Your new book about the lifetime works of M.C. Escher comes to you at a special time - almost exactly 50 years from his passing, on March 27th, 1972. The introduction is a rare chance to read Escher’s own words about his art and life.
It won’t surprise you to hear that Escher was deeply engaged with mathematics in his work. He routinely corresponded with mathematicians, some of whom were inspired by his inquiries to try their own hand at art, including Nobel Prize winner Roger Penrose. After viewing Escher’s work while attending a mathematics conference in 1954, he was inspired to create the famous Penrose Triangle illusion - which Escher himself then used in his famous Waterfall illustration, the very last image in your book on page 95.
You can see how the Penrose Triangle and other famous illusions work on Vsauce3 in Jake's video: Impossible Objects.
For a number of years, Escher conducted his own mathematical research, most of which (as you’d guess) focused on geometry, and how to fit shapes into each other. He even discovered two unique mathematical theorems: one was about congruent lines in a triangle, the other about diagonals in hexagonal patterns. A number of subjects that Escher investigated were later topics of formal investigation by scientists and mathematicians, but only decades after Escher had already incorporated them in his work. For a full list, we encourage you to read this article.
Interestingly, Escher never considered himself an artist, instead always referring to himself as a designer. He also insisted that his prints didn’t have any hidden meanings, and were just illustrations of mathematical concepts, though art historians have noted how they are often rooted in his life experiences, including the loss of his close friend and mentor to the Nazis, along with personal issues in his marriage and relationships. While the works in this book cover the entire breadth of his career, they particularly focus on two subjects that have a geometric, mathematical connection: tessellations and knots. Escher himself called tessellations “the richest source of inspiration I have ever struck.”
Tessellations are repeating geometric patterns that cover a surface with no gaps or overlaps. They’re also called “tilings,” because that’s where they came from; the original tessellations were Roman floors covered in ceramic tiles. In honor of that origin, the shapes in a tessellation are still technically called tiles. There are several ways to manipulate each tile to fit them together and form a tessellation:
- Reflection (flipping the shape over)
- Rotation (turning the shape around)
- Translation (sliding it around)
- Glide reflection (a translation followed by a reflection, in other words sliding it around and flipping it)
You can see some of these manipulations at work in the tessellations in your book; for example, Circle Limit III (page 42) is an example of rotation and Horseman (page 27) is an example of glide reflection. There are a total of 17 types of symmetrical tessellation patterns that can be created by combining these four manipulations together
It’s fun - and surprisingly easy - to make your own tessellation! With a piece of paper, or in your favorite computer illustration program, just follow these simple steps:
1. Take a square or rectangle, and draw a line through it
2. Cut and separate the shape along the line:
3. Take the left side of your divided shape, move it to the right side of the shape, then reconnect them along the flat edges:
4. You’ve now created a tessellation tile! You can copy this shape over and over and create your own tessellated surface:
M.C. Escher is just one of many artists who influenced, and were influenced by, mathematical concepts; if you’re interested, check out works by Robert Fathauer, Ada Dietz, and Simon Beck. Escher’s illustrations are engrossing, thought-provoking, and fun. We hope you enjoy this fantastic book, and it encourages your own artistic journeys!