were I to go on a walkabout in the town's streets
with only one map I would take
Beck's map (1933) of London Underground or his 1951 Métro
de Paris one , based on
the concept of an electrical diagram
Think about it, 1933, and
so modern in concept.
( see this maps's history and the London Transport from which these maps are 'quoted'
- I send
them a mail asking for their permission and never got an answer so
judiciary principle that "he who says nothing in opposition is
I am quoting their
maps here while honestly giving credit
where credit is due.
By the way "quoted" is better form than "lifted"
but baring the acknowledgment it is all the
same : using someone
work. May be that is a little bit what this climber on a forum
See a Métropolitain ( Métro for short ) de Paris
map for comparison.
No doubt topography
of much less use for
understanding of the essential
relations between stations than
Instead of stations, for knots
think of crossings.
will, I hope, convince you of
the superiority of topology over topography to
understand the essential
relations between elements .
This superiority apply to knots and their diagrams, compared to a full
artistic drawing or a
photography, which are of course the
which can give a notion of the actual
What is topology ?
Simple !................................. it is that which makes that this
is still Bugs Bunny.
Like it or not, believe it or not it is
This is what is called "rubber sheet geometry"
Of course the geometry
of angle and distance
Only the relations between elements/parts are
That is why in a morphing
software you can reverse the process manually - done quicker
'undo' button though-
relations been destroyed it would no be possible to
reverse, to back track.
When in an attempt to disguise your appearance you pad your cheeks with
you respect the
topology but you so alter the geometry (topography
too, if you have to
depict the new face) that you hinder fast human visual identification.
Topology is what make that, essentially, the uncreased flat
paper is 'not
topologically altered but stay 'equivalent', when you
are crumpling it. (only crumpling, no
no gluing please -
just a tiny
and it is not any more
Well, in everyday life that apply with no trouble
incurred if it is a
banknote you crumple, it
will not detract its value but though you will have committed no
topological damage I
advise against doing it with this exceptional Durer engraving at the
If you do not know the magnificent work of D'Arcy Thomson
( Sir D'Arcy Wentworth Thompson 1860 -
On Growth And Form published in 1917
here is an
illustration of it.
You see now that 'topology' is not reserved to mathematicians, it lives
the real world.
Why not use this tool on 'real life'
REVERSE AND MIRROR
Not trying to be finicky and difficult but still hoping to built "help"
classification of knots.
Starting stance : would you accept to hear your
lawyer, surgeon, architect, tree surgeon,
car mechanic use
words from their station in life in the way every one of us
ill-treat and mis-use words in mundane conversation ?
I hope not!
Well then why accept to hear it or to write it when
seeking clarification about knots.
Knots are geometric entities and as such should get the
having a highly
guarded language used when speaking about them.
Mundane, everyday mishandling of language should be given the
"leave town before sunset" or rather before discussion begin.
Failing that nothing
is possible without the greatest potential confusion and ambiguity.
Hence this 'introducing' topic and some of the following ones.
It looks to me that is is very important not to use 'REVERSE(D)',
without due precautions.
'Reversed' is sometimes used instead of 'MIRROR(ED), and sometimes as
direction' which is it meaning.
Too often people are mis-using one for the other.
Ashley at times made a somewhat 'not very guarded' use of
some words, 'reverse' being
one of the victims.
Page 49 of ABOK about #278 he wrote:
"...that is identical with ....(#1676), although reversed...."
Utterly untenable on the ground of elementary logic : if it is
'reversed' there is absolutely no
way it can be 'identical'.
By now, Reader my other self (to paraphrase RIMBAUD), you should be
always keeping in mind
these notions if by
happenstance you were not before.
This will serve only as an introduction for now : Page
6 you will find the rationale for a much more
use of some words.
Geometric transformations , Image manipulation and propriety of