For those of you who do not know, M.C. Escher is one of the most famous graphic artist.  Maurits Cornelis Escher was one of those unique artist who extensively used woodcuts and lithographs for his drawing and sketches. He was also famous for the application of mathematics in his art.

Fortunately I got the opportunity to experience his art at the Exhibit organised by World Chess Hall of Fame in Saint Louis, MO on May 21st, 2019. The extended exhibit was in Saint Louis University Museum. In this article we will cover mathematical relation to M.C. Escher’s art leading us to understanding of deformation grid and finally presenting our own 3D piece of M.C. Escher’s drawing “Relativity”.  Escher’s early work was based on the concept of tessellation. Tessellation is a technique of pattern design where the space is equally divided and a transformation from one object to another takes place.  Escher first started relating mathematics to art during his travel to Alhambra in Granada, Spain. It was marveled at the wealth of decoration in majolica tiles with the illustration of Islamic art. Later he started understanding the filling of planes from articles by Hagg. As he progressed in his mathematical realm, he came across the work of Sir Roger Penrose known as “Impossible Objects: A Special Type of Visual Illusion” published in British Journal of Psychology. Penrose’s paper described the illusion of impossible structure through three dimensional figure. This included the Penrose Triangle and Penrose staircase (infinite staircase). Both of which are vividly used by M.C. Escher in his multiple drawings. Most noted ones are: Waterfall and Relativity.  Where “Waterfall” covers the mathematical aspect of making the wheel move using the falling water which is again send back to the top, “Relativity” depicts existence of multiple dimensions in one plane -perhaps an attempt to discover the fourth dimension. If viewed at a certain distance the observer’s eye can interpret the formation of Penrose Triangle as well as infinite staircase respectively. Escher not only took Roger’s work as an inspiration, he very well incorporated it with the general idea of relativity by Einstein. Escher brought the idea of impossible objects from academia to social realm.

As Escher’s work progressed he started playing with geometrical side of objects. This initiated the journey of “Deformation Grid” . Escher’s expertise to render his imagination on paper required some background preparation and Escher choose deformation of the drawing to illustrate his understanding of the world.  Escher very artistically focused on Dorset effect. Seemingly not very appealing piece of art in black and white become the target of research for mathematicians and Philosophers. This piece of art was called “Escher’s Print Gallery”.  It was even left with the mysterious blank space in the middle to fill in. To understand “Escher’s Print Gallery” , lets first understand Dorset effect and Deformation Grid.

  1. Dorste Effect: It is the effect of picture recursively appearing within itself. Amazingly it can be produced through graphic design software.
  2. Deformation Grid: Its a grid drew by Escher for the lithograph of his piece ” Print Gallery”. It is deformed in a way to produce Dorste effect since graphic design software were not available in Escher’s era.

Dr. Hendrik Willem Lenstra led a project to unfold the secret of the blank space in “Escher’s Print Gallery”. He proposed the idea of filling the space with image within an image and proved it mathematically. In his study he talks about con formality property that was used by M. C. Escher. According to this property the angle is preserved during transformation for the image. It can be represented as :

transformation w = f(z)

An analytical function is con-formal at any point where it has a non zero derivative. The Deformation Grid made by M.C. Escher preserves the angle of each block of image while changing its shape. In his drawings M.C. Escher tried to produce Dorste effect but as he reaches to the end of it , he leaves it blank either for the readers to understand or discover the filling in the blank spot. His Grid replicates the spiral movement. To achieve Dorste effect if any image is rotated horizontally producing spiral effect and then zoomed in or out at certain scale , a image inside the image is visible. 2pi is the measurement for horizontal movement of the image and is subject to change . The vertical scale is the zoom in and zoom out factor. For perfect repetition of Escher image , zoom in factor is 22.4 with rotation of 160 degree where 2pi sits at the diagonal. At the same time he used circular expansion where everything increases by the factor of four that is 256.This is more explainable through visual effect:

Here we have made an attempt to 3D print Escher drawing called “Relativity”. The drawing is quite intricate and took around 17 hours 35 minutes for printing. It has 607 layers with support type of everywhere. The software we are using is Cura.  We will be capturing beginning of the built.

Final Product: