​The Impossible Triangle (also known as the Penrose Triangle or the Impossible Tribar) was first created by Oscar Reutersvärd (1915 - 2002), a Swedish graphic artist known as the ‘father of the impossible figure’. It is anecdotally, but widely reported that he created it in 1934, aged 18 while doodling as a student in his Latin class. The illusion was independently discovered later and popularised by Lionel Sharples Penrose (1898 -1972), a British psychiatrist, geneticist, and mathematician, and his son Sir Roger Penrose (1931 -), a British mathematician, physicist and philosopher of science. Penrose and Penrose published the illusion in the British Journal of Psychology in 1958.

The Penrose Triangle is an impossible figure (or impossible object or undecidable figure): it depicts an object which could not possibly exist. It is impossible for the Impossible Triangle to exist because in order for it to exist rules of Euclidean geometry would have to be violated. For example, the bottom bar of the tribar is represented as being spatially located to both the front of, and, at the same time, the back of the topmost point of the tribar.
​The Impossible Triangle has three sides. There are versions of impossible figures that have four sides and more, and in a number of different configurations and contexts. (See the figures below and explore other impossible figures in the Illusions Index.) Artists such as Oscar Reutersvärd (see below) and M. C. Escher (e.g. see Belvedere) have frequently used impossible figures of varying types in their work.

Mathematicians have studied the mathematical and computational properties of impossible figures to try and develop formulas and algorithms for modelling impossible objects, for use in such things as computer vision. Cognitive scientists have been interested in the processes involved in continuing to see impossible figures as possible even when we know them to be impossible. Why, for instance, do we not see the Impossible Triangle just as some lines on a page once we realise that it can’t exist in three dimensional space? In answering this question, debates about modularity and cognitive penetration are of central importance. To explain: on the hypothesis that the mind is modular, a mental module is a kind of semi-independent department of the mind which deals with particular types of inputs, and gives particular types of outputs, and whose inner workings are not accessible to the conscious awareness of the person – all one can get access to are the relevant outputs. So, in the case of impossible figures, a standard way of explaining why experience of the impossible figure persists even though one knows that one is experiencing an impossibility is that the module, or modules, which constitute the visual system are ‘cognitively impenetrable’ to some degree – i.e. their inner workings and outputs cannot be influenced by conscious awareness. Philosophers have also been interested in what impossible figures can tell us about the nature of the content of experience. For example, impossible figures seem to provide examples of experiences with content that is contradictory, which some philosophers have taken to count against the claim that perceptual states are belief-like (Macpherson 2010).

Oscar Reutersvärd has produced many other beautiful and compelling versions of impossible figures:

Richard Gregory (1968) produced a wooden object that, when, and only when, viewed from one position in space appears to be a real three-dimensional impossible triangle, but it is really just three struts of conjoined wood as one can see when one views it from other angles:

An Impossible Triangle sculpture was designed by artist Brian McKay and architect Ahmad Abas, which was built in Claisebrook Square in East Perth, Australia. It is 13.5 metres high, and was commissioned after being chosen as the winning entry in a competition for East Perth redevelopment: ​   ​   ​ 
Momument Valley is game published by Ustwo Games, released in 2014, based around a series of impossible figures. The player guides a princess across the surfaces of impossible figures while manipluating those objects to reach various locations.