In the vast world of geometry and art, there are figures that challenge our perception and understanding of space. One of these figures is the Penrose Triangle, also known as the Tribar. This object, considered an “impossible figure”, is formed by three bars that appear to connect at right angles, forming a triangle.
However, this structure defies several laws of Euclidean geometry, such as the sum of the interior angles of a triangle, which should always be 180°. Perceiving this figure requires a constant reinterpretation of the distances and positions of its parts in the represented space.
The Tribar was invented in 1934 by Swedish artist Oscar Reutersvärd, although his works remained largely unknown until the 1980s.

Independently, British mathematician Roger Penrose also conceived it in 1954 after attending an international mathematicians’ congress in Amsterdam, where an exhibition of works by Dutch graphic artist M.C. Escher sparked his interest in impossible figures.
Penrose, along with his father Lionel Penrose, published an article on the Tribar in the British Journal of Psychology in 1958, which helped popularize the figure.
Figures applying principles similar to the Tribar have existed in art since the invention of perspective. A notable example is the “Carceri” by Italian artist Giovanni Battista Piranesi, which depicts partially impossible architectures.

In the 20th century, Reutersvärd had already begun experimenting with impossible figures, including the Tribar, since 1934. However, it was in the 1980s when his work gained greater recognition, being honored with the issuance of three postage stamps in Sweden featuring his impossible objects.
Escher, informed of the Tribar by Penrose, integrated this figure into his work. His piece “Belvedere” was created before knowing Penrose’s article, but it inspired him to create “Treppauf, Treppab” which uses Lionel Penrose’s infinite staircase, and later “Waterfall”, an image depicting a watercourse flowing in a zigzag along the long sides of two elongated Penrose triangles, so that it ends two floors above where it started, evidently based on the Tribar.
There are various sculptures based on the Tribar around the world, such as the one created by Brian MacKay and Ahmad Abas in Perth, Australia, or the model in a playground in Sankt Margareten im Rosental, Carinthia. Also notable is a stainless steel sculpture by W.A. Stanggaßinger in the German Museum of Technology in Berlin.

Penrose explained in his article that each individual part of a figure is acceptable as a representation of a normal object in space; however, accepting the entire object leads to a deceptive effect of an impossible structure due to the incorrect connections between the parts.
An impossible figure meets two conditions: first, it consists of individual parts that are possible in two-dimensional space; and second, these parts connect in a way that, although possible on a two-dimensional surface, is impossible in the represented three-dimensional space.
Gestalt psychology plays a crucial role in explaining these figures, emphasizing that perception is not a passive process but an active interpretation of what is seen.

Gestalt suggests that the perception of a whole is distinct from the sum of its parts, and even though we know the figure is impossible, we cannot escape the illusion.
The idea of the Tribar can be extended to other polygons and figures, such as the Penrose Square, the Penrose Pentagon, and the Penrose Torus. These variants demonstrate the versatility of the principle behind the Tribar, applying it to different geometric figures.
The Tribar is a fascinating visual paradox that reminds us of the complexities and wonders of art, human perception, and geometry.
Sources
L.S. Penrose, R. Penrose, Impossible objects: a special type of visual illusion. British Journal of Psychology, vol.49, issue 1. February 1958. doi.org/10.1111/j.2044-8295.1958.tb00634.x | Impossible figures in the real world (Impossible World) | Francis, G.K. (2007). The Impossible Tribar. In: A Topological Picturebook. Springer, New York, NY. doi.org/10.1007/978-0-387-68120-74 | Wikipedia
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