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From: Veronica Karlsson <e93-vkn@nospam.sm.luth.se>
Subject: Re: The Waterfall
Date: 1997/07/22
Message-ID: <33D4D1AB.446B9B3D@nospam.sm.luth.se>
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References: <33CE7C9D.3F54BC7E@nospam.sm.luth.se>
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Organization: Junk e-mail gladly reported (I have got credit for chucking
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Newsgroups: alt.ascii-art


DaSmurf wrote:
> 
> Veronica Karlsson wrote:
> > Michael Naylor wrote:
> > [ 8< ]
> > > WOWZERS!  I wanted to add the waterwheel, but it seemed awfully tough.
> > > Nice job Veronica!
> >
> > :) *blush*
> >
> > I just looked at the picture.... (and I happened to know that it is
> > based on two "impossible triangles" placed above each other.
> > Understanding how it works probably made it a little bit easier for me
> > to draw it)
> >
> > The idea behind it looks something like this:
> >
> >       __        __
> >      /\ \      /\ \
> >     /  \ \    /  \ \
> >    / /\ \ \  / /\ \ \
> >   / / /\ \ \/ / /\ \ \
> >  / / /__\_\/ / /__\_\ \
> > / / /______\/ /________\
> > \/_____________________/
> >      -> waterfall ->
> >
> > (you'll have to turn it 90 degrees to get it right)
> > --
> 
> Have to change a little in your explanation... ;)

No, you misunderstand, the discussion was about a specific painting by
M.C. Escher where water runs away in a canal system into the distance
and then it falls down a waterfall and back to the beginning of the same
canal system. That particular picture is based on a "double impossible
triangle" such as I have drawn above as an attempt to _illustrate_ my
words. This little sketch-like illustration is not what we were
interested in, we have seen that one before.

You can admire Escher's picture here:
http://www.cs.utah.edu/~dmcallis/Pic/Escher/waterfall.jpg

> 
> The first picture made that way was like this:
>       __
>      /\ \
>     /  \ \
>    / /\ \ \
>   / / /\ \ \
>  / / /__\_\ \
> / / /________\
> \/___________/

I know that, but that is not what we were talking about. We were talking
about this:

The Waterfall
Michael Naylor (7/97)
   
             ,__--"^-_.
      ,__--"" ;__--_ : "-_.
  _--",~__--""  .:|:"-_.~.`"-_.
 !-_`. "-_    .:;;|:::::"-_.'`."-_.
 |::"-_`. "-_'::::"-_::::::"-_.'.  "-_.
 |:::::"-_))))" '::::"-_:::::::"-_. . '"-_.
 |::::::::;!)!)   ':::(()-_::::::::"-_. `  "-_.
 |::::::::;!!!)      ((o))::"-_::::::::";-_.` ;"-_.
 |::::::::;'!!!   ..  (()):::::"-_::::;_-"  ',_-" |
 |::::::::; !!!  ::::..!!  '::::::"_-" , ,_-"     |
 |::::::::;.!!!   ''          ;_-"'  ,_-"         |
 |::::::::; !!!           ,_-"  ~,_-"             |   ()
 |:::::::.( !!!')     ,_-" . ,_-"                 |  (())
 |::::::(:(`!!!!'),_-".  ,_-"                  _-"  ((()o)
 |::::::.((,!!!_)~.,~,_-"            ()     _-"  ... (())
 |:::::::',((!'),~;_-"              (o)) _-"    ::::..!!
 |::::::(.:~(_;_-" )            :  ((()o)     () '''
 |:::::::::"::|  .)            :::,_(o))     (o))
 |::::::::.:::|                 ;-:..!!     ((()o)
 |::::::::::::|              _-"        ...  (())
 |::()):::::::|           _-"          :::::..!!
.."(()o)::::::|        _-"              '''
..((o)))):::::|     _-"
...((())"-_:::|  _-"
..  '||    "-_|-" mn
:::..!!                
''

of which I made my own version (complete with big wheel and power
station):


            _.-'\
        _.-'     \
    _.-'          \
_.-_-'\         _.-'`-._
  |\ _-|    _.-'_.-'`-._`-._
  | |  |   |_.-'        `-._`-._|`-._
\ \ |  |  /    /     _      ``---._  `-._
 \ \|_-' | / /    .-' `-._|`-._    ``-._ `-._
  \     || ||  / /|-'|`-._     `-._     `--._`-._|`-._
   \_.-'   | |      |||   `-._     `-._      ``---._  `-._
   |    |   | | |   |||    |||`-._     `-._|`-._    ``-._ `-._
   |        |    __ |||    |||    `-._          `-._     `-._ `-._
   |    |    |  |_.-|||.   |||        `-._       _.-'|       `-_.'|
   |    |   |  ||/  ||| \  |||          _.-'|_.-'_.-'      _.-'   |
   |     | | |  .   |||  . |||      _.-' __.---''      _.-'       |
   |    |   |  ||   |||    |||'|_.-'_.-''     _.-'|_.-'        _.'
   |    ||    ||    |||  |'|||----''      _.-'             _.-'
   |       | |  |   |||   '|||   _.-'|_.-'             _.-'
   |    |  ||  _.\  ||| /  |||.-'                  _.-'
   |    | |  '  |_`.|||' _.-'                  _.-'
   | _.-||  |  |    ||`-'                  _.-'
   |`-._     |   _.-'                  _.-'
   |    `-._ _.-'                  _.-'
   |        |                  _.-'
   |        |              _.-'
-._|        |          _.-'
   `-._     |     _.-'
       `-._ | _.-'
           `|'


And that is what Michael Naylor was commenting on. I then explained that
the picture by Escher that is obviously the idea behind our, somewhat
simpler, pictures is based on two impossible triangles placed above each
other in the way that I tried to describe above and that that is why my
picture looked better than his "three-and-a-half-sided" version...

Then, a day or two later, I made this picture, which looks more like the
original by Escher:


                 _.-"""-._    
               .'         `.  
              /             \ 
             |               |
             |               |
             |               |
              \             /                              _.-"""-._
             _.`.         .'.                            .'         `.
            |`-._`-.._..-'_.-|                          /             \
            |  _ `-._ _.-'   |                         |               |
            | | `-._ " _.-'| |                         |               |
            | |    || |    | |                         |               |
            | |    || |    | |                          \             /
            | |    || |    | |                          _`.         .'.
            | |    || |    | |                         |`-.`-.._..-'.-'|
            | |    || |    | |                         |    `-._.-'    |
            | | _.-'| |_   | |                         | |`-._   _.-'| |
            | |'_.-'| |_`-.| |                         | |   || |    | |
           |_.-'    | | `-._ `._|`-._                  | |   || |    | |
          /    /    | |     ``---._  `-._              | |   || |    | |
         | / /    .-' `-._|`-._    ``-._ `-._          | |   || |    | |
        || ||  / /|-| |-._     `-._     `--._`-._|`-._ | |   || |    | |
           | |   ' || |\  `-._     `-._      ``---._  `| |   || |    | |
        | - | | |  || | \  | |`-._     `-._|`-._    ``-| |`-.|| |    | |
    _.-'    |   |  || |  \ | |    `-._          `-._   | |-._ | |._  | |
_.-'     |     |   || |   \| |        `-._          `-.| |   `| |_ `-| |
        |    |  |  || |    | |            `-._       _.| |    | | `.-| |
        |   |  ||  || |    | |              _.-'|_.-'_.-'     | |_.-'  |
         | | |  .  || |    | |          _.-' __.---''       _.| '      |
        |   |  ||  || |    | |\_.-'|_.-'_.-''      _.-'|_.-'        _. |
         | |  |    || |    | |'      _-        _.-'             _.-' | |
        ||    ||   || |_.-'| |_.---''      _.-'             _  |     | |
\          | |  |  || |_.-'| |    _.-'|_.-'             _.-'|| |     | |
 \      |  ||    .'_| |    | |_.-'                  _.-'| | || |     | |
  \     | |     |._ | |    | `-._               _.-'    | | || |     | |
   \    ||  |  |   `| |    ```-._`-._|`-._  _.-'        | | || |     | |
   |\_.-|       |      `-._      ``---._  `-._          | | || |     | |
   |     |                 `-._|`-._    ``-._ `-._      | | || |     | |
   |    || ||  |                    `-._     `--._`-._|`| | || |     | |
   |       | |  |                       `-._      ``---.| |`|| |     | |
   |    |   | | |`-._                       `-._|`-._   | |-.| |-._  | |
   |        |      ||`-._                            `-.| |  | |._ `-| |
   |    |    |  |  || |  `-._                         _.| |  | |  `-_| |
   |    |   |  ||__ | |    | `-._            _.-'|_.-'_.-'   | |_.-'   |
   |     | | |  ._.-| |.   | |   `-._    _.-' __.---''      _|         |
   |    |   |  ||/  | | \  | |  _.-'|_.-'_.-''     _.-'|_.-'           | 
   |    ||    ||    | |  .'| |-' __.---''      _.-'                    |
   |       | |  |   |.|   '| |-''     _.-'|_.-'                        |
   |    |  ||  _.   | |  | | |    _.-'                                 |
   |    | |  '  |   | |    | |_.-'                                     |
   |    ||  |  | \  | | / _| '                                      _.-'
   |_.-'     |   _`.| |'-'                                      _.-'
   |`-._          _.|                                       _.-'
   |    `-._  _.-'                                      _.-'
   |        |'                                      _.-'
   |        |                                   _.-'
   |        |                               _.-'
   |        |                           _.-'
   |        |                       _.-'
   |        |                   _.-'
   |        |               _.-'
_  |        |           _.-'
 `-|_       |       _.-'
     `-._   |   _.-'
         `-.|.-'


If you look at it you should be able to find the "double impossible
triangle" (the waterfall itself is the bar that is mutual between them).
It even has an extra side to it so maybe it should actually be described
as a "triple impossible triangle":

       ____________
      /\  ________ \
     /  \ \______/\ \
    / /\ \ \  / /\ \ \
   / / /\ \ \/ / /\ \ \
  / / /__\_\/ / /__\_\ \
 / / /______\/ /________\
 \/_____________________/
      -> waterfall ->

(turned 90 degrees because that made it easier to draw in ascii)

[ 8< smurfs 8< ]

-- 
:)
Veronica Karlsson
( e93-vkn@sm.luth.se  http://www.ludd.luth.se/~vk/ )