Nautile aka Charles Hamel's personal pages
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It is best to not make any confusion between topoLOGY (study of...) and topoGRAPHY
 (depiction of...),
I will let go of choroGRAPHY. Think topography as local and  chorography as regional.

Which illustration do you think is the clearer of the pair shown if the objective is to get a
quick grasp of the essential relationship between 'parts'?

Is it this map respecting the topology or this one respecting the topography ?

There is no doubt in my mind, I will take topology any time for understanding the relationship.

But were I to go on a walkabout in the town's streets with only one map I would take
topography !

Beck's map (1933) of London Underground or his 1951 Métro de Paris one , based on
the  concept of an electrical diagram are marvels.

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 following the
judiciary principle that "he who says nothing in opposition is agreeing"  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 else's work. May be that is a little bit what this climber on a forum
defined as "cross-pollination")

See a Métropolitain ( Métro for short ) de Paris map for comparison.

Compare it with Beck's (1951)

No doubt  topography is of much less use for a quick understanding of the essential
relations between stations than topology is.

Instead of stations, for knots think of crossings.

This 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 only one which can give a notion of the actual
external aspect.

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 Bugs !

This is what is called "rubber sheet geometry"

Of course the geometry of angle and distance is destroyed.
Only the relations  between elements/parts are respected.

That is why in a morphing software you can reverse the process manually - done quicker
by hitting the 'undo' button though-
Had the relations been destroyed  it would no be possible to reverse, to back track.

Want to see the rubber sheet in action ?

When in an attempt to disguise your appearance you pad your cheeks with cotton wads
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 sheet of paper is 'not
topologically altered but stay 'equivalent', when you are crumpling it. (only crumpling, no
cutting, no tearing, no gluing please - just a tiny pin-point hole and it is not any more
topologically equivalent.)

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
strongly advise against doing it with this exceptional Durer engraving at the Museum of Art.

If you do not know the magnificent work of D'Arcy Thomson
( Sir D'Arcy Wentworth Thompson    1860 - 1948 )
 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 in the real world.

Why not use this tool on 'real life' knots.



Not trying to be finicky and difficult but still hoping to built "help" toward a
classification of knots.

Starting stance :
would you  accept to hear your lawyer, surgeon, architect, tree surgeon,
car mechanic  use technical 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 honour of having a highly
guarded language used when speaking about them.

Mundane, everyday  mishandling of language should be given the warning :
"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 'in opposite
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'.

It is not a reverse but clearly a mirror.
Reverse is quite different
If mirror image it is then it cannot be identical.

Confusion of words and concepts should be avoided.

Trefoil and fig-8 prior mirroring
Trefoil and fig-8 post mirroring
Trefoil mirror
Fig-8 mirror
Mirroring Fig-8
Another mirror-back
Yet another mirror-back
#525 mirror-back

By now, Reader my other self (to paraphrase RIMBAUD), you should be perfectly at
ease with 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 surveyed use of some words.
Geometric transformations , Image manipulation and propriety of language.

Go to page 4 of  bat's belfry

Copyright 2005 Sept - Charles Hamel / Nautile -

Overall rewriting in August 2006 . Copyright renewed. 2007-2012 -(each year of existence)

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