Field, M. & Golubitsky, M. (1992): Symmetry in Chaos: A Search For Pattern in Mathematics, Art and Nature, The Society for Industrial and Applied Mathematics, PA, USA.
Field and Golubitsky explore patterns created by computers that mimic in great detail and precision the naturally occurring patterns found in nature. They argue that the complex geometric patterns are created by merging symmetry and chaos; order and disorder; yin and yang. The challenge is trying to give a simple explanation of how symmetry and chaos dynamics merge to create structure and chaos at the same time. I found the following section interesting as it relates to other sources who argue that we look for order as a way of structuring our lives and making sense of the world around us:
“We often name issues of situations that are too complex to understand, and then feel at ease when these names are repeated. Well, mathematicians are sometimes as guilty as others in naming away their lack of understanding.”
Field and Golubitsky also discuss the nature of symmetry as a motion, which applies to many geometric patterns and designs which follow a repetitive theme and often carry a sense of movement. Movement within a design is something I explore in some of my own work, particularly in my mandala designs.
Hagood Slegelis, Maralynn (1987): A Study of Jung’s Mandala and Its Relationship to Art Psychotherapy. The Arts in Psychotherapy, vol. 14. pp. 301-311. Pergamon Journals, USA.
Hagood Slegelis discusses Carl Jung’s claim that the mandala, a type of concentric geometric design, has a calming effect on its practitioners. I found this resource interesting because of my reading into geometric art used as therapy. Hagood Slegelis argues that Jung used mandala drawing in his own personal therapy and believed that the result was a visual representation of his current mental state. This made me think of other similar exercises such as stream-of-consciousness writing and auto-writing. Furthermore, Hagood Slegelis writes that Jung believed that symbolism appeared in many of the mandala designs, creating a bridge between the consciousness and subconsciousness. This allowed patients to interpret their own designs and not rely as much on Jung as their therapist—essentially he created a way for them to conduct self-therapy.
The calming effect could be linked to repetitive motion and other tactile exercises (drawing, art as a way of expression, origami, etc.), areas I have encountered in other text sources throughout my research. This also relates to my own explorations into mandala design and echoes my own experiences with its calming, mesmerising effects.
Kandinsky, W. (1979). Point and line to plane. New York: Dover.
Kandinsky developed an interest in non-objective painting after an epiphany in his thirties which changed his approach to art. He began painting for painting’s sake rather than to express information. Through his new practice he developed a way of looking at the geometric point as both the silence that happens between two elements (the silent full stop within a sentence) and the literal point, or mark, a tool makes on a canvas. Changing the size, shape and placement of the point changes both the nature of the point itself and alters the elements surrounding it. A longer silence has more gravitas or impact than a shorter one. Kandinsky also notes that the point on a canvas digs itself into the plane and asserts itself, presenting a short, fixed assertion. He writes:
“Therefore, the point, in its outer and inner sense, is the proto-element of painting and especially of the “graphic.”
I found it fascinating how Kandinsky draws parallels between the shape of the full stop and its purpose, as well as the shape an artists’ tool makes on a canvas—to me, they are opposing marks, one representing the negation of elements, and the other Kandinsky’s positive assertion. But I have come to believe from reading Point and Line to Plane that the point is more than just a symbol: it is a powerful tool, and the basis upon which all other elements are supported.
In my own practice I predominantly focus on shapes and how they work together in a design to create a visually-pleasing message. Rarely do I consider how a short, simple assertion might convey the message just as creatively, by better utilising negative space (silence) alongside the positive (noisy) elements.
Mandelbrot, B. (1983). The Fractal Geometry of Nature. New York: W. H. Freeman and Company.
Mandelbrot’s book is about fractal geometry found in nature and he argues that we should not limit ourselves to a tight set of rules but remain open to the possibilities presented to us in art, design and science. Mandelbrot believed that many of the theories that dominated thinking in the past—though still perfectly valid—did not push exploration and open-mindedness enough. In studying fractals in nature Mandelbrot discovered a wealth of new possibilities spanning many fields, including mathematics, science, computing, and art and design.
One of the things that will help guide me through my next projects is Mandelbrot’s passion and enigmatic approach to challenging the world and the boundaries we have set for ourselves. What I believe now might not be what I believe after conducting more extensive research and I must remain receptive. Mandelbrot was keen to bring together existing scientific theories of geometry with the naturally occurring geometry in nature, hoping to bridge some of the gap between a subject that is built on questions and trying to find answers, and something as vast as the universe with innumerable possibilities.
“But we do deal with a new form of the controversial but ancient theme that all graphical representations of mathematical concepts are a form of art, one that is best when it is simplest, when (to borrow a painter’s term) it can be called “minimal art.” (Mandelbrot, 1977)
A number of sections in The Fractal Geometry of Nature were beyond me and delve deeply into mathematics and programming, but there were ideas and information in the book that I can apply to my own creative practice—mainly to remain open and conscious of the natural geometry around us and the possibilities it can bring to my work.
Maor, E. and Jost, E. (2014). Beautiful Geometry. Princeton, N.J.: Princeton University Press, USA.
The purpose of Beautiful Geometry is to consider how closely tied art and mathematics are and how visually stunning patterns and colours can help us better understand geometry theorems. I found this book fascinating particularly as a creative thinker and someone who has not come from an analytical or mathematical background. Throughout my research I have struggled to comprehend some of the maths-based theories, but this book enabled me to approach the topic on a more artistic and creative basis. Beautiful Geometry is described as:
“The result is a delightful and informative illustrated tour through the 2,500-year-old history of one of the most important branches of mathematics.”
Jost and Maor wonderfully prove that art is not simply based in intuition and emotion, and that maths is not simply based in cold rationality. The two compliment each other as dual working aspects of our brains. The book is accompanied by numerous illustrations of geometric shapes and formations, covering everything from Euclidean constructions and the Golden Ratio to the Snowflake Curve and infinity. They believe that “The search for pattern is indeed the common thread that ties mathematics to art” and this basic human tendency to look for patterns and order is something I have encountered many times in my research.
Müller, Lars (2015): Josef Müller-Brockmann Pioneer of Swiss Graphic Design. Lars Müller Publishers, Zurich, Germany.
Josef Müller-Brockmann (shortened to “MB” in Lars Müller’s book) has featured heavily in my research for both Research and Enquiry and Practice 1. This book highlights MB’s creative journey, from his early days searching for his creative path and trying a number of different creative practices, to his most recent work. The book features many of MB’s designs and illustrations, charting the development of the style MB became famous for: International Typographic Style (Swiss Style).
This book provides examples of a wide range of Swiss Style design and discusses the reasons behind MB’s decisions. One part that stood out in particular was when Müller wrote about MB removing some of his earlier illustrative work from his portfolio once he had settled on his objective style. This is a practice I initially did not agree with: hiding earlier work because it does not fit with current work. I felt that it was destructive to history and perhaps even rude to past customers who were still using older designs. A graphic designers portfolio is a visual map of development, and although early designs and drawings might not be relevant in the present they were relevant at the time of their creation and held just as much meaning and purpose.
With this in mind I looked back through some of my earliest web and graphic design. I could clearly see the flaws and acknowledge where I would do things differently now, but the dominating emotion was how different my early work was to my current design. I started to understand that MB did not necessarily remove older work out of embarrassment, but rather to give prevalence to Swiss Style and bring it into the spotlight.
Schouten, K.A.; Niet, Gerrit J.; Knipscheer, Jeroen W.; Kleber, Rolf J.; Hutschemaekers, Giel J. M. (2014): The Effectiveness of Art Therapy in the Treatment of Traumatized Adults. Sage Journals – Trauma, Violence & Abuse 2015, Vol. 16 (2), 220-228.
This article aims to evaluate evidence of the effectiveness of art therapy for trauma treatment. Schouten et al. suggest that art therapy is relevant to trauma (PTSD) patients because it fits with the wordless and nonverbal nature of traumatic memories (Schouten, et al. 2014). This, I felt, tied in with my research into repetitive geometric patterns and visual symmetry. Drawing, painting, collage and sculpting are used to shape and express feelings, thoughts and memories and eventually lead to change, development and acceptance (Schouten et al. 2014).
Schouten et al. provide statistics based on the 223 participants who took part in their study, with sub-groups of participants engaging with different types of art, including mandala drawing, plaid colouring and free-form colouring. There was no evidence that art therapy alone is more effective than art therapy combined with other treatments, but there was evidence that art therapy was effective in reducing trauma symptom severity.
This study was a useful resource because of my research into drawing patterns—specifically mandalas, although it relates to any repetitive drawing exercises—for mental wellbeing. This is an activity I have participated in before personally, with mandala and zentangle drawing. These drawing methods are designed to help relax, focus and expand the mind. There are a number of reasons why someone might draw mandala or zentangle: for therapeutic reasons (to overcome a trauma, to help relieve stress, to take their mind off something) and for creative purposes (drawing practice, creative focus and exploring symmetry/geometry/pattern-making).
Skinner, Stephen (2006): Sacred Geometry – Deciphering the Code. Sterling Publishing Co., Inc. New York, NY, USA.
In Sacred Geometry Skinner discusses geometry rooted in sacred art, spiritual architecture and in nature. The book covers how many spiritual buildings and constructs (churches, temples, tombs, and mosques) carefully reference mathematics—specifically geometry—because a harmonious design was thought to be more sacred, and thus bring you closer to god. This practice dates back as far as Ancient Egypt, and similarly through ancient Greek and the Dark Ages, and up until the present day. Skinner argues that ancient civilisations attempted to bring harmony to their structures using repetition and pattern, symmetry and proportion—considered divine measurements. In the case of Islamic architecture it was not permitted to use human or animal forms in religious art, so instead geometric patterns were used and it was believed that the purity of geometric pattern made these buildings sacred.
This book was the first time I encountered the notion of sacred dimensions, and it opened my eyes to the relevance of geometry in spirituality. I considered the incorporeal nature of belief and compared it to the solid, analytical nature of mathematics, with nature acting as the bridge between the two. Although at face value these two ideas seem a million miles apart I can now better understand how they complement one another.
Tondreau, Beth (2011): Layout Essentials: 100 Design Principles for Using Grids (Design Essentials).Rockport Publishers, 1st Edition.
Mathematical principles have been used for centuries in art and design, from Renaissance paintings to the Swiss Style of graphic design that developed in the early part of the 20th Century. In Layout Essentials, Beth Tondreau walks us through many of the different ways to layout information on a page using grid systems, from classic newspaper and magazine styles to more contemporary poster designs. She includes tips on how to avoid overcrowding, how to change the boundaries of the columns for effect, or the benefits of creating space to draw attention to certain areas of a design. I found this an invaluable resource as the book covers a hundred different approaches and also contains visual examples of each one and the impact those approaches have on the page and content. One section that stood out to me in particular is toward the end of the book, where she closes with the notion that it is fine to break the rules, providing you have taken the time and done the work to learn the rules first. This is where the designer makes conscious and creative choices to push boundaries—something I strive for in my own work.
Welch, R. and Welch, H. (2012): Art Deco and Geometric Stained Glass Pattern Book. Dover Publications, New York.
This book does not contain a great amount of textual information, but it does show many examples of geometric patterns and formations from Art Deco style and from stained glass windows.
I had not considered stained glass windows as inspiration until I found this book, but it has given me a wealth of ideas about how to develop my geometric designs (particularly when working with logos, patterns and layout) and also search for more examples of stained glass artwork. I am considering a series of graphic designs that re-make popular media in stained glass style. This idea is only in the beginning stages but could be interesting as part of my geometric MA project going forward.
I am also keen to see if I can modernise or play with Art Deco style and create something inspired by it but new. Initially I thought about creating shapes and patterns with text itself, although this is something I need to experiment with first.