Geometry in Spiritual Art and Design

Geometry is not only associated with mathematics, art and nature. One application that I have wanted to explore in depth since choosing my keyword is geometry as it occurs in spiritual art and design, a concept I encountered first hand while travelling in Southern India in the year 2000.

I was lucky enough to arrive in India at the start of the Pongal Festival, a Tamil harvest festival in which people celebrate the sun and appreciate the gods and goddesses for a successful harvest. This festival takes place every year in January. Generally the women in a household rise each morning and draw intricate geometric patterns on the ground outside their houses or on the floors within the houses. These patterns are created from various materials: rice flour, chalk powder, rock powder or synthetically coloured powders.

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The Tamil people of Southern India believe that by drawing the kolams and paying homage to the harvest, the goddess Mariyamman will grant them blessings and increase the prosperity of their home and family (Laine, 2009). At the time of seeing these patterns—of walking among and across them with my bare feet—I had no idea that they would leave a lasting impact on my creativity in years to come. As a young woman, I was not so keenly aware that this practice is only carried out by women and that it is a way for them to invite well-being, express their spirituality, but also measure their creativity and dexterity (Ascher, 2002). The tradition of kolam creates a space for Indian women to experiment art and also science. It is the meld of religion, spirituality, creativity, mental discipline and mathematics. Kolam drawing connects women with the wider community (the harvest) and with the family (the blessing), but it is also a personal and internal exercise.

This is an interesting practice that relates to my work on the MA because most of the kolams are represented using geometric shapes and formations. The repeating, symmetrical patterns range from basic white kolams to elaborate, kaleidoscopic designs.

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Kolam is a form of visual language, and interestingly computer scientists have become interested in the algorithmic nature of some of the designs as a way of creating picture languages (Ascher, 2002). This only reinforces the idea that many of the patterns, passed down from mother to daughter and spanning countless generations, are a highly skilled practice and require a taught technique.

My experience with kolam and later with mandala has led me to create my own concentric geometric designs, paying homage to the things that are important to me. The following design was created in 2017 using mandala as inspiration. This is one of a four-part series work in progress based on the seasons.

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A mandala (Sanskrit for “circle”) is another form of concentric geometric pattern drawing that is conducted across the world in different countries. Specifically healing mandalas are created by cultures including Tibetan Buddhists and Navajo people in North America. In his article Mandala Constructing Peace Through Art, Tom Anderson discusses the drawing and deconstruction of mandalas as a way to explore the idea of “reconstructing” —a caring, cooperative and self-reflexive community project (Anderson, 2002). They are intended to protect or repair either people or the environment. Their aim is to restore balance and harmony—a theme that plays a large role in other areas of my research into geometry. These processes also bring a social harmony and reinforce shared beliefs, morals and values (Anderson, 2002). In its own way, it is a form of art therapy and is seen as a positive practice.

This relates to art in general as a form of visual messaging, a shared experience but an experience that is at the same time deeply personal to whoever is viewing the art.

In Sacred Geometry, Deciphering the Code, author Stephen Skinner writes that geometry is the archetypal patterning of many things, perhaps all things (Skinner, 2006), and this notion has become more and more evident to me as I have read, researched and written about geometric shapes in their many forms and applications. Over the months my sense of geometry being universal has grown stronger: it is a form of common language between humans, a way of reasoning in the sciences, a way of communicating within our belief systems, and a way for us to figure out where we are in relation to the rest of the world.

Skinner goes on to discuss geometry dating back to ancient times, most likely beginning in Egypt, with temples and other sacred spaces designed around geometric formations and scientists using geometry to map the movement of the heavenly bodies and the seasons (Skinner, 2006). These geometric buildings were intended for people to use to communicate more directly with their deities. The harmony and precision of the structures made them sacred. I believe this ties in with a common thread I have encountered throughout my research, which is that people take comfort and solace in the structured, organised, harmonious nature of geometric shapes.

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With this knowledge and my own personal interest in pattern-making and mandala / kolam design, I am eager to explore geometric design more in my work. It is a strong language and an effective way of communicating ideas and messages across the world, something that language cannot alone achieve. In this regard, art transcends other forms of communication and touches people from all countries and all walks of life.

References

Anderson, Tom (2002): Mandala Constructing Peace Through Art. Art Education Journal, 55:3, 33-39.

Ascher, Marcia (2002): The Kolam Tradition. Sigma Xi, vol. 90, No. 1. The Scientific Research Honor Society. North Carolina, USA.

Laine, Anna (2009): In Conversation with the Kolam Practice: Auspiciousness and Artistic Experiences Among Women in Tamil Nadu, South India. University of Gothenburg, School of Global Studies. Sweden.

Skinner, S (2006): Sacred Geometry, Deciphering the Code. Sterling Publishing, New York, London.

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Artist Spotlight: Isabella Conticello

Isabella Conticello is a graphic designer and illustrator whose work I have encountered on Behance. I will specifically look at her series A GEO A DAY where she created a colossal body of different minimalistic geometric designs, which she published as a series in its entirety in 2015.

One of the things I find most remarkable about this project is Isabella’s dedication to such a massive solo project. On first glance the inclination is to assume she must have grown bored of the same type of design day after day, but the interesting thing is that the challenge would have only increased over time to design new and unique pieces and generate new ideas. Isabella did this by assigning colours to different areas of her life and representing those areas in her art (Osso Magazine, 2015).

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The designs are reminiscent of modern minimalist art which rose to prominence in the 1960s, even down to her modest colour palettes. Minimalist art utilises geometric shapes and patterns in place of natural or organic forms (Smigel, 2012), and here many of these shapes combined with their colours clearly represent the world that Conticello lives in. By reducing the number of elements and simplifying each idea, she invites viewers to study the raw shapes, lines and curves in the world around us, unhindered by additional objects or other clutter that can distract.

There is also a definite element of mathematics in her work; Isabella herself admits in her 2015 Osso Magazine interview that the Bézier curve helps her to realise her artistic goals and it is clearly evident in some of her pieces.

Her minimalist approach, careful selection of geometric shapes and thoughtful composition makes it possible to extract entire landscapes from some of the designs. There is a beauty and vastness in their simple shapes and soft colours. I particularly like how she captures the essence of nature in the following examples, where I see the ocean and sweeping sand dunes, respectively.

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While not all of the designs are easy to interpret, each one is telling a story about something—a place, or an object, or a concept. Isabella herself loves yellow, blue and red and believes they are the perfect triad (Osso Magazine, 2015), and these colours feature heavily in her designs, often as pastels but also sometimes as richer variations. She creates many pieces using a paler palette, which lends an ambience to the flow of the gallery. I found while scrolling through and examining her work that it was as relaxing as it was fascinating.

Isabella also occasionally plays with texture and three-dimensional elements in her designs, giving them a stronger sense of depth than a lot of the flat, two-dimensional pieces have:

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Overall her aim was to create a cohesion that spans all of her work (Osso Magazine, 2015) and it is possible to see this achievement if you scan her Behance gallery, where many of the themes, styles and colours found in the A GEO A DAY  set are present.

I admire Isabella’s dedication to this project, because as a fellow graphic designer I know that it is not always easy to simply sit down and create something that you like or that you are happy to show to people. Some days it is a struggle to start a piece of work and other days it is a struggle to complete a work-in-progress. Regardless, Isabella has managed to create a connection between these pieces that ultimately tell a story of her life over the course of many months, observing her surroundings and paying homage to her favourite things.

I would like to develop this type of dedication and discover a way to represent my own interests, favourite places and ideas. While I do not think I would use a style similar to Isabella’s, hers is a prime example of how simplicity can speak volumes and communicate a concept with just a few basic shapes. This ties in well with my findings from other areas of geometric design, right down to my research and work with logo design: more often than not, simplicity is key.

References

View Isabella’s full project on Behance here, 2015.

Interview with Isabella in Osso Magazine, 2015.

Article about Isabella’s work at Partfaliaz, 2015.

Smigel, Eric (2012): Lessons That Bear Repeating and Repeating That Bears Lessons: An Interdisciplinary Unit on Principles of Minimalism in Modern Music, Art, and Poetry. General Music Today (journal), issue 1, pages 5-10.

Artist Spotlight: Louise Nevelson

“Some of us come on earth seeing,
Some of us come on earth seeing color.”
– Louise Nevelson

Louise Nevelson (1899—1988) was a Russian-born American sculptor who created intricate, abstract geometric art out of pieces of wood and debris that she found on the street of New York City where she lived. Her sculptures held a grand and intimidating appeal, many of them towering walls made up of individual boxes, which were filled with the objects she found.

I am drawn to Nevelson’s work because it is comprised of detritus that most people discard, ignore and believe to be useless. She scavenged most of the wood used in her sculptures and from this scrap Nevelson created deeply personal works that told many stories of the city. Their structures were often chaotic and busy, emulating the structure of New York City with its districts and blocks and the myriad lives contained within each one, all of them unique.

Nevelson01Black Wall, 1959.

Nevelson began painting her sculptures one uniform colour in the 1940s, usually matte black, although she occasionally deviated to white or gold. When I look at her work, the single, solid colour helps to unify the many different shapes and aspects of the piece, as if she was saying that all of these components might be different but they ultimately work toward a common goal, or share the same playing field. This also speaks to me of the city life.

Nevelson02Dawn’s Wedding Feast, 1959.

Not all of Nevelson’s work were about her surroundings. A number of them were deeply personal to her and related to her past. Dawn’s Wedding Feast was the first sculpture she painted white. This piece was symbolic of Nevelson’s failed marriage, a bittersweet homage to a traumatising event in which she separated from her partner and also abandoned her son so she could travel to Europe to study art, but she also considered it the dawning of a new chapter in her life and work—signified by the change from black to white (Rapaport, B. K, et al. 2007). This sudden change is striking and an extremely conscious choice. In Louise Nevelson: A Passionate Life, Laurie Lisle, who spent extensive time interviewing Nevelson, states that Nevelson was not the sort of person to dwell on the past but preferred to look at what was important in the present (Lisle, 1989). However, I would argue that Nevelson might have instead been constructing a maze-like structure in which to bury her past, rather than process it and move on.

Troubling events in Nevelson’s early life led her to become interested in the Cubist movement. Nevelson herself said:

“The Cubist movement was one of the greatest awarenesses that the human mind has ever come to. Of course, if you read my work, no matter what it is, it still has that stamp. The box is a cube.”

Cubism gave her a sense of structure, offering her sanctuary and order when everything else in her life seemed chaotic (Rapaport, B. K. 2012).

In her article Geometric Abstraction, Ana Franco writes that although Nevelson’s art followed a geometric structure and hinted at a profound desire for order, Nevelson herself was interested in the magical powers of art (Franco, A. 2012). It is easy to see, when looking at Nevelson’s complex sculptures, that there could be a hidden spiritual message within. Her wall sculptures are maze-like, or like giant mythical puzzles, with countless dark crevices and nooks for shadows to settle. When I look at Nevelson’s work I see both the mathematical nature of her composition but also the sense of monumental curiosity, secrets and stories captured within the details.

Nevelson was also acutely aware that, at the time she was working to develop her art, she was existing in a male-dominated profession.

“Throughout her career, her beauty and flamboyance made it difficult for many people to take her seriously as an artist. She was often depressed or enraged as she struggled to exhibit and sell her work. Yet while she attacked the art world for being male-dominated, she saw herself as unique—not a model for other women.” (Lisle, 1989)

It is interesting to me to find an artist who wanted to simultaneously break down the gender imbalance in the art world and also stand apart from and above it. Nevelson’s wanting to be unique and not a model for women to aspire implies a superiority complex, although this seems strange given her hardships and the incredibly gritty experiences she had been through years before. Again I wonder if it could be possible that in her complex geometric structures, Nevelson was trying to escape the human aspects of her life—all the trials and struggles—rather than process and let go of them.

Nevelson03An American Tribute to the British People (1960-1964).

Nevelson’s style was borne from an intuitive art style. She once said:

“I hate the word ‘intellect’ or the word ‘logic’—logic is against nature, and ‘analysis,’ another vulgar word.” (Lisle, 1989).

I admire the notion of allowing intuition to dominate the creative process, although I also believe that to be afforded this type of creative freedom, and for people to trust your instinct, you have to have had an extensive background in your field and have done the research, trial-and-error, and learned all that you can from many different sources. According to Nevelson’s son, nobody really knew how much she knew about art because she never revealed the extent of her knowledge (Lisle, 1989). I wonder if her tendency to play her cards close to her chest was due to her not being widely accepted by her male contemporaries. Her documented bravado and flamboyance could have also been a screen for her to hide behind. Furthermore, if Nevelson thought herself above most other humans, she might not have seen any reason to impart all of her wisdom.

Nevelson04Mrs. N’s Palace (1964-1977).

Nevelson’s work is important to my research because of her use of geometric structures—whether they are meant to tell stories, or be a hiding place for truths. The last piece, Mrs. N’s Palace, strikes me as the embodiment of a mental fort that Nevelson might have constructed, but her use of palace speaks of her confidence and outspoken attitude and her tendency to set herself apart from everyone else.

References

Franco, Ana M. (2012): Geometric Abstraction. The New York-Bogotá Nexus. American Art, vol 26, No. 2 (Summer 2012). The University of Chicago Press.

Lisle, Laurie (1989): Louise Nevelson – A Passionate Life. Summit Books, New York, NY.

Rapaport, B. K. (ed.); Danto, Arthur C.; Senie, Harriet F.; Stanislawski, Michael (2007): The Sculpture of Louise Nevelson: Constructing a Legend. Yale University Press, New Haven and London.

Louise Nevelson at the Tate: https://www.tate.org.uk/art/artists/louise-nevelson-1696

Louise Nevelson photos: Maine, An Encyclopedia – http://maineanencyclopedia.com/ and Biography.com – https://www.biography.com/people/louise-nevelson-20854319

Artist Spotlight – Manolo Gamboa Naon

Manolo Gamboa Naon is a Creative Coder from Argentina. He creates interactive installations, video games, data visualisation, websites and tools for digital advertising. He describes himself as “obsessed with generativity” and likes programming images in which he works with geometric patterns, textures and overloading.

I found Manolo’s work a few months ago while browsing Behance, and was immediately drawn in to his designs. They spark with colour, shapes and motion, and there is often an incredible sense of depth in his work thanks to his relentless layering techniques.

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For me his work stands out because it seems like every geometric shape has been placed carefully and thoughtfully to reach the overall outcome. It must be a painstaking and time consuming process putting some of these works together, and Manolo does it prolifically—indicating that he simply loves creating these deeply complex compositions.

In his essay Nature of Abstract Art, Meyer Schapiro writes that in stripping away the literal subject and the bias of the artist, we are left with pure aesthetic elements, shapes and colours.

“The new styles accustomed painters to the vision of colors and shapes as disengaged from objects and created an immense confraternity of works of art, cutting across the barriers of time and place.”

This way of thinking meant that all art was suddenly valid, from children’s paintings to the scribbles of people with mental disorders, and all deserved of consideration (Schapiro, 1937). Art is subjective, as are most creative fields, and perhaps more so when it comes to abstract, where shapes and colours exist as their own entities and remain open to interpretation.

To me, Manolo’s work feels almost stream-of-consciousness that produces beautiful complicated works. They are not uniform patterns and there is not often perfect symmetry, but the use of geometric shapes gives off an illusion of order and precision. This is an interesting balance to me: abstraction and order. I love how Manolo achieves this and makes it look effortless.

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A number of his pieces are reminiscent of colourful fantasy cityscapes. One in particular I am drawn back to time and time again is this piece posted in July 2018:

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My interpretation is that I am looking down onto many circular buildings, the faint grey lines and intersections signifying the many pathways and possible directions people can travel, with the larger circular shapes housing their own networks and pathways within. It could also represent a person’s brain, with its innumerable neural pathways and connections, the colours representing the different areas of the brain working for different purposes. Manolo is a creative coder who also makes video games, which is possibly why when I look at this piece I see a landscape full of areas to traverse and explore.

While I enjoy most of Manolo’s pieces, there are one or two that do not work so well for me. For example:

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This piece is far more minimal than his other work and in its own right fine, but when viewed in his gallery sitting next to his other pieces it almost looks like a mistake. While I appreciate the simplicity of many geometric shapes and their different meanings this piece does not speak to me or tell me a story. Within the context of his other pieces I feel this one does not fit. Many of Manolo’s other creations feel like well-rounded stories or places for great exploration. Unfortunately he rarely adds a commentary along with his art so I am not sure what this represents to him or where the idea came from.

References

Schapiro, Meyer (1937): Nature of Abstract Art. Published in Marxist Art Quarterly, from the American Marxist Association.

Visit Manolo’s website | Manolo on Behance

Naturally Occurring Geometry

One fascinating resource I encountered in my research is the book The Fractal Geometry of Nature, by Benoit B. Mandelbrot. It has been important to me to write about naturally occurring geometry as it crops up in a lot of my reading and is something I am inspired by in my own work.

fractal-nature

Nature is fragmented and multi-faceted. This is where Mandelbrot got the idea for the term fractal, from the Latin frangere, meaning “to break” (Mandelbrot, 1982). But the fragmentation of nature means that there are an infinite amount of possible geometric shapes and formulas, surpassing Euclidian theories that there is a finite number of geometric shapes available to us. In art and design, Mandelbrot suggests that we are more likely to accept abstract art that resembles fractals because it is subconsciously familiar to us.

M. C. Escher, whose famous tessellations were inspired by the Islamic tiles that he saw while on a trip to the Alhambra in Spain (Van Dusen, Scannell and Taylor. 2012), is often described as a fractal artist with a keen interest in the properties of patterns found in the natural world. His pattern Circle Limit III repeats at various sizes and scales without losing its geometric formation (Fig. 1) and is reminiscent of naturally occurring geometry in plants and flowers. Similarly, the Koch Snowflake, a fractal curve that was presented by Helge von Koch in 1904, is a scalable geometric pattern that can repeat indefinitely (Fig. 2).

escher-koch

Mandelbrot’s vision of geometry was unrestrained by the mathematical theory that preceded him, although this is not to say he does not respect those theories that came before; rather, he builds upon them and tries to bring them into better harmony. He believed that mathematicians shied away from the possibilities that nature presented, and attempts to meld science with philosophy. This blending of precision and boundless creativity is a pairing that mirrors the job of a designer working in the present day.

“One must also recognize that any attempt to illustrate geometry involves a basic fallacy. For example, a straight line is unbound and infinitely thin and smooth, while any illustration is unavoidably of finite length, of positive thickness, and rough edged. Nevertheless, a rough evocative drawing of a line is felt by many to be useful, and by some to be necessary, to develop intuition and help in the search for proof. And a rough drawing is a more adequate geometric model of a thread than the mathematical line itself.” (Mandelbrot, B. 1982. P22)

Here Mandelbrot expresses the difficulty in representing something as broad and infinite as a geometric line, but appreciates that we must find ways to root our understanding of geometry—in the form of drawings and art—and pay homage to the shapes present around us. As an artist and designer, this is interesting to consider: shapes are merely representations of things we might otherwise consider unfathomable. We are trying to make visual sense of something as vast and infinite as the universe.

Mandelbrot also points out that while many believe that mathematics, music, art and architecture seem to be related to one another, it is only a superficial connection, as—in the case of architecture—one building can follow strict Euclidian geometry, but another building can be more rich in fractal aspects (Mandelbrot, 1982). I see the similarities drawn between geometry and art, mathematics and music often, most recently when studying Swiss Style graphic designer Josef Müller-Brockmann’s poster for Beethoven’s symphony, where the geometric choices are believed to have represented the structure of the symphony. In my Practice 1 assignment I look at modern geometric Swiss Style art used to portray grunge band posters from the 1980s-1990s, although I struggle to draw the connections and similarities between something that is rigidly structured, ordered and precise with something that stood for distortion, being true to yourself and individuality.

Mandelbrot also writes: “It is widely held that minimal art is restricted to limited combinations of standard shapes: lines, circles, spirals, and the like. But such need not be the case. The fractals used in scientific models are also very simple (because science puts a premium on simplicity). And I agree that many may be viewed as a new form of minimal geometric art.” (Mandelbrot, B. 1982)

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These snowflakes are from a winter-themed graphic design project I worked on last year for a local restaurant and shop. At the time I was not thinking in terms of geometry, but I was faced with the problem of deciding on the shapes and formations of the snowflakes, as there were so many possibilities. Even before I began reading about geometry I was aware of the challenge of representing natural complexity in art.

snowflake

In Fractal Architecture: Organic Design Philosophy in Theory and Practice, James Harris explores the fractal geometry of nature as the inspiration for some forms of architecture where the natural “recursive mathematical derivation” of form creates a new structure that is removed from the more logical and rigid Euclidian geometry (Harris, J. 2012). Harris believes that philosophers and mathematicians created geometric theories to cope with the unfathomable randomness of the universe, which was beyond their comprehension (Harris, J. 2012). I believe this is a natural human behaviour—the need to make sense of the vastness of the universe and all the endless possibilities within. We search for and even yearn for our own kind of structure within the study of complex, naturally occurring structures.

In the paper Is the Geometry of Nature Fractal? written for Science Magazine, David Avnir, Ofer Biham, Daniel Lidar and Ofer Malcai offer an alternative idea:

“Fractals are beautiful mathematical contructs [sic] characterized by a never-ending cascade of similar structural details that are revealed upon magnification on all scales. Over the past two decades, the notion has been intensively put forward that fractal geometry describes well the irregular face of nature.” (Avnir, O. et. al. 1998)

While it is difficult to believe that all geometry in nature holds a perfect symmetry and fractal behaviour, I do not think that theorists like Mandelbrot were trying to constrain nature within the definition “fractal”, but rather offer another view of something that cannot easily be understood or contained. Again, a way for us to wrap our minds around this deeply complex and mysterious world surrounding us.

As part of my Practice 1 assignment, I am looking at the multi-faceted nature of geometry and how low-poly art emulates the geometric shapes found in crystals and ice—neither of which are uniform in size and shape, and yet both of which form a structure with many faces that reveal different colours and reflections. Mandelbrot raises the question in the opening of his book “Why is geometry often described as cold and dry?” and I would like to explore geometry used to expose a depth and intricacy behind the “cold, dry” surface.

References

Avnir, David; Biham, Ofer; Lidar, Daniel; Malcai, Ofer (1998): Is the Geometry of Nature Fractal? Science, Jan 2nd – Vol. 279, Issue 5347, pp. 39-40.

Harris, James (2012): Fractal Architecture: Organic Design Philosophy in Theory and Practice. University of New Mexico Press.

Mandelbrot, Benoit B. (1982): The Fractal Geometry of Nature. W.H. Freeman and Company, New York.

Van Dusen, B; Scannell, B. C; Taylor, R. P: A Fractal Comparison of M.C. Escher’s and H. von Koch’s Tessellations. Published in Fractals Research, School of Education, University of Colorado, Boulder, Colorado, USA.

The Scutoid Has Landed

Just a short post about something that happened recently which relates to my chosen keyword geometric. A paper written by Pedro Gómez-Gálvez and Pablo Vicente-Munuera was published in Nature Communications in July 2018 introducing a new geometric shape that is linked to the metamorphosis of tissue cells during organ development. This shape is called a Scutoid*.

scutoid

“As animals develop, tissue bending contributes to shape the organs into complex three-dimensional structures. However, the architecture and packing of curved epithelia remains largely unknown. Here we show by means of mathematical modelling that cells in bent epithelia can undergo intercalations along the apico-basal axis. This phenomenon forces cells to have different neighbours in their basal and apical surfaces. As a consequence, epithelial cells adopt a novel shape that we term “scutoid”.” (Gómez-Gálvez, P. et. al. 2018)

Nature has found a way to conserve energy and pack cells more efficiently, and we are only now uncovering this phenomenon. While the science of the scutoid falls outside of my main stream of research, I do find it fascinating that yet again geometry plays a major role in nature as much as it does in mathematics. It is this blending of science and the creativity of nature that arises time and time again. The more I read about geometry the more I feel that it links everything–from art and design to the fabric of living creatures and the universe itself.

Gómez-Gálvez and Vicente-Munuera also write:

“In addition to this fundamental aspect of morphogenesis, the ability to engineer tissues and organs in future critically relies on the ability to understand, and then control, the 3D organization of cells.” (Gómez-Gálvez, P. et. al. 2018)

Discovering this deeper level of our biology will no doubt have an impact on how effectively and efficiently we approach medical and scientific problems in the future. The new shape could also play a role in art and design. 

References

Pedro Gómez-Gálvez, Pablo Vicente-Munuera, Antonio Tagua, Cristina Forja, Ana M. Castro, Marta Letrán, Andrea Valencia-Expósito, Clara Grima, Marina Bermúdez-Gallardo, Óscar Serrano-Pérez-Higueras, Florencia Cavodeassi, Sol Sotillos, María D. Martín-Bermudo, Alberto Márquez, Javier Buceta, Luis M. Escudero (2018): Scutoids are a geometrical solution to three-dimensional packing of epithelia. (www.nature.com/articles/s41467-018-05376-1) Nature Communications, 2018.

* According to Science Daily (28th July 2018), the new shape was labelled “Scutoid” because of its resemblance to the scutellum, the posterior part of an insect thorax or midsection. (http://www.sciencedaily.com/releases/2018/07/180728084136.htm)

Architectural Geometry

“In an architectural context, geometry is often used as an ordering device, that is, a way to link spaces to functions.” (Williams, K; Ostwald, M.J. 2017)

In the introduction to vol.19 of the Nexus Network Journal, Manifestations of Geometry in Architecture, Kim Williams and Michael J. Ostwald highlight that the geometry of architecture is not only a functional measure to ensure the most efficient and structurally sound building, but also to create a more appealing aesthetic.

“Once a structure is codified and built, geometry can add beauty to a design.” (Williams, K; Ostwald, M.J. 2017)

This idea rings true to me as a creative who has explored geometric graphic design for a long time without paying mind to the precise mathematical foundations that fortify such design. I rarely associate mathematics with beauty, not as much as with its creative cousins art and music, but in the precise geometry of buildings there can be found a great sense of rhythm, movement and intent.

Architectural design can range from the simple to the complex, just like with any other creative field. Examples of creative architecture include the Corpus Museum in the Netherlands which advertises itself as a “journey through the human body,” and the Spiraling Treetop Walkway in Denmark, created to give tourists a bird’s-eye view of the forest without disturbing the natural surroundings. Both embrace nature in their own unique way, and both attempt to represent the purpose (subject) of the building (Fig. 1).

corpusmuseum-denmarkobservatory

Geometric architecture also encompasses the bizarre. For example, the Dancing House in Prague (Fig. 2, left). Designed by Frank Gehry and Vlad Milunič, this unusual building that leaps over the intersection symbolises the direction the Czechoslovak people have taken to move away from the totalitarian rigidity of the past.

dancinghouseandothers

The other two photos (Fig. 2, right) are buildings that do not conform to conventional architecture but rather deviate to create a new and unexpected environment and an entirely different experience. I thought about how this relates to contemporary design and my own work. I feel that we are often prone to working within pre-defined boundaries and expectations, but sometimes it is necessary to not just step outside the box but take a positive leap. The above architects were not afraid to break tradition; to them, it was extremely important to do so.

Architect Frank Lloyd Wright designed impressive buildings that worked alongside their surroundings rather than dominated them. Lloyd Wright himself described his buildings as “organic architecture” (Pfeiffer, B. 2004) and you can find examples of this in his work:

franklloydwright

In Geometry And Architecture, Steven Firth writes:

“The uses of geometry in architecture tend to be instrumental, dovetailing with the process of building as a species of order. The modulation and arrangement of plans and the surfaces of elevations go hand in hand with this fiction, but also depend upon geometrical play.”

The implication here is that geometry exists to bring order and structure to a landscape or a people, but I cannot help but feel when looking at Lloyd Wright’s harmonious meshing of man-made and nature that there is a synergy, with the one blending seamlessly with the other. The water flows beneath the house and out over the rocks, much in the same way that people flow around and through the house, and in the way the creative ideas flow through the architect or designer.

The notion that geometry can influence a person’s perception of a building cropped up frequently in my reading. In Aesthetics of Simulated Soiling Patterns on Architecture, Carlota M. Grossi and Peter Brimblecombe explore the reactions in people to different soiling found on buildings, in an attempt to see if there was a preferred type of pattern. The idea is that if there is a less obnoxious soiling of architecture then building managers wouldn’t have to have their buildings cleaned so often. To me this is a study with only a tenuous connection to my own research, but what I found interesting was the results: people generally prefered a more uniform type of soiling–a pattern that they could make sense of–rather than a randomised formation.

The more I dig into geometry and shapes, the more it seems that we are drawn to pattern, repetition and order.

Finally, I looked at geometry at work in architecture from cultures other than my own. One fascinating article written by Carol Bier is Art and Mithāl: Reading Geometry as Visual Commentary. Bier explores the idea that the decoration on Islamic monuments is often viewed through a Western lens (Bier, C. 2008), but the pervasive use of pattern must be an expression of something more than simple ornamentation. The implication is that elaborate decoration somehow implies a lack of deeper meaning and significance. Bier also writes:

“It now appears that aperiodic patterns with five-fold symmetry related to that of quasicrystals may have been understood by craftsmen in Iran, if not mathematicians, hundreds of years before this phenomenon was discovered in the West.”

It might be naive of us to assume that geometric pattern relates to decoration or mathematics.

In the year 2000 I spent three months in India, and had the opportunity to see the Taj Mahal up close. The mausoleum itself is comprised of countless geometric shapes and precise lines and symmetry, but inset into the white marble are row upon row of black onyx script. The two details combined created an incredible sense of peace, flow, history and reverence.

tajmahal

Again, it fascinates me how geometry can be used in architecture to evoke different feelings from those viewing it. I do wonder how an opinion might shift if the story of a building is changed: if you transformed the Taj Mahal, built as an elaborate mausoleum by emperor Shah Jahan for one of his wives, into the stately home of a wealthy oil family, built to show off status and power. Suddenly the romantic spirit of the building is changed. The geometric components remain the same, the patterns are no less intricate and the work that went into it was no less laborious, but now it becomes a structure of arrogance and vanity.

This relates to my Practice 1 project, where I want to look at how small changes made to a geometric design can change people’s opinions and perceptions. My project relates to branding and logo development, but from my research into architectural geometry I can take away that geometry does not always have to be rigidly ordered to work; it can be combined with more fluid design (elements surrounding it) and they can work together harmoniously.

References

Bier, Carol (2008): Art and Mithāl: Reading Geometry as Visual Commentary. From Iranian Studies, Vol. 41, No. 4, Sciences, Crafts, and the Production of Knowledge: Iran and Eastern Islamic Lands.

Firth, Stephen (2010): Geometry and Architecture. Architectural Theory Review, Taylor & Francis Online.

Grossi, C. M.; Brimblecombe, P. (2004): Aesthetics of Simulated Soiling Patterns on Architecture. School of Environmental Sciences, University of East Anglia, Norwich.

Pfeiffer, Bruce Brooks (2004): Frank Lloyd Wright, 1867-1959: Building for Democracy. Taschen, Germany.

Williams, Kim; Ostwald, Michael J. (2017): Manifestations of Geometry in Architecture. The Nexus Network Journal, Architecture and Mathematics, vol. 19. Birkhäuser Publishers, Switzerland.

The Geometry of Textiles

In her symposium paper Textiles and the Body: The Geometry of Clothing, Madelyn Shaw talks about the mathematical principles of weaving material used in clothing:

“Interlaced threads create square or triangular grids, techniques such as knitting or crocheting can make grids of any shape, from triangular to polyhedral. Those who make clothing transform flat fabric planes into three-dimensional forms through a variety of means.”

Patterns occur not only on the finished fabric as a visual design—they are integral to the structure of the garment itself. A single thread can generate complex geometric patterns that most people wearing the finished garment will never see with the naked eye or look for in the first place. In the Textile Research Journal article Hierarchy of Textile Structures and Architecture of Fabric Geometric Models (Lomov, S.V; Huysmans, G; Verpoest, I; 2001), it is posed that the hierarchical structure of fibrous materials influence the mechanical behaviour of textiles. A well-developed weave can result in sturdier fabric and a longer lifespan of the garment. Additional fabrics can also be added to make the inherent geometry of the weave varied and more complex, although this raises the possibility of structural inconsistency.

There are a number of recognisable garments that utilise geometric shapes: the kaftan, the poncho, or the Pakistani jumlo (Shaw, M. 2006), and a number of high fashion designs draw on geometry to enhance and bring attention to different areas of the body (Fig.1).

An interesting point to note about the inherent geometry within textiles are the changes that take place throughout the course of a garment’s lifespan, from the moment it is woven to the moment it is discarded. Force and pull can lead to stretching at various stages and none of the shapes that are created during the sewing will remain in a solid state—they will constantly fluctuate with wear and even become permanently distorted in some cases of intense wear. The force applied to yarns and fibres creates a deeper and ever-changing level of geometry. In the paper Mechanics of Textile Composites: Micro-Geometry (Miao, Y; Zhou, E; Wang, Y; Cheeseman B. 2007) this is referred to as “micro-geometry.”

I decided to look more closely at how pressure and force can change the nature of geometry within textiles. I spread one of my cardigans across a table and then took a photo of the fabric in its unworn, untouched state. Fig.2 shows the change in the geometry of the weave when the jumper was pulled at either side.

A more obvious type of geometry that can be applied to a garment is the visual design and pattern, often created by using different colours or different materials. Repetitive pattern brings structure, gives flow and rhythm, and can be appealing visually and aesthetically (Perkins, M. 2015). During my research I wanted to know why we are drawn to geometric patterns and what it is about patterns that we find pleasing. Mathematician Ian Stewart says that we live in a universe of patterns (Stewart, I. 1998), from naturally occurring patterns in the makeup of plants, the formation of clouds in the sky, in topography, to genealogical patterns and the structure of our families. The patterns that surround us every day provide our lives with symmetry, repetition, order, movement and rhythm (Kraft, K. 2015), and it is hard to believe that patterns do not emotionally and intellectually influence us all in some way.

Perhaps this is why we are drawn to structure and order; is necessary in society, and for many people it is necessary in day to day planning and living. Geometry, being present in many aspects of our lives, even down to the clothes we wear, could play more of a role in how we feel and what we think than we realise. 

In Design Aesthetics: Principles of Pleasure in Design, Paul Hekkert discusses the meaning of aesthetics, and argues that works of art are mostly produced for the purpose of gratifying the senses (Hekkert, P. 2006).  This could go doubly for textiles, whose tactile and visual experience combines to produce a stronger reaction. But is there a deeper logic at work in the things we are drawn to? Hekkert also raises the question of why we like certain objects—what is it about a pattern or the feel of a product that reaches us on a deeper level than the purely visual? Something well-structured could subliminally tell us that it is sturdy and safe. A geometric pattern can also imply stability and structure, and bring comfort in its repetition and order. Hekkert writes:

“As demonstrated, adaptations have evolved to serve functions beneficial to our survival. It would have been helpful for the development of these adaptations if things in the world around us that contribute to these functions were reinforced (Tooby & Cosmides, 2001). In other words, it must be beneficial for humans to seek cues or patterns that serve these adaptive functions. We therefore (have come to) derive (aesthetic) pleasure from patterns or features that are advantageous to these functions.”

If this is the case, then we instinctively look for recognisable or pleasing patterns as a way of choosing the safest, most beneficial course of action. It would be an interesting experiment to see if somebody wearing a geometric pattern and somebody wearing a randomised pattern influences how trustworthy or approachable others perceive them.

Through my research into geometry used in textiles I have discovered that there are a number of different ways geometric shapes can play a part in our garments and the fabrics surrounding us. Geometry is worth considering when creating a textile and pattern is important for a designer working with textiles. Additionally, geometry and pattern can be used to evoke different meanings and responses and also influence the integrity of a textile.

As a result of my research I am keen to develop patterns that I can use in my own work and share as resources for other designers. In the past I have created patterns for clients (one was printed on a wedding shirt, which relates to the topic of this post) but I would like to approach pattern design with more clarity and focus, using the knowledge I have picked up from reading about patterns in textiles.

References

Hekkert, P (2006): Design Aesthetics: Principles of Pleasure in Design. Delft University of Technology, Netherlands.

Kraft, Kerstin (2015): Textile Patterns and their Epistemological Functions. Textile: The Journal of Cloth and Culture.

Lomov, S.V; Huysmans, G; Verpoest, I (2001). Hierarchy of Textile Structures and Architecture of Fabric Geometric Models. Leiden University, Netherlands.

Miao, Yuyang; Zhou, Eric; Wang, Youqi; Cheeseman, Bryan A. (2007): Mechanics of Textile Composites: Micro-Geometry. Department of Mechanical and Nuclear Engineering, Kansas State University, USA.

Perkins, M. (2015): Print & Pattern : Geometric. 1st edition, Laurence King Publishing, London.

Shaw, Madelyn (2006): Textiles and the Body: The Geometry of Clothing. Textile Society of America Symposium Proceedings, University of Nebraska – Lincoln.

Geometry in Origami

Many of us are exposed to origami at some stage in our early years, either at school or through the media we consume. It can be used as a tool for learning, as an aid for focusing, a way of embracing our creative side, or simply for relaxation and pleasure. In this post I explore origami and its various uses.

In the paper A Note on Intrinsic Geometry of Origami, Koryo Miura, Professor and Director of the Research Division of Space Transportation, says: 

“The mathematical expression of an origami process is a transformation of a flat piece of paper into a polyhedral surface which expresses something.”

The word origami means “fold” and “paper” in Japanese, and can range from basic shapes to extremely intricate folded designs and formations. While Koryo Miura’s paper focuses on the mathematical aspects of origami, this sentence jumped out at me immediately as a creative practitioner, and I can relate it to my own creative processes and results.

I have always imagined origami to be a personal venture and it is often referred to as an art rather than simply a hobby. This could be because it provides both mental and physical challenge and problem-solving, introduces you to geometry, symmetry and spatial thinking, and deals with shapes and clean lines. It instills the ability to follow instructions and listen carefully, but it can also bring a deep sense of satisfaction and catharsis if followed through, and it helps cultivate creative thinking. The element of trial-and-error teaches patience and persistence (if folded wrong, a piece can become difficult or even impossible to continue). It is a timeless practise, accessible to anyone with access to a sheet of paper, and holds appeal to both children and adults alike.

In the article Analysis of Design Application on Structural Model of Origami, by Hsun-Yi Tseng, it is believed that: 

“Origami started out as a traditional art form found in religious rituals and folk customs before evolving into a creative art for leisure and recreation.”

jentriangularbox

My own attempt at origami. I went for something fairly simple: a trinket box design I found a few years ago. It has been a while since I’ve folded this box and it took a degree of fiddling, studying the tutorial, and testing my own dexterity before I could finish it. While there are areas that need improvement (corners not quite closing), I am overall happy with the result and it was worth the time invested. This type of origami also has its practical uses – the box will hold paper clips, pins and other small pieces of stationary on my desk.

Returning to my earlier statement that I can relate expression through origami to my own creative processes, I found that folding this box focused my attention immediately. This is something that happens frequently when I begin a new design, whether its a practical or digital piece, particularly if there are intricate elements to it. Many designs rely of different parts coming together to create meaning and resolution.

The box was not without some frustration, with some of the flaps awkward to slot together, but once the triangular shape started to come through these frustrations lessened. Perhaps this is because I could see the box taking shape. 

In the book The Use of the Creative Therapies with Sexual Abuse Survivors, editor Stephanie L. Brooke writes:

“Another important therapeutic aspect of Origami is physical and psychological effect of the act of folding paper, which allows us to use the left and right spheres of our brains at the same time (Shumakor, 2000).”

Much like zen drawing and art therapy, origami can become a form of catharsis with its repetitive motion and tendency to focus attention. I frequently engage in ASMR (Autonomous Sensory Meridian Response) exercises as it relaxes my mind and body, and one of my main triggers is the folding, cutting, touching and manipulation of paper and card. Paired with the joy of solving a puzzle, creating a complex or beautiful design, and focusing attention, I can see why origami is used in art therapy and mindfulness practises to enhance a sense of wellbeing or help process a traumatic experience.

One of my favourite origamists is Ekaterina Lukasheva, whose manages to merge incredibly complex and precise modular designs with a great sense of movement and an organic feel:

ekatrina

I can’t help but think that she was inspired by naturally occurring geometry in the world around her, a topic I will blog about in a separate post.

Tomoko Fuse’s modular origami boxes are also worth noting, combining basic geometric design with gentle motion and peacefulness. The structure of the box itself creates the pattern, with no need for additional decoration or intricacies:

tomokofuse

The Sonobe Cube Lamp is another lovely and practical piece of origami, based on the Sonobe module (believed to have originated from Mitsunobu Sonobe and Toshie Takahama), and illuminated by Judith at Origami Tutorials:

sonobe-lamp

Also the beautiful Hydrangea Tessellation, originally created by Shuzo Fujimoto:

hydrangeatessellation

References

Brooke, Stephanie L. (2006): The Use of the Creative Therapies with Sexual Abuse SurvivorsCharles C. Thomas Publisher; 1st edition.

Miura, Koryo (1989): A Note on Intrinsic Geometry of Origami. First presented at the First International Meeting of Origami Science and Technology, Ferrara, Italy.

Tseng, Hsun-Yi (2017): Analysis of Design Application on Structural Model of Origami. Published in 2017 International Conference on Applied System Innovation (ICASI).

Learn how to fold the origami triangular box hereSee more of Ekaterina Lukasheva’s origami hereTomoko Fuse Modular Box hereOrigami Tutorials.

Discover ASMR. There are some incredible ASMR artists out there and they are easy to find on YouTube, but the ones I am always drawn back to are Charlotte Angel and Rhianna at ASMR Magic.

Geometry at an Early Age

Most of us are introduced to geometry at a young age, and it is taught formally in many schools around the world. Geometry is an ancient practice and it is connected to numerous mediums and areas of study—many of which I will explore in this blog. This post focuses on geometry we are taught during our formative years and the difficulties many schools are faced with when trying to create a universally valid curriculum.

In The Place of Experimental Tasks in Geometry Teaching: Learning From the Textbook Designs of the Early 20th Century, Taro Fujita and Keith Jones write:

“A characteristic feature of geometry is its dual nature, in that it is both a theoretical domain and perhaps the most concrete, reality-linked part of mathematics.”

It is this duality that creates difficulties for teachers who are faced with trying to bridge the gap between theory and practice within the classroom, two parallel studies that many children struggle to fathom at the same time (Fujita, Jones, 2004). It is interesting that the two threads are difficult to weave together in a mathematical setting when theory and practice so often converge in nature and in art and design.

In The Role of Intuition in Geometry Education: Learning From the Teaching Practice in the Early 20th Century, Fujita Taro, Keith Jones and Shinya Yamamoto question the role that geometry has in a teaching curriculum and suggest that a slight reshuffle of the specification of geometry might be in order, specifically focusing on intuitive geometry–where children show an ability to “see” geometrical shapes and manipulate them in the mind to solve problems (Fujita, Jones, Yamamoto, 2004). Interestingly, their article highlights an address given by J. Perry, Professor of Engineering at the Royal College of Science, in 1901, who questioned the value of teaching Euclidian geometry to children (Fujita, Jones, Yamamoto. 2004). Perry was an advocate of experimentation and a more intuitive approach to geometry. This is fascinating to me as a creative; the idea that mathematics can be approached in a more creative way, by being encouraged to simply observe our surroundings. This method would likely appeal to children who do not think analytically or struggle in a scientific or mathematical setting. It also opens the floor to a wider selection of children being brought into geometry study and develops the ability to think outside the box or push expectations.

This is another aspect of my own graphic design that I want to develop further: rather than simply designing how I think I should design (trying to anticipate the expectations of others), I’d like to grow my own ability to “see” design in new ways, try new approaches and not concern myself with only doing what is popular in the moment. Seeing a subject as far removed from art and design as geometry in a new and creative way encourages me to push my own boundaries and methods.

This intuitive ability to see shapes and use them to solve problems ties in with the paper Modularity and Development: The Case of Spatial Reorientation by Linda Hermer and Elizabeth Spelke. Hermer and Spelke explore children who, during development, share some similarities with animals in how they spatially place themselves in their surroundings using geometric cues. Their study revealed that the developing brain is more in tune with room geometry than with lesser-defined patterns or other defining factors (like odour, colour, etc). At an early age, we reorient ourselves in accord with the shape of our surroundings (Hermer, Spelke. 1996), meaning that geometry is playing a large part in our natural development, before it is formally introduced or taught. That we are subconsciously aware of geometry relative to ourselves is an exciting concept and something that could be explored in advertising–particularly advertising aimed at children.

Attempts at tying geometry with a more creative, intuitive practice have been made in a number of key learning equipment. Not all of the following toys were intended for children but have been adopted over the years as useful developmental tools.

The Spirograph, whose origins date back to the mid 19th Century, was originally created to help prevent bank note forgery, but it eventually became a popular 2D geometry drawing game. Spirographs involve the creation of hypotrochoid (a circle rolling inside a fixed circle) and epitrochoid (a circle rolling around the outer edge of a fixed circle) roulettes. A similar method known as Guilloché is still used on paper currencies, passports and security seals today, which means we have all come into contact with spirograph patterns.

The Lego Group was founded in Denmark in 1932 by Ole Kirk Kristiansen and, since then, Lego has become one of the most popular toys of all time. Providing children and adults alike with Automatic Binding Bricks of different shapes and colours, it teaches us to think not just creatively but also geometrically and in 3D. Lego is an effective puzzle-solving tool which exposes us to science, technology and engineering. For example, constructing a complex object like a bridge takes thought, development and trial and error. These small plastic geometric shapes teach children to patiently think through problems to reach a desired outcome. Through playing with different coloured Lego bricks, children can also learn about symmetry, colour combinations and patterns, and really let their imaginations run wild.

The Helix Oxford Maths Set was launched in 1935 by Helix and offered its famous geometry set. It wasn’t until 1959 that they began to mould their own plastic tools, which are the tools I used when growing up. The set includes a ruler, set squares, protractor, compass and pencil, pencil sharpener and eraser.

Finally, the paper Fortune Teller was first introduced in the book Fun with Paper Folding (Murray and Rigney; 1928) and is a more informal geometric toy that many will recognise from their school days.

fortuneteller01

The fortune teller was originally called the “salt cellar” and was intended to stand on a table while the four pockets contained condiments. Fortune Tellers were a popular game at school and I remember creating them with my friends (usually asking most pressing questions such as “Which member of Take That will I marry?”).

The conclusion I draw from my reading into educational geometry study is that I need to be more observational when I am developing concepts and designs, and consider how those designs fit in with their surroundings. In the same way a child could spatially place themselves in relation to geometric objects around them, I am keen to consider how I might approach, say, a logo design in relation to the elements that will surround it, which will involve digging deeper into the client’s intended application of the logo.

References

Fujita, T; Jones, K (2004): The Place of Experimental Tasks in Geometry Teaching: Learning From the Textbook Designs of the Early 20th Century. International Congress on Mathematical Education (ICME-10). Copenhagen, Denmark.

Fujita, T; Jones, K, Yamamoto, S (2004): The Role of Intuition in Geometry Education: Learning From the Teaching Practice in the Early 20th Century. 10th International Congress on Mathematical Education (ICME-10). Copenhagen, Denmark; 4–11 July 2004.

Hermer, L; Spelke, E (1996): Modularity and Development: The Case of Spatial Reorientation. Elsevier – Cognition 61. Department of Psychology, Cornell University, Uris Hall, Ithaca NY.

Murray, William D., Rigney, Francis J (1928): Fun with Paper Folding. Fleming H. Revell Company.

Spirograph | Denys Fisher | Science Museum

Guilloché Patterns | Ed Pegg Jr., February 9, 2004

Lego

Helix Oxford Maths Set