​Escher's Ascending and Descending
Licence: Fair Use;
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​Impossible Staircase by Oscar Reutersvard;
Licence: Public Domain;
source
Impossible Staircase by Oscar Reutersvard;
Licence: Public Domain;
source

The Penrose Stairs Figure was created 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. It was first published in the British Journal of Psychology in 1958. 

Shortly afterwards, in 1960, Escher produced his lithograph print Ascending and Descending, which contained such a staircase. Penrose and Penrose cited Escher's work as part of their inspiration for creating the staircase, and sent a copy of their paper to Escher. Escher subsequently wrote back to Penrose and Penrose (see Ernst 1992):

A few months ago, a friend of mine sent me a photocopy of your article... Your figures 3 and 4, the 'continuous flight of steps', were entirely new to me, and I was so taken by the idea that they recently inspired me to produce a new picture, which I would like to send to you as a token of my esteem.

However, an impossible staircase was first created many years earlier, in 1937, by Oscar Reutersvärd - unbeknown to the Penroses and Escher.

The Penrose Stairs is an impossible figure (or impossible object or undecidable figure): it depicts an object which could not possibly exist. It is impossible for the Penrose Stairs to exist because in order for it to exist rules of Euclidean geometry would have to be violated. For example, if one were to complete a circuit of the stairs, one would end up back at the same level that one began, even though each flight of the stairs continuously rise (or fall, depending on the direction of travel). It is one of many types of impossible figures which you can search for in the Illusions Index.

Artists such as Oscar Reutersvärd and M. C. Escher have frequently used impossible figures of varying types in their work, and 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 Penrose Stairs simply 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 challenge the claim that perceptual states are belief-like (Macpherson 2010). They also prove problematic for sense-data accounts of perception that posit that corresponding to every experience that we have there are mental objects that we are aware of that have the properties that the objects that our experiences tell us they do. They problem is that sense-data would have to be impossible objects. But, surely, impossible objects can't exist!