# Why is a Penrose triangle impossible

## Penrose triangle

The Penrose triangle, also Tribar called, is a so-called "impossible figure". It shows three bars that seem to be at right angles to each other and yet are connected to form a triangle. This violates several laws of Euclidean geometry, including the one that says that the sum of the interior angles in a plane triangle is always 180 °. The viewer of a Tribar representation is confronted with the difficulty of having to repeatedly reinterpret its distance to the parts of the Tribar and their location in the space shown.

### The tribar in art

Figures formed according to principles similar to that of the tribar have existed in art since the invention of perspective. One example is the “Carceri” by the Italian artist Giovanni Battista Piranesi, on which an architecture that is sometimes impossible is shown.

In the 20th century, Reutersvärd had been experimenting with depicting impossible figures since 1934, including those of the Tribar. However, it wasn't until the 1980s that his work gained the attention of the masses. At that time, the Swedish Post also published three stamps with impossible objects from Reutersvärds, based on principles similar to that of the Tribar.

Escher was made aware of the Tribar by Penrose. He had already designed his picture "Belvedere" when Penrose sent him a copy of his article. This inspired him to his picture “Treppauf, Treppab”, in which he took over the infinite staircase from Lionel Penrose without major changes, and a little later to his picture “Wasserfall”, which shows the viewer a waterfall that feeds itself and its Representation based on the tribar.

### Is an impossible object like the tribar possible?

Penrose gives the simplest answer in his article: “Every single part of a figure is acceptable as a representation of an object that stands normally in space; however, accepting the entire object leads, as a result of incorrect connections between the individual parts, to the deceptive effect of an impossible structure. "

An impossible figure thus fulfills two conditions:

1. It consists of individual parts that are possible in the pictorial space without contradiction.

2. These parts are connected in a way that is possible on the two-dimensional picture surface, but impossible in the three-dimensional space shown.

For the explanation of such figures, the findings of Gestalt psychology play an important role, in particular that seeing is not a passive process, but always the active interpretation of what is seen, that the whole of perception is something other than the sum of its parts and that we are under illusion can not withdraw, even if it represents something seemingly impossible.

### Creation of the tribar from partial figures

Creation of the tribar (right) from two real perceptible sub-figures

If you move the left part of the figure parallel to the right until its upper horizontal edge is aligned with the upper horizontal edge of the middle part of the figure, the tribar (right) is created by the overlap of the two part figures.

The first two partial views of the tribar are individually perceptible, whereas the resulting tribar represents an impossible figure.[2]

### Individual evidence

1. ↑ Sharpless Lionel Penrose, Roger Penrose: Impossible objects: A special type of visual illusion. In: British Journal of Psychology. Volume 49, No. 1, 1958, ISSN0950-5652, pp. 31-33, doi: 10.1111 / j.2044-8295.1958.tb00634.x.
2. ↑ Impossible Figures (Escher) from mathe-werkstatt.de (based on the creation of the Escher figures), accessed on May 9, 2021