display | more...

                  ::::::::                  
                 ::::::::++                 
                ::::::::++++                
               ::::::::++++++               
              ::::::::++++++++              
             ::::::::++++++++++             
            ::::::::++++++++++++            
           ::::::::++++++++++++++           
          ::::::::++++++++X+++++++          
         ::::::::++++++++XXX+++++++         
        ::::::::++++++++XXXXX+++++++        
       ::::::::++++++++XXXXXXX+++++++       
      ::::::::++++++++ XXXXXXXX+++++++      
     ::::::::++++++++   XXXXXXXX+++++++     
    ::::::::++++++++     XXXXXXXX+++++++    
   ::::::::++++++++       XXXXXXXX+++++++   
  :::::::::::::::::::::::::XXXXXXXX+++++++  
 :::::::::::::::::::::::::::XXXXXXXX+++++++ 
:::::::::::::::::::::::::::::XXXXXXXX+++++++
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX+++++ 
 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX+++  
  XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX+   

The impossible triangle was first conceived by the Swedish artist Oscar Reutersvärd in 1934. Although at first glance it may appear to be a regular three-dimensional object, with each corner looking perfectly normal, when considered as a whole, the paradox becomes apparent. The sides of the triangle look as if they both recede and come towards you simultaneously. Yet, somehow, they meet in an impossible configuration. The drawing itself isn't impossible. Just your three-dimensional interpretation of it.
The Impossible Triangle is an optical illusion, first attributed to Swedish artist, Oscar Reutersvärd, in 1934. Originally, he drew the shape with strings of cubes, rather than solid bars.

The more familiar version (shown below) was created independently by Roger Penrose after attending a lecture by Dutch artist, M. C. Escher in 1954.

        __
       / /\
      / /  \
     / / /\ \
    / / /\ \ \
   / / /  \ \ \
  / /_/____\ \ \
 /__________\ \ \
 \_____________\/
Escher, in turn, was inspired to create images, including his famous "Waterfalls" image, by the work of the Penroses.

In 1980, the Swedish government honoured Reutersvärd by using his original image on a stamp between 1982 and 1984.

How to build an impossible triangle

Yes, it is possible to actually build an impossible triangle. It's all a question of perspective, as you will see.

  1. The first stage is to build the three bars that will make up the triangle. Take a piece of A4 or legal size card or stiff paper. Cut out three rectangles of about 12 x 25 cm.

  2. Put these three pieces together and fold them in half along their long axis and the do it again. The stack of paper should be a quarter of the width you started with.

  3. Crease and unfold. You should have three bits of paper that look like this:
    +---------------------+
    |                     |
    |- - - - - - - - - - -|
    |                     |
    |- - - - - - - - - - -|
    |                     |
    |- - - - - - - - - - -|
    |                     |
    +---------------------+
    
  4. Fold each of these into a tube and tape them. You now have three square cross section tubes of paper.

  5. Now we will tape them together. The first taping is easy. Just tape one end of on of the tubes to another making sure you form a right angle. Make the end look a little like this:
          /\/\/\
         |      |
         |      |
         |      |
         |______|_______
         |              \
         |              /
         |______________\
    
    
  6. Hold this construction so that it looks like an L. Now tape the third tube onto the bottom right hand side of the L orientated facing away from you. Like so:
          /\/\/\
         |      |  L shape
         |      |     |           End of newest tube     
         |      |     v                  |
         |______|_______     ____________v__
         |              \    \       |      |
         |              /    /       |      |
         |______________\    \_______|______|
    
    
    Or in 3D (ish)
    
     X          ____     
               /__a/|
              |   | |                     ___
              |   | |                   b/  /|
              |   | |                   /  / |
              |   | |                  /  /  /
              |   | |                 /  /  /
              |   | |                /  /  /
              |   | |_______________/  /  /                
              |   |/_______________/__/  /
              |   |                |  | /
              |___|________________|__|/
    
  7. That's it. You now hold in your hand an impossible triangle, Ha!


  8. No, wait, before you click that vote button: Close one eye and look at your weird construction from the X so that 'a' and 'b' are on top of each other. It will look like a right angle between the 'a' and 'b' tubes if you built it with sufficient accuracy. The other two angles will also look like right angles (because they are). Three right angles in a triangle? Surely that's impossible!

It is also possible to create other impossible figures, such as the impossible cube, using methods like this. Experiment, write up, enjoy impossibility!


Sources:
Bruno Ernst, "Adventures with Impossible Figures", ISBN: 0906212545

Log in or register to write something here or to contact authors.