PLAYING WITH ABOK #525 and Figure-of-9
( this possibility was
evoked without detail in a post on Igkt forum, Dan_Lehman it must
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
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
Point to keep in mind : if you can go 'smoothly', 'continuously,
in other words only making
Reidemeister's moves from one knot appearance to another one then they
(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,
A VERTICAL MIRROR IS INVERTING NEITHER LEFT/RIGHT
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
physiologically - does not make you less prone to have it !
on the wall in front of you into which you are looking at
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
between these two extremities.
Yet the bilateral left/right
fooling us into the psychological illusion of seeing a
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
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
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
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
you see looking at the back of
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
Still not convinced ?
Draw on a transparency an
arrow with a sharp point at one extremity and a feather at the
Hold it in front of the mirror.
Both arrows, the one on the paper and the one in the mirror,
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
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
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
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
Not convinced ?
Take one pencil sharpened at one side and blunt at the
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
Anyway, writing like what I found it done on a US Coast Guard site, "tie a second
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
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)
if the mirror is lying
in the ( y, z) plane - vertical on one of your side - then it
is (x) that
Just remember this Cartesian coordinates so as to be at ease
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 :
image in a
mirror is placed such that the image distance 'behind the
mirror' ( depth ) is equal as the
object distance in front of the
Fermat's Principle: Light follows the path of least time.
speed is constant, the
minimum time path is in fact the minimum distance path