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Following on from the impossible figures node, this node contains further examples and explanations of the most common types of impossible figures. The first kind of impossible figure to be considered is the dual and multi-level flat planes. Below is a simple example of this kind of figure.

It shows a flat ring with a set of stairs going down from one level to the same level. This is possible because in two dimensions there is plenty of room for a seemingly descending set of lines. You might think that this is a possible figure because you could simply tilt the stairs. Look closer at the lines making up the stairs. They are all parallel with the other lines of the figure. Tilting the stairs is a good solution but the resulting picture would not look like this.

             _______________________________
            /\                              \
           /  \      __________________      \
           \   \     \              _ /\      \
            \   \     \___________ /\ \ \_     \
             \   \     \       _/\ \ \_\/ /\    \
              \   \     \  _ /\ \ \_\/   /  \    \
               \   \     \/\ \ \_\/     /\   \    \ 
                \   \     \ \_\/       /  \   \    \ 
                 \   \     \/         /    \   \    \
                  \   \     \________/______\___\    \
                   \   \                              \
                    \   \______________________________\
                     \  /                              /
                      \/______________________________/


The next figure is a classic Escher figure that takes this idea a little further. The explanation for the impossible triangle in impossible figures is about right angles that cannot be right angles. The right angles between the apparently horizontal and vertical areas of this figure are not possible. Indeed, if you were to cut this figure in half and join up the lines in the right way you would have two impossible triangles. The horizontal plane in this figure extends out into one axis (Z) and then re-joins the vertical bar from an opposing axis (X) without apparently changing orientation.

                                             Y
             _                               |
           _/ \_                             |
         _/     \__                          |
        /          \__                     _/ \__ 
       |\__           \__                _/      \__Z
       |   \__           \__           X/
       |      \__           \__
       |         \__           \__
       |            \__           \__
       |     __        \__           \__
       |     | \__        \__           \__
       |     |   |\__        \__           \
       |     |   |   \__        \         _/|
       |     |   |      \__    _/       _/  |
       |     |   |         \__/       _/    |
       |     |   |         _/       _/    _/
       |     |   |       _/       _/    _/
       |     |   |     _/       _/    _/
       |     |   |   _/        /    _/
       |     |   |  |\_        \_ _/ 
       |     |   |  |  \_        \_
       |     |   |  |    \_        \_
       |     |   |  \_     \_        \_
       |     |   |    \_     \_        \_
       |     |   |      \_     \_        \_
       |     |   |        \_     \_        \
       |     |   |          \_  __/         |
       |     |   |           _\/         __/|
       |     |   |        __/         __/   |
       |     |   |     __/         __/      |
       |     |   |  __/         __/      __/
       |     |   |_/         __/      __/
       |     |   |        __/      __/
       |     |   |     __/      __/
       |     |   |  __/      __/
       |     |   |_/      __/
       |     |         __/
       |     |      __/
        \_   |   __/
          \_ | _/
            \|/

The logical conclusion of this kind of deception is the impossible staircase. You can walk forever up or down this staircase an never reach the end. This figure was discovered by L. S. Penrose and while it does not originate with Escher he used it to great effect in "Babel" and "Ascending and Descending". The key to this figure is the number of stairs on each side of the figure and the fact that its sides are not parallel.

M.C. Escher said about "Ascending and Descending", "That staircase is a rather sad, pessimistic subject, as well as being very profound and absurd. ... Yes, yes, we climb up and up, we imagine we are ascending; ... and where does it all get us? Nowhere. ... And descending, running down with abandon, is not possible either. ... How absurd it all is. Sometimes it makes me feel quite sick."


                               _____
                             _/___  \
                          __/  / |\__\
                       _ /  /|/  | |__|
                    __/  /|/     | \__\
                 __/  /|/        |  |__|
              __/  /|/           |  \__\
           __/  /|/              |   |__|
          /  /|/                 |   \  \
         |\  \__                  ___|\__\
         | \|\  \__           ___|\__\|  |
         |    \|\  \__    ___|\__\|      |
         |       \|\  \__|\__\|          |
         |          \____\|              |
         |           |                   |
         |           |                   |
         |           |                   |
         |           |                   |
         |           |                   |
         |           |                __/
          \_         |             __/   
            \_       |          __/
              \_     |       __/
                \_   |    __/
                  \_ | __/
                    \|/


Have a look at the next figure. It is a perfectly possible block with a hole through it.

          ___________________________________
        /|                                   |
       / |    ___________________________    |
      |  |   |                          /|   |
      |  |   |_________________________/ |   |
      |  |   |                        |  |   |
      |  |   |________________________|__|   |
      |  |                                   |
      |  |___________________________________|
      | /                                   /
      |/___________________________________/

Next we cut the figure in half.

          _________                  __________
        /|         |                /|         |
       / |    _____|               / |_____    |
      |  |   |    /               | /     /|   |
      |  |   |___/                |/_____/ |   |
      |  |   |                          |  |   |
      |  |   |_____                  ___|__|   |
      |  |         |                /|         |
      |  |_________|               / |_________|
      | /         /               | /         /
      |/_________/                |/_________/

Then take the right half and copy it and rotate the copy through 180o. There's no way we can join these two up in three dimensions. The faces we would have to join up have completely different orientations.


           __________            __________
          /         /|          /|         |
         /_________/ |         / |_____    |
         |        | /         | /     /|   |
         |    ____|/          |/_____/ |   |
         |   |  |                   |  |   |
         |   |  |_____            __|__|   |
         |   | /     /|         /|         |
         |   |/_____/ |        / |_________|
         |         | /        | /         /
         |_________|/         |/_________/

Fortunately all the lines line up in two dimensions so we can just join them up in two dimensions. This is yet another case of the parts of the figures being in incompatible axes but still being joined up.

           __________________________________
          /                                  |
         /_______________________________    |
         |                              /|   |
         |    _________________________/ |   |
         |   |  |                     |  |   |
         |   |  |_____________________|__|   |
         |   | /                             |
         |   |/______________________________|
         |                                  /
         |_________________________________/


The final example is a simple one. It is a concave, convex illusion. Escher was also known for this type of illusion although he mostly used it for confusion rather than impossibility. It should be pretty obvious by now how this is done. It is a yet another case of individual consistency but overall inconsistency, the hallmark of an impossible figure.

         _________________________________
        |\                                \
        | \________________________________\
        |  |                                |
        |  |________________________________|
        |   \                            |
        |    \___________________________|
        \  |\ |                           \
         \ | \|____________________________\
          \|________________________________|


Sources:
Bruno Ernst, "Magic Mirror of M. C. Escher", ISBN: 1886155003
Bruno Ernest, "Adventures with Impossible Figures", ISBN: 0906212545
M.C. Escher, "29 Master Prints", ISBN: 0-8109-2268-1
Also some of the figures in this write up were based on the examples at http://www.brisray.co.uk/optill/oreal.htm

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