Magic Forest

The Penrose Triangle

 

Introduction

The Penrose stairs is an impossible object devised by Lionel Penrose and his son Roger Penrose and can be seen as a variation on his Penrose triangle . It is a two-dimensional depiction of a staircase in which the stairs make four 90-degree turns as they ascend or descend yet form a continuous loop, so that a person could climb them forever and never get any higher. This is clearly impossible in three dimensions; the two-dimensional figure achieves this paradox by distorting perspective.

The best known example of Penrose stairs appears in the lithograph Ascending and Descending by M. C. Escher , where it is incorporated into a monastery where several monks do penance by ascending continuously, but are allowed to turn around and descend occasionally.

The staircase had also been discovered previously by the Swedish artist Oscar Reutersvärd , but neither Penrose nor Escher were aware of his designs.

The Challenge

To build a Penrose Staircase..even if impossible

Budget

Nil

Items used (Version 1)

  1. Cardboard
  2. Paper Glue
  3. A4 paper
  4. White emulsion or poster paint
  5. A little bit of magic

Build Time

45 minutes

Items used (Version 2)

  1. Lego
  2. More magic

 

Build Time

20 minutes

Method (Version 1 )

 

On the left you can see the finished product. I am not totally happy with the result but with the constraints it gives quite a pleasing illusion but needs work in the future.

I started by cutting out sections of card as shown below.I initially drew the diagram on graph paper and I found it easy to place the paper over the cardboard and prick each corner with a compass point and then draw lines with a metal ruler and cut with a sharp craft knife.

The squares depict the number of layers of card required for each stack or staircase. Instaed of doing individual squares , cut 4 layers containing all the squares, remove one square from the long side and create a shorter layer and continue around until you have just one square, the very last step on stack 18.

For example

  • 4 layers of squares 4 to 18
  • 1 layer of squares 5 to 18
  • 1 layer of squares 6 to 18...etc

 

You should now have 18 pieces of card in various sizes. Simply glue them together, neatly and so that they are square on

The sides

Simply get some white paper and glue it to the sides, for this illusion you only need to cover the sides that are visible in the above picture.

Wait for it to dry and then paint the visible areas white emulsion or poster paint.

This is best viewed with strong lighting from all sides.

Method (Version 2 )

I am lucky as my daughters still have Lego bricks lying around and this was a quick and easy option for the build.

There is no science involved in this but note that the stairs are not rising by a whole brick height , but at a quarter brick height. These shallow layers are normally found in special sets of Lego, as was this from a Harry Potter box set.

Using the diagram above you can build the layers.

I have used Macromedia Fireworks to slightly blur the outer part of the picture and the internal quadrangle to hide the perception of depth. If anything this one is the better of the two but I still reckon it would be even better if the bricks were all one colour.

 

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