What Makes An Impossible Figure Impossible?
By: Areeba Merriam
Tags: Math, Penrose, Geometry, History, Science

What Makes An Impossible Figure Impossible?

Impossible shapes are the ones that form in an inconsistent way to create a spooky optical illusion. That deceives our brain on a very first impression by creating a false reality. A few of them are discussed in this article.

Penrose Triangle

The Penrose triangle or the impossible triangle (Penrose tribar) was first designed by a Swedish artist Oscar Reutersvärd in 1934. He was known as the “father of the impossible figures”. He created that picture at the age of 18 while drawing randomly in his Latin class.

The triangle gained more widespread attention in the 1950s when a psychiatrist Lionel Penrose and his son Sir Roger Penrose, who is now a prominent Nobel Prize-winning mathematician, independently expressed it as a “pure impossibility”. They published a paper entitled “IMPOSSIBLE OBJECTS: A SPECIAL TYPE OF VISUAL ILLUSION” in The British Journal of Psychology in 1958. Moreover, the objects with impossible geometry are also featured in the works of Dutch graphic artist M. C. Escher.

According to Wiktionary, the Penrose triangle is defined as: “An optical illusion depicting an impossible solid object made of three straight beams of the square cross-section which meet pairwise at right angles at the vertices of the triangle they form.”

Hence the lines are connected in such a manner as to illustrate an impossibility. In order for these shapes to exist, the rules of Euclidean geometry must violate.

Figure: 1 The Penrose triangle

By observing the above figure, you can have the perspective that the right side of a triangle is extending away from you and the left side is extending towards you. Yet they are connected to the same rectangular bar and appear on the same plane.

Penrose attended a lecture in 1954 by Escher and felt motivated to illustrate that concept. Later, Escher incorporated the effect of the Penrose triangle as an inspiration to create the famous impossible construction of the “Waterfall lithograph”.

Figure: 2 This lithograph has the structure of a watercourse with two Pensore triangles. The aqueduct begins at a waterwheel to to the observer it feels as if it is slanted upwards. The cycle of a waterfall off the aqueduct is an impossible cycle here. The two towers also have polyhedra at their top showing Escher’s interest in mathematics.

Based on a similar concept another impossible shape can also be constructed known as the Impossible/Irrational cube.

Figure: 3 The impossible cube

The parts of the cube are appearing to be in front of and behind at the same time which is not possible geometrically.

The Penrose Staircase

Figure: 4 The Penrose staircase

The Penrose staircase is a never-ending wind of the steps going up in one direction while back down in the other direction and so on and so forth. At a first glimpse, it makes a person wonder about how it works. But the moment we rotate the object and behold the other side then the illusion is no more.



The paper published by Penrose consisted of a description of the impossible staircase also. The structure is “acceptable as representing a flight of steps but the connections are such that the picture, as a whole, is inconsistent”.

M. C. Escher created the image of the impossible staircase in his lithograph print “Ascending and Descending” two years later in 1960.

Penrose cited Escher’s work in their paper as an inspiration for constructing the impossible staircase and forwarded a copy of their paper to Escher. Escher in a due course responded:

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.

Although an illustration of the impossible staircase was first created by Oscar Reutersvärd in 1937, Penrose and Escher were unfamiliar with it.

Figure: 5 Impossible staircases drew by Oscar Reutersvärd

The Penrose Stairs is simply an impossible figure because if one were to complete a loop of the stairs, one would end up again at the same level even though each step of the stairs continuously rise or fall, depending on the clockwise or anti-clockwise direction.

Figure: 6 M. C. Escher’s “Ascending and descending” lithograph. The stair in that drawing paradoxically accends while going in one direction of the round and descends while going in the other direction. Still one would end up at the same level.

Figure: 7 M. C. Escher’s “Belvedere” lithograph. This drawing includes 3 impossible aspects. First of all, observe the impossible arrangement of pillars (the way they begin at the top and end at the bottom). Secondly, observe the impossible positioning of a stairway (it starts and ends at the same level with a little slope). Lastly, notice the man on the bench, outside the building, holding the impossible cube.

The concept of Penrose steps/staircase (the infinite staircase) was also shown in an action/sci-fi movie directed by Christopher Nolan Inception in 2010. Also, in the movie The Avengers in 1998, a person is appeared to be walking down a Penrose stairway who ends up being at the same place after walking.

Moreover, Monument Valley is a video game developed by Ustwo Games, released in April 2014, is based on a network of optical illusions and impossible figures.