From exhibitions to fame, Escher attracts huge crowds of enthusiasts who are intrigued with his impressive, optical illusion, and mind-bending mathematical artworks. [2]

Escher explored tessellations of the hyperbolic plane through his five works: Circle Limits I-IV and Snakes. A tessellation is tiling of a surface with a pattern of an entire plane without any overlap nor gap.[3] Hyperbolic geometry follows Euclid's postulates but negate the 5th rule or the parallel postulate. [4]

Now, let’s analyze Escher’s last piece in the Circle Limit series called Circle Limit IV.

In this tessellated work (Figure 1), Escher made use of negative spaces to  create hyperbolic patterns with angels (white) and devils (black). 

To understand and describe the pattern that underlies Circle Limit IV, Douglas Dunham used {p,q}, where p-sided polygons meet q at a vertex. In Figure 2, p is twice the number of angles/devils that are joined by feet while q is the  number of wing tips that meet at a point. At the meeting point of feet, the lines of bilateral symmetry intersect. Hence to obtain pe, double the number of angles/devils. [6]

Many of Escher’s geometric figures are subjects of Polyhedra—geometric figures that are results of integrating polygons joined at their edges. [7] 

As seen in the pictures above (Figure 3), polyhedra are pieces made out of multiple geometric pieces stacked together. [8] 

One of the most interesting polyhedron works of Escher is the Stars (Figure 4) where an octahedron, tetrahedron, and a cube are put together to act as a cage of chameleons. Because he was fascinated to incorporate mathematics into his art, why not add a spice? That’s why a seemingly random chameleon is caged in the piece above triggering a sense of alarm in us.

Fundamentally, the visual elements suggest celestial order with the polyhedra  and biological chaos through the chameleons. Inspired by geometry and astronomy, the Stars challenge viewers to perceive Escher’s work with fresh eyes. [11] 

The genius Maurits Cornelis Escher’s out of this world imaginations makes viewers question their perspective in mathematics, reality, and life. He has more than hundred piles of lithographs, woodcuts, and wood engravings, along with over 2,000 drawings and sketches waiting to be seen by you. One doesn't need to be an Escher 2.0 to beat impossible patterns of this universe because it only takes a discovery of what language one truly speaks to defy the limits. 

This embedded quiz is best viewed in a desktop format. In case the quiz does not work on your device, you can go here!