Point to keep in mind : if you can go 'smoothly', 'continuously, in other words only making
use of Reidemeister's moves from one knot appearance to another one then they are
(will tackle the problem of 'sameness' which under many disguises  later confuse unaware
Continuous moves are to be opposed to the 'discrete' moves that are re-threading, cutting,
tearing, gluing



Inspired by what seems to a me a not so rare confusion :
the one made between REVERSE and MIRROR about knots.

Shocking is not it ?  
Contrary to every day experience ?

Well that may be but nevertheless it is still true.

A playful guess : what is the difference between error and illusion ?

Well when you are shown how the error was made you are less prone to fall into it.
Knowing an illusion - think about so many of the visual illusions we are plagued by
physiologically - does not make you less prone to have it !

The mirror on the wall in front of you into which you are looking at yourself  (beware
beautiful flower, recall what happen to Narcissus ?) gives no more left/right inversion than it
gives an up/down one.

When a person looks at their image in a mirror they never are under the illusion that they are
 head down and feet up.
That is so because there is a very evident non-symmetry between these two extremities.

Yet the bilateral left/right symmetry is fooling us into the psychological illusion of seeing a
left/right inversion.
We believe that it put left where right was and right where left was.

Apparently the 'virtual one' in the mirror is doing with what is taken as a left hand what the
'in the flesh' one is doing with the right and vice-versa.

This is a mistaken impression.

Think about it !

You sport a wristwatch on your left wrist, your mirror-self  is doing it too.
It is a fact, image has no watch on the right wrist.

Not convinced ?

Well just put your left watch wearing hand flat on the mirror and you will easily see that
where you are touching  your image there is a watch too. 
No left/right inversion.

Still not convinced ?  

Write with one of those special pens on a transparent sheet.
Put what you have written in front of the mirror : you can readily see that what you see  in
the mirror is what (it strictly correspond to) you see looking at the back of the
No left right inversion .

Another thought :  look in your driving mirror and you should see that in the car behind the
driver is not on the side your passenger seat is, no inversion left/right

Still not convinced ?
Draw on a  transparency an arrow with a sharp point at one extremity and a feather at the
other. Hold it in front of the mirror.
Both arrows, the one on the paper and the one in the mirror, are pointing to the same side
of you. There is no left / right inversion.
Or much simpler just point with one of your hand toward one of the side walls of the room
you are in , you will see your image pointing to the same wall.

Not convinced ?
Well I will gladly leave you with your certitudes and keep mine.
Do not lose any more time as you had better cease reading at this point if you still hold that
a mirror is inverting left / right.

The geometric reality of the mirror effect  is  no more a left / right inversion than it is an
up / down inversion, but there is for sure an inversion somewhere :
it is an inversion on the depth axis.  The axis that is perpendicular to the mirror surface.

Not believing it ?

Make a cross with a blue and a red pen and hold it in front of your mirror.
Say the blue pen in the cross you hold is the nearest of you, then this blue pen will be the
furthest of you in the image of the cross. Depth inversion.

Not convinced ?
Take one pencil sharpened at one side and blunt at the opposite side.
Point it toward the mirror sharp point first and approach to touch the mirror with the point
of the pen, point of the real pen will touch the point of the image of the pen while the blunt
extremity that is nearest to you is now in the mirror  furthest to your real self.

Our illusion of left / right inversion is due to our bilateral symmetry.
Just take  a photography of  a landscape with no feature of symmetry and look at it in a
mirror : no illusion of left / right inversion because no bilateral symmetry.

Anyway, writing like what I found it done on a US Coast Guard site, "tie a second overhand
on top so that it mirrors ( right and left reversed) the first one...." is really not to be done.

Even if they are writing about the Reef Knot and are far from demeaning it and  also
hold the bowline in good esteem I cannot accept their notion of mirror inverting right and

Summary : in a set of 3D Cartesian coordinates the mirror is lying in the plane of 2 axis and
the inversion of depth is on the third axis perpendicular to the plane of the mirror.

If mirror is on ( x, y)  plane - vertical like your wall mirror -  then the (z) becomes (-z),

if the mirror is lying in the (  x, z) plane - horizontal like the surface of a lake - then the (y)
becomes (-y),

 if the mirror is lying in the ( y, z) plane  - vertical on one of your side - then it is (x) that
becomes (-x)

Just remember this Cartesian coordinates so as  to be at ease with  the next topic.

For knots the 'mirror' image is in a mirror held perpendicular to the horizontal plane on
which the knot is drawn, that is mirror is on the ( y, z) plane  while the knot is on the ( x, z )
plane hence an inversion along the  (x) axis.

Hard physics facts :
The reflected image in a plane mirror is placed such that the image distance 'behind the
mirror' ( depth )  is equal as the object distance in front of the mirror.

Fermat's Principle: Light follows the path of least time.
As the speed is constant, the minimum time path is in fact the minimum distance path

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Copyright 2005 Sept - Charles Hamel / Nautile -
Overall rewriting in August 2006 . Copyright renewed. 2007-2012 -(each year of existence)

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