What is tessellation MATH art?
Tessellation MATH art uses repeating geometric shapes to fill a plane with no overlaps or gaps.
Perhaps the best known examples of math art in a tessellation pattern are the works of MC Escher, who created drawings inspired by the Moorish use of symmetry in tile patterns. Tessellations have historically been used as tiling PatTerns to create decorative mosaics in Moorish, Roman and Greek art and architecture. Tessellation patterns use geometry in art to create mosaics, tessellation artwork, quilt patterns, origami designs, architecture and more.
Tessellations can be seen all around You
Take a look around the next time you go out for a walk! There are many tessellations in nature, such as the structure of honeycomb in a bees’ hive and the placement of petals in many flowers. Tessellations are often used in architecture and design, and have been used for centuries to create mosaic tiling for decorative purposes. Traditional quilt patterns are often based on tessellations and you can even create tessellations with toys!
math Art by Luke Rawlings
When I was young, I was fascinated with the prints of MC Escher. I wanted to understand how he made them so I started drawing - first, tracing his prints to understand the patterns, and then starting to create my own drawings. I wasn’t good at math, and had no idea that I was using math to draw.
I started a notebook with sketches and wrote rules for myself on how to change a square into an animal. Eventually I discovered repeatable design motifs. Art was a refuge for me throughout high school and college. I continued drawing, and continued to develop my designs.
- For Math Educators -
Coming to an Understanding of Math Through Art
I came to mathematics because I was an artist and I was obsessed with MC Escher’s regular division of the plane drawings. It was mathematics that helped me understand this art better, and even improve on the way I create designs that fill the plane with no gaps or overlaps. I was lucky to be able to use my own drawings when I wrote a dissertation on the group theoretic concepts that I had no idea could be found in the art I had been producing in both notebooks and elsewhere since college. Coming up with a design using some underlying mathematical structure is fascinating to me because it is so similar to problem solving.
Early in my life, mathematics helped me come to an understanding of art. And now, I continue to discover mathematics through the process of creating the artwork. The artist of these tessellations intersects the mathematician by the process of solving a problem. Anyone who has endured a mathematics course has encountered challenging problems whereby the experience of trial and error may have contributed towards the journey to a successful solution. Something as simple as solving a quadratic equation, which requires an algorithm, might become challenging if the student makes a mistake in arithmetic, arrives at an inaccurate answer and, thus, is prompted to investigate the process to detect what went wrong. Proving a theorem in advanced mathematics is an activity that most likely does not come with an algorithm, which compels a student to unravel a truth that seems locked inside a riddle.
As an example of a problem to solve in the type of artistic endeavor involved in tiling the plane by hand, consider the following process in the creation of a design:
The artist chooses a tiling on which to create a design.
The tiling comes with several geometric rules that explain how the tiles fit together, and the artist must either discover these rules or know them beforehand.
After using trial and error in the creative process, the original tiling transforms into a new tiling.
Refinements are made such that the new tiling satisfies the artist wishes culminating in the finished work.
One encounters many difficulties and challenges when creating a pattern: a shape is formed that fills the plane, and the artist decides what form (recognizable or not) that shape will take. Similar processes occur in mathematics: given a challenging problem, by what means and struggles does one move toward the realization of its solution?
Hungarian mathematician Zoltan Dienes compares the work that a mathematician does to that of an artist. Though any pool of artists might disagree on what is art, Dienes defines art as “an expression of an urge to create constructions which externalize internalize inputs from the environment”. Work in mathematics parallels this definition, he argues, and the educator’s challenge is to “harness the incredible energy released by what we describe as ‘play’ in both artistically and mathematically meaningful ways”, creating an environment in which students feel safe to perform according to their own nature. Dienes’ view of mathematics as a form of art leads inevitably to his belief that learning mathematics “should ultimately be integrated into one’s personality and thereby become a means of genuine personal fulfillment”.
LOOKING FOR CURRICULUM?
Grunbaum and Shephard’s Classification of Escher-Like Patterns with Applications to Abstract Algebra
by Dr. Luke Rawlings
Abstract
This study investigates a link between art and mathematics. It attempts to show that patterns in the Euclidean plane, such as those made popular by the artist M.C. Escher, can function as inspiration for the transmission of mathematical knowledge at the college level. The study is broken up into two parts. The first part revisits a result of Branko Grunbaum and G.C. Shephard, and focuses on how the seven Frieze groups associated with infinite strip patterns of the plane can be classified into 15 pattern types by way of recognizing the relationship between a motif and the symmetry group of the infinite strip to which it belongs. The second part of the study sheds light on rich mathematical concepts associated with monohedral tilings of the plane, otherwise known as tessellations. Concepts found in college mathematics courses that cover group theory, linear algebra, and geometry surface in the work through the language of symmetry. Throughout the work, the study places the mathematics educator as its primary audience, signaling the possible impact that visual inspection of artistic patterns in the plane might have in the learning process. It is asserted that this art is useful as extra examples in several college mathematics courses, among them abstract algebra, linear algebra, geometry, and liberal arts mathematics, where the use of such patterns aids in understanding abstract concepts often encountered in these courses. Over 100 figures are used to show that learning through visuals is an important part of the work. During the study, a bank of problems forms from both the classification of pattern types and study of tessellations, and is presented in an appendix. The work represents a first form of a handbook which educators might use as a source for examples in a college mathematics classroom.