patterns – Fountain Magazine

The Mathematical Patterns around Us

Louima Cunningham — Fri, 01 Mar 2019 01:27:10 +0000

“How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”
Albert Einstein

I have always been pretty sure that mathematics and science are the best languages to explain the natural phenomena. While for many mathematics is too abstract, for me, it is the beautiful language of the universe.

There is no upper limit to the numerical abilities of humans. First, we discovered fire, to get warm. Then, we needed light, so we invented electricity. When we needed to talk to someone 10,000 miles away, we invented the internet. Behind all these inventions, was mathematics.

Of course, the universe cannot speak or think. However, we, the people, can read the universe. There are many scientifically and mathematically inclined people who can read the universe and find answers and then describe them. We, the normal people, can also use our imagination as an apparatus to read the universe and nature. If we can read, hence, something is written. In order to write, a language is always needed. So, the universe should have a language. The letters are circles, triangles, hexagons, etc.

Everything in life has mathematical patterns. Think of the wild animals with stripes or patterns for the purposes of camouflage. But why does a leopard or cheetah or tiger have a particular design?

The Enigma codebreaker, Alan Turing, had a mathematical theory about leopard’s spots. Turing suggested in his paper “The Chemical Basis of Morphogenesis” (published in 1952) “a mathematical schema for the formation of the patterns found in animals and plants.” This was 60 years ago [1].

Stars have patterns. Astrologists have been looking at the outer space searching for patterns to better understand life. Whatever it may be that they find, it is always about mathematics.

Seasons have patterns. They come and go. And they influence nature: the climate changes, animals migrate north or south, rain comes, snow melts, the earth changes color, etc.… Of course, seasons cannot make these miracles. They can only have mathematical patterns.

Einstein had pondered for years on how mathematics works so perfectly. He knew that mathematics is the bridge or the language that connects humans with the universe. And being a connection between us and the universe makes mathematics the greatest achievement of mankind.

If you take a closer look at the patterns of our world, you will witness the language of mathematics. Let me give you some specific examples.

Fibonacci, the golden ratio, spiral, cabbage…

Our universe is filled with spiral designs. Spirals can be found in the shapes of the DNA double helix, flowers, elephant tusks, sunflowers, hurricanes, draining water, animal horns, a nautilus shell, a snail shell, a pinecone, a cabbage, a fingerprint, algae, galaxies… the list goes on and on. Tons of lifeless and living things have spiral designs. And they are not random spirals. They have something in common: the golden ratio! And surprisingly, “there is a strong case that this so-called ‘Golden Ratio’ (1.61803…) can be related not only to aspects of mathematics but also to physics, chemistry, biology and the topology of space-time” [2].

All these spirals in nature tell us there are numbers all around us. Let’s observe the numbers of petals on some flowers. When you count the number of petals of the flowers in your garden, you will get the numbers 3, 5, 8, 13, 21, 34, or 55. These numbers are not random numbers. These are very unique numbers; they are part of a sequence developed by Fibonacci, a 13th century mathematician, by adding up the last two numbers starting from 1:

1+1= 2, 1+2= 3, 2+3= 5, 3+5=8 …

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

F_n= F_n-1+F_n-2, F₁=1, F₂=1

But, why are those Fibonacci numbers so important? The key is, the relationship between the progression of growth and the proportion. There is a harmonic proportion hidden in the Fibonacci sequence.

A fact: If you divide one number in the sequence by the previous number, the answers result in or come closer to phi:

For example: 5/3 = 1.6666;

13/8 = 1.6250; 377/233 = 1.61802575; 317811/196418 = 1.618033399

Definition: In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities [3].

These numbers can be demonstrated with the spiral of the florets in a sunflower. The florets in a sunflower head also form two spirals. If you count the clockwise and counterclockwise spirals that reach the outer edge, you’ll usually find a pair of numbers from the sequence: 34 and 55. If it is a very large sunflower, you will get 89 and 144 [4].

These spirals are not only in sunflowers. You can see them if you look at a pine cone or a daisy. If you mark the spirals and count them, you will always get a number from the Fibonacci sequence. And if you count in the other direction, this time you will find an adjacent Fibonacci number.

The nautilus shell, the golden mean

What makes the nautilus shell so special for mathematicians? Having the Golden Mean. But how do we know that the nautilus shell has the Golden Mean?

First of all, we will start with drawing a small, one unit square. Then we will draw another square which is larger than the previous one. We need to add in a counterclockwise direction. The length of each square has a value from the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 … Then we can draw spirals, starting with the smallest one, outward through the largest one. Then the Golden Mean will appear.

Flowers, plants, or objects have no idea about mathematics, yet they manifest the best of mathematical patterns. This marvelous mathematical art has been placed in their nature for us not to be fascinated only but also to explore the mysteries behind it.

Note

Liberating Theology from Alphabetic Writing’s Reductionistic Reign

Louima Cunningham — Thu, 01 Mar 2012 00:00:00 +0000

The words on this page are squiggly black ink pressed onto white paper. You could run your fingertips across these letters and feel their slick texture, but the ink you touch as you read is separate from the keys I click as I write. It would be impossible to tell – from these letters alone – whether or not their author is a withered war hero, a teenage suburbanite, or a patient in a psychiatric ward. The words are physically divorced from my Euro-American, male, twenty-something body. Alphabetic writing removes bodies from communication. But does alphabetic writing’s disembodied nature really matter, and – what is more – should it unnerve theologians?

Multiple factors shape thought, but communication is especially heavy-handed. Communication not only restructures the contents of the mind,; it re-configures the brain’s very neurological patterns (Rotman 2008: 51-52). Brian Rotman explains four ways people communicate selfhood: self-pointing, a spoken “I,” an alphabetically written “I,” and “self-enunciation within contemporary media” (Rotman 2008: xxxiii). The written “I” is the only disembodied method, but this separation subliminally trickles beneath conscious awareness. Writing has been a major component of communicating theological concepts because writing has been the easiest form of mass communication for centuries. Theologians 1,000 years ago probably did not ponder the disembodiment of writing any more than landing spaceships on the moon. Television – which mediates communication through bodies – was not on their radar. But contemporary theologians have options. Gunther Kress is clear that, “The former constellation of medium of book and mode of writing is giving way… to the new constellation of medium of screen and mode of image” (Kress 2003: 9). These moving constellations have forced a fundamental shift in the way social brains engage thought. Theology that fails to take this into account will find itself hopelessly outmoded and no longer equipped to engage the study of God in fresh situations.

This article addresses the cognitive and neurological effects stemming from shifts in “culturally mediated systems of external memory” (Rotman 52) – such as writing and images – and how theological communication could expand in response. Writing reinforces left-brain, systematic, and linear thought (Armstrong 1995: 115). Images, on the other hand, encourage more right-brain, intuitive, and spatial reasoning. It would be simplistic to hold that writing is only mediated through the right hemisphere and images only through the left, but communicative modes create lateral biases. Theological minds emerge, – not only from the “linear, logical, detail-oriented left hemisphere”, – but from embodied, whole brains. The first section of this paper shows writing’s left-brain bias. The second section turns a corner by examining emotion as altered body states and how this leads to a right-brain bias. Visual mediums are the most direct way to communicate right-brain thoughts, so the last step is to show how visual mediums could add a holistic, embodied element to theology.

WRITING AND A DISEMBODIED, LEFT-BRAIN BIAS

Writing forces neurons to march in line with time (Kress 2003: 2). The left hemisphere attributes meaning to words and strings those words together to form sentences, paragraphs, and full discourses (Hogue 2003: 87). In a sentence, one… word… follows… another, so the brain follows a preset “reading path” (Kress 2003: 3). A dog bit Karen means something entirely different than Karen bit a dog. Reading paths necessitate choices about which words make the most sense in which temporal order (Kress 2003: 3), so writing creates “the subliminal appeal of reducing wholes to constituent parts” (Jahandarie 1999: 57-58). It entrenches linear (temporal) neurological patterns through the repetitive, reductive decision-making (Jahandarie 1999: 54).

This leads to what could be called: alphabetic reasoning. Daniel Siegel explains that the left hemisphere is responsible for “more slowly acting, linear, sequentially active, temporal (time-dependent) processes. Verbal meanings of words… are a primary mode of processing for the left side” (Siegel 2001: 179). The left-brain deals in “monosemantic ‘packets’… which are then processed in a slower, linear mode” (Siegel 179). Writing strengthens linear thought patterns in the left hemisphere. Siegel holds that, “Repeated activation of specific neuronal pathways reinforces the strength of connections between groups of neurons” (p. 194), so neurons fire in similar patterns in the future (p. 24). The brain functions like a malleable muscle, but writing only exercises the left-brain. The brain is sculpted like a bodybuilder who only lifts with one arm. That arm ripples, but the other atrophies. Plastic makes perfect.

The “linear, logical,” and “linguistically-based” left-hemisphere (p. 161) evaluates the contents of written words through syllogistic reasoning (p. 197), prejudicing the brain to evaluate their contents through the brain’s semantic operator. The semantic operator deals with “propositional representations – symbols of external or internal facts that can be… assessed as ‘true’ or ‘false’” (p. 35 [emphasis mine]). In alphabetic reasoning, ideas are filtered into categories of: right or wrong, – fact or fiction. Then, alphabetic reasoning reinforces the brain’s quest for causation through sequencing. Reading paths communicate meaning through cause-effect relationships, so the ensuing left-brain thought patterns in the semantic operator create a syllogistic lens. Wholes are eliminated in the effort to snap details into linear, cause-effect relationships. Meaning is confined to these necessary but limited ways of processing information.

THEOLOGY AND LEFT-BRAIN BIAS

As writing has become increasingly more accessible in punctuated bursts over the past few millennia, theologians have become more and more linear and syllogistic in thought. But neurons fire in strikingly complex and highly plastic patterns (Bennett and Hacker 2008: 137). Linear rules for systematic theology are only engrained through repetition. The dark side of alphabetic reasoning is that it can produce theologians who tend to exercise linearity while alternative methods atrophy. Recall that the semantic operator can only qualify concepts as right or wrong. Linear patterns do not leave cognitive space for multiple details to be processed simultaneously. In the study of God, this cognitive capacity is absolutely necessary.

So far, I have examined several important consequences of alphabetic writing: it is inherently linear; in committing to using particular words in particular sequences it automatically eliminates other options; it coerces thought into cause-effect analysis, and it externalizes, or disembodies, thought. Of these consequences, the disembodied nature of theological writing is the most problematic.

EMOTION AS ALTERED BODY STATES

Emotions are physiological changes in a person’s body. These are mediated through combinations of gestures, postures, and behaviors (Damasio 2003: 63). External images – or stimuli – are mediated to the brain through the senses, and internal body and brain states refract the external world (Damasio 195). Marshall McLuhan writes, “The use of any medium or extension of hu/man/ity alters the patterns of interdependence among people, as it alters the ratios among our senses” (Jahandarie 51). The higher cognitive processes involved in developing theological concepts are pushed and pulled by the body’s constant emoting (Siegel 143).

The signals passing in and out of bodies generate emotion, so change signals can make emotions contagious (Siegel 143, 50). Siegel explains that, “Complex neural/bodily aspects of emotional processes are not easily translated into words. Nonverbal expressions, including those of the face, tone of voice and gestures, can transfer information about internal states more fully to the outside world than words can do” (Siegel 150). When it comes to altering the emotions correlated with belief – belief that can override higher cognitive functions – bodily gesture trumps writing.

Because this emotional processing is primarily non-conscious, people do not qualify intuition as true or false by sequencing details into rational arguments. For example, anger is typified by “dilated pupils, widened orbital area, raised eyebrows, furrowed brow, and pursed lips” (Siegel 128). The brain absorbs all of these signals at once when someone unexpectedly sees this:

Through non-conscious processing, faces can change emotions and minds in a flash.

Intuition of other people’s emotions occurs within the Mirror Neuron System (MNS). This system draws most heavily from facial expressions (Siegel 290), and it tells the viewer’s brain to respond to the viewed face in kind (Siegel 129). The MNS non-consciously indicates: whatever is happening in that person is also happening in me. This is why people constantly imitate the “motor acts, postures, and gaze of their co-citizens” (Kanwisher and Duncan 2004: 463-64). When people talk, their body language loops without their awareness as emotions harmonize. Yawning becomes contagious. When the MNS causes altered body states and facial expressions, people influence each other’s emotional-cognitive looping, and thought itself is steered.

There is scientific evidence that the entire body is engaged in understanding and remembering. James McGaugh demonstrated that the Vagus nerve – which connects the brain stem to the stomach – directly unites visceral and emotional experience with mental processing. McGaugh presented subjects with two stories containing similar amounts of data. One story had high emotional content and the other had low emotional content. People with functioning Vagus nerves recalled more data from the stories with high emotional content than from the stories with low emotional content. People with damaged Vagus nerves had identical recall from both stories. This suggests that certain types of thought are crippled by the absence of full-bodied emotion. Siegel explains that, “conceptual representations are nonverbal. They form the fundamental building blocks of our thoughts, beliefs, intentions, and aspects of our explicit memories” (Damasio 167). Written theological method is often left wanting embodied communication to close the cognitive loop.

EMBODIED COMMUNICATION AND RIGHT-BRAIN BIAS

The body’s indicators of emotion – somatic markers¬ – interface with the right hemisphere more than the left (Damasio 144, 82). The right-brain simultaneously correlates multiple details rather than sequencing them in cause-effect relationships. Siegel describes how the right-brain deals with nonverbal communications through “fast-acting, parallel (simultaneously active), holistic processes” (Damasio 179). This means that the right-brain is the default processor of metaphor and paradox. Imagistic representations in the mind do not force choice, categorization, or analysis. They exist simultaneously without being truncated by “top-down” processing as emotions ebb and flow (Siegel 165).

Kress calls this type of right-brain embodied thought “spatial reasoning,” and he places it in opposition to the “temporal reasoning” sublimated by alphabetic thought (Kress 45). Spatial reasoning portrays multiple details simultaneously, so spatial reasoning is correlational rather than sequential. This opens up possibilities for theologians by beginning with building blocks of thought before they are sequenced into categories of justified and true, fact or fiction.

The right-brain is functionally nonverbal, so its contents have to be externalized in “non-word-based ways, such as drawing a picture or pointing to a pictorial set of options” (Siegel 327). Alternative mediums, such as video and visual art, could generate entirely new forms of theological thought.

MEDIA’S ROLE IN SPATIAL REASONING

Video and visual art are viable possibilities for developing a right-brain, spatial reasoning into theological method. Jahandarie poetically explains, “Electric media are extensions of our central nervous systems and thus put us in touch with the totality of experience” (Jahandarie 99). Jahandarie overstates his case; – for example, video does not communicate through odors, for example, – but his intended meaning is helpful. New technologies make it possible to create right-brain mass communication. The MNS is 80-85% as effective at influencing emotion through video as it is in person (Kanwisher and Duncan 469), so gestures captured on video have a startlingly high ability to steer the emotional side of the cognitive loop. Emotions can be altered and details can be expressed simultaneously if theologians were to use a wider set of cognitive and communicative tools.

Rotman explains that people listen, “not to speech sounds as such… but to the movements of the body causing them; we focus on what happens between the sounds, to the dynamics of their preparatory phases, pauses, holds, accelerations, fallings away, and completions” (Rotman 23). Through the MNS, we get hunches, – not only about what another person is thinking, – but how the person is thinking and feeling. The dynamics of gesture in film add an intuitive, interpretive element to its communicative ability. In split-brain studies, the left-brain cannot even register facial expressions or emotional states (Rotman 151). While the left-brain might be incredible at reading and writing books, it is inept at spatial reasoning. Siegel writes that the left-brain’s “monosemantic ‘packets’” operate through neurological patterns and processes “quite distinct from the analogic representations seen, for example, in an artist’s painting or in a photograph,” so “the right hemisphere more fully ‘sees the world for what it is,’ whereas the left hemisphere must reduce the world much more into mentally defined, often socially constructed chunks of information” (Rotman 179). The right-brain’s holism and simultaneity could be invaluable to theologians, but it is too often dampened by the limitations of past technologies.

WHEN THEOLOGY IS EMBODIED

Siegel mentions the “theory of nonlinear dynamics of complex systems” in regard to personhood (Siegel 7). The firing of every synapse in every thought exponentially complexifies what it means to be human. Perhaps God is not only linear, either. Images are not filtered through the linear critiques applied to writing because they function differently in the brain. This does not make imagistic, spatial reasoning irrational. It makes spatial reasoning non-rational, and theologians desperately need non-rational ways to discuss a nonlinear God. Spatial mediums are not able to comprehensively encapsulate God’s personhood any more than writing, but they could generate new perspectives to better triangulate theological depth in conjunction with it.

In the syllogistic line that runs from rational to irrational, the concept of a trinity is inherently problematic. The semantic operator only deals with logical, sequential coherence, so the multiple, simultaneously existing details involved in conceptualizing the trinity do not get cognitive space. In this case, it might be advantageous to take a right-brain, imagistic approach to theology.

Depictions allow for multiple details to coexist without sequence. Images can be worth far more than 1,000 words. Imagistic reasoning allows space for entire persons to subject themselves to concepts. The “events and actions” lost in writing’s “principles and concepts” could be filled out through images (Jahandarie 17). Instead of only understanding theology in terms of right and wrong, the theologian could also trust a Vagus nerve when it communicates that an image or gesture is good or bad. The details could remain simultaneous in the right-brain without being sorted, and persons could experience theologies to a fuller degree in addition to critically evaluating them.

CONCLUSION

As neurological patterns become more spatial and less linear, theologians have the opportunity to construct new theological paradigms by diversifying their communicative tools. Creating theology through imagistic mediums would be bounded by its own “epistemological commitments,” (Kress 57) though. Critical thought is a crucial element in theological conversations, for example. This suggests that – when used in isolation – either alphabetic or imagistic methods are reductionistic. Theology has to be alphabetic, linear, and critical to be constructive, but it also needs to be imagistic, embodied, and holistic to avoid reductionistic sequencing of nonlinear concepts. The multiple natures of cognition cannot be mixed or separated. Language and experience, left-brain and right-brain thinking, knowledge and belief, are locked into cognitive loops (Eichenbaum and Bodkin 179), and it is impossible to have a one-sided loop.

The advent of new mediums makes the creation of an embodied theological method – complete with shifting neurological patterns – a mandatory opportunity. The next step is to synthesize multimodal theologies with alphabetic reasoning. A critically holistic methodology would emerge as gestalt. Full-blown theological paradigms could materialize if both hemispheres were used in conjunction. However progressive, however gestalt, and however critically holistic theological method becomes, it can never be comprehensive or complete. Theologians must do theology at all times. When necessary, they should use words.

Jonathan Camery-Hoggatt is a freelance writer in Seattle. He has MA in divinity from Princeton Theological Seminary.

BIBLIOGRAPHY

Armstrong, David F., William C. Stokoe, and Sherman Wilcox. Gesture and the Nature of Language. Cambridge ; New York, NY, USA: Cambridge University Press, 1995.
Barth, Karl, Geoffrey William Bromiley, and Thomas F. Torrance. “Barth’s Church Dogmatics (14 Volumes).” In Logos Bible Software series 3. [Bellingham, WA]: Logos Bible Software ; Libronix Corp., 2008.
Bennett, M. R., and P. M. S. Hacker. History of Cognitive Neuroscience. Chichester, U.K. ; Malden, MA: Wiley-Blackwell, 2008.
Bradt, Kevin M. Story as a Way of Knowing. Kansas City, MO: Sheed & Ward, 1997.
Bultmann, Rudolf Karl. Theology of the New Testament. 2 vols. New York,: Scribner, 1951.
Camery-Hoggatt, Jerry. Reading the Good Book Well : A Guide to Biblical Interpretation. Nashville: Abingdon Press, 2007.
Cone, James H. God of the Oppressed. Rev. ed. Maryknoll, N.Y.: Orbis Books, 1997.
Damasio, Antonio R. Looking for Spinoza : Joy, Sorrow, and the Feeling Brain. 1st ed. Orlando, Fla.: Harcourt, 2003.
———. The Feeling of What Happens : Body and Emotion in the Making of Consciousness. 1st ed. New York: Harcourt Brace, 1999.
———. Thinking About Belief. Edited by Daniel L. Schacter and Elaine Scarry, Memory, Brain, and Belief. Cambridge, Mass.: Harvard University Press, 2000.
Deacon, Terrence William. The Symbolic Species : The Co-Evolution of Language and the Brain. 1st ed. New York: W.W. Norton, 1997.
Eichenbaum, Howard, and J. Alexander Bodkin. Belief and Knowledge as Distinct Forms of Memory. Edited by Daniel L. Schacter and Elaine Scarry, Memory, Brain, and Belief. Cambridge, Mass.: Harvard University Press, 2000.
Ekman, Paul, and Erika L. Rosenberg. What the Face Reveals : Basic and Applied Studies of Spontaneous Expression Using the Facial Action Coding System (Facs). 2nd ed, Series in Affective Science. Oxford ; New York: Oxford University Press, 2005.
Hodgson, Peter Crafts. Liberal Theology : A Radical Vision. Minneapolis: Fortress Press, 2007.
Hodgson, Peter Crafts, and Robert Harlen King. Christian Theology : An Introduction to Its Traditions and Tasks. Newly updated ed. Minneapolis: Fortress Press, 1994.
Hogue, David. Remembering the Future, Imagining the Past : Story, Ritual, and the Human Brain. Cleveland: Pilgrim Press, 2003.
Jahandarie, Khosrow. Spoken and Written Discourse : A Multi-Disciplinary Perspective, Contemporary Studies in International Political Communication. Stamford, Conn.: Ablex Pub., 1999.
Kanwisher, Nancy, and John Duncan. Functional Neuroimaging of Visual Cognition. Oxford ; New York: Oxford University Press, 2004.
Kress, Gunther R. Literacy in the New Media Age, Literacies. London: Routledge, 2003.
Minear, Paul Sevier. Images of the Church in the New Testament, New Testament Library. Louisville, Ky.: Westminster John Knox Press, 2004.
Newberg, Andrew B., Eugene G. D’Aquili, Vince Rause, and Judith Cummings. Why God Won’t Go Away : Brain Science and the Biology of Belief. 1st ed. New York: Ballantine Books, 2001.
Niebuhr, H. Richard. The Meaning of Revelation. New York,: The Macmillan Company, 1946.
Rotman, B. Becoming Beside Ourselves : The Alphabet, Ghosts, and Distributed Human Being. Durham: Duke University Press, 2008.
Siegel, Daniel J. The Developing Mind : How Relationships and the Brain Interact to Shape Who We Are. New York London: Guilford Press, 2001.
Stenning, Keith. Seeing Reason : Image and Language in Learning to Think, Oxford Cognitive Science Series. Oxford ; New York: Oxford University Press, 2002.

A Pristine Spring

Louima Cunningham — Tue, 01 Mar 2011 00:00:00 +0000

A skyline of lively hills and purple mountains appeared

As sleepers floated in their deepest slumbers

In purple and pink, successive dawns were swelling

While in waters swans were elegantly gliding

We felt the shivering breaths of a new spring.

A cerulean serenity had enveloped our dreams

And the sky winked with hope from a distance

In a moment, we were left without defenses

At the point where providence meets resolve

God made our dreams bloom.

The overflowing fountain has come to life again

Breezes high and low promising a bright future

Everywhere embraced by a blessed joy

And the happiest days of life I longed for

Were being lived in the deepest sense, once again.

A pristine spring, with its fresh verdant hues

Like the patterns and colors of the past

The people found the touchstone they had lost

Making up for the great loss they sought

Purple rosebuds on heavenly hills.

With the breeze blowing, now grasses atone

In the lands, once struck lifeless by autumn, are new roses

As if the Trumpet has blown, the dead are resurrected

Along with them in this azure dream

Grieving hearts attain reunion.

The Influence of Islamic Art on M.C. Escher

Louima Cunningham — Thu, 01 Jul 2010 00:00:00 +0000

Although most of us do not remember or have not ever heard the name M.C. Escher, we are probably familiar with the world-famous illustration shown below:

Figure 1: Drawing hands, 1948

Maurits Cornelis Escher (1898–1972) is one of the world’s most famous graphic artists of impossible structures (e.g. “Ascending and Descending”) and transformation prints (e.g. “Metamorphosis I-II-III”). Currently, one can see his work on posters, book covers, calendars, wall hangings, and many web sites enjoyed by millions of people all over the world (1).

His exquisite and mind boggling pictures are drawn from the mathematical world of symmetry, topology, transformational geometry, and regular divisions of the plane. At the same time, they exhibit a rich and artistic talent unrivaled by most. Furthermore, respected scientists have realized that his works are simple illustrations of sophisticated theories (2). For instance, mathematician D.J. Lewis indicates that Escher’s prints entail a systematic approach combined with an ingenious argument similar to the most beautiful results in algebra. In 1952, Herman Weyl, a Princeton mathematician, used Escher’s famous work “Symmetry” for his book cover. Escher’s rendering of “Horseman” was used by Chen Ning Yang, a physicist and Nobel Prize winner, to illustrate his new hypothesis involving symmetry and its application to quantum physics (3). Escher has also inspired scientists in their academic studies. For example, some of his sketches helped his half-brother B.G. Escher, a professor of geology, in solving crystallography problems (4).

Tessellation of a plane, also called tiling, is the mosaic formed by filling the plane with no gaps and no overlaps. A person who is familiar with Islamic art immediately notices the deep connection between Escher’s transformational geometry and tessellations, and that of Islamic patterns. One can even use Islamic art and tessellation techniques to generate Escher-like drawings. In fact, Escher’s 1922 visit to the Alhambra Palace in Spain was the turning point in his life. He was fascinated and inspired by the spiritual significance of the tile work at the palace, and Islamic patterns played a key role in transforming his art (5). This article will explore the intimate relationship between Islamic art and Escher’s work, in particular the significance of themes with “flat surfaces” and “flat surfaces with respect to pictorial representations.”

Brief summary of Escher’s art

Escher produced 448 lithographs, woodcuts, and wood engravings and over 2000 drawings and sketches during his lifetime (1). His understanding of mathematics was largely visual and intuitive, and his works display a strong mathematical component (6). More than 150 of colorful works testify to his ingenuity in regular division of plane. He was very successful at depicting the real world in 2-dimensional plane as well as at translating the principles of regular division onto a number of 3-dimensional objects such as spheres, columns, and cubes. Some of his prints combine both 2 and 3-dimensional images with a startling effect as demonstrated in “Reptiles” (7).

Figure 2: Reptiles, 1943
Upon further examination, one finds three dominating themes in Escher’s works (2):

• Spatial structures: His work before 1937 aims solely to depict realistic structures or scenes composed of mostly landscapes and portraits, and reveals no analytical interest. In contrast, after 1937, he combines these themes with the others described below.

Figure 3: Atrani, Coast of Amalfi, 1931

• Flat surfaces: The subject matter of later works encompasses his major area of expertise: “regular division of plane” including regular tessellations; symmetry and order; identical, congruent figures; or those with graduated surface dimensions. Upon his visit to the Alhambra in 1922, Escher was deeply influenced by the art works of the Moors and worked out a system for periodic drawings. These periodic drawings portray surfaces filled with similar shapes and often illustrate approaches to the infinite. Escher mastered his skills on geometric grids and used them as the basis for his sketches, later improving them with additional designs, mainly animals such as birds, lions, and reptiles.

Figure 4: Day and Night, 1938
• Flat surfaces with respect to pictorial representation: Escher’s final and most famous type of work is his portrayal of “impossible structures.” He was very skilled at illustrating three-dimensional conflicting situations in two-dimensional spatial representations (8).

Figure 5: Waterfall, 1961

Patterns in Islamic art
In Islamic art, the spiritual world is regarded as being reflected in nature through geometry and rhythm. Hence, Islamic artists used geometry as an aid to raise their spiritual understanding as well as the viewer’s:
“Muslim intellectuals recognized in geometry the unifying intermediary between the material and the spiritual world. These patterns may be seen as symbolizing the Islamic principles of ‘Tawhid’ (the unity of all things) and ‘Mizan’ (order and balance), which are the laws of creation in Islam.”(9)
Tessellations are one of the major components of Islamic art. Islamic artists mastered regular division of plane using, in particular, circles on triangular or square grids, because the circle – which has no beginning and no end and thus symbolizes infinity – was considered to be the most perfect geometric form. In mosques, where a wealth of these geometric patterns could be found, one could contemplate the infinite nature of God simply by looking at the walls or ceiling. In short, these geometric forms expressed Islamic artists’ fascination with mathematics as a metaphor for divine order and presence (10). Figure 6, Figure 7, Figure 8, and Figure 9 are examples of triangle and square grids and produced patterns adopted from (11):

Figure 6: Triangular grid (a) and examples of patterns: (b) 6-pointed star-hexagon (c) Ceramic wall panel – Iran – 13-14th centuries.
Figure 6 demonstrates the 6-pointed star-hexagon pattern that can be obtained by coloring a triangular grid whereas Figure 7 integrates circles to produce more complicated patterns and an increase in variety. The examples shown in the figures are real tiles mounted in mosques around the world.

Figure 7: Triangular grid and examples of patterns.
Figure 8 and Figure 9 illustrate the usage of circles on a square grid in two different ways. The square-hexagon pattern in Figure 8(c) is commonly used on the ceilings of mosques whereas variations of the star-cross pattern in Figure 8(c) have mostly been used on walls.

Figure 8: Square grid and examples of patterns: (a), (b) and (c) square-octagon pattern, (d), (e) and (f) star-cross pattern.

Figure 9(c) shows a scallop pattern, often used for fences. The I-Bar pattern in Figure 9(e) is sometimes used in tiling walls, but is more commonly used for floors, pavements, and paths.

Figure 9: Square grid (a) and examples of patterns: (b) and (c) scallop pattern, (d) and (e) I-bar pattern.

The Alhambra’s influence on Escher

Figure 10: (a) A view of the Alhambra, (b) the Lion’s Court in the Alhambra, which inspired Escher’s drawing.

Escher became fascinated by the regular division of the plane in 1922 when he first visited the Alhambra, a fourteenth-century Moorish castle in Granada, Spain. (1). He then studied Polya’s seventeen plane symmetry groups, (thirteen of which are displayed in the Alhambra), and Haag’s mathematical definition of the division of the regular plane (12). But the real metamorphosis in his art began in 1936, with his second visit to Alhambra, which he described as “the richest source of inspiration” in his writings (5). Like many Islamic artists, Escher believed that repetitive patterns indicated a higher source of knowledge that existed before mankind. He considered order, regularity, cyclical repetitions, and renewals to be the “laws of the phenomena” around us; accordingly, the structure of his designs was a simple reflection of these laws from his own perspective (9).

Escher studied, took detailed notes, and made sketches of the tile patterns at the Alhambra. In his writings, he described his fascination with the double use of contours and divisions of the plane as follows:

“The Moors were masters in the filling of surface with congruent figures and left no gaps. In the Alhambra, in Spain, especially, they decorated the walls by placing congruent multicolored pieces of majolica together without interstices.”(8)

Figure 11: Examples of patterns in the Alhambra (adapted from http://www2.spsu.edu/math/tile/grammar/moor.htm)

In his later work, Escher used genuine techniques devised from triangular and square grids, applying reflections, translations, and rotations to obtain great variety of patterns in his tessellations. The simple trick of modifying the grids utilized in Islamic art to ensure the perfect fit of patterns, which Escher used in his tessellations, is demonstrated below:

Figure 12: An example of an Escher-like pattern obtained from the rectangle (adopted from (11).

The idea is to start with a pattern such as a rectangle in Figure 12(a). The tessellation pattern is created by cutting portions of the pattern as in Figure 12 (b) and (d), and mounting them to the correct locations of the pattern considering the rotations and reflections as in Figure 12 (c) and (e). Finally, the pattern is rendered in tile as illustrated in Figure 12(f). One can imagine how easy this novel technique was to apply, yet how complicated it was to discover.

Figure 13: Escher’s symmetry drawings produced from (a) diamond, (b) rectangular, and (c) I-bar patterns. The grids are placed on the original images to reveal the relationships.

Figure 13 presents examples of Escher’s symmetry drawings. Figure 13(a) utilizes a diamond pattern by converting it to a man applying a rotation of 120. whereas Figure 13(b) was created from a rectangle with a rotation of 180. In Figure 13(c), Escher was able to take the I-Bar pattern and adapt it to the totally dissimilar motifs of angels and devils.

In many of Escher’s tessellations, not only the patterns, but also the entire scene is inspired by the circle and eternity as in Islamic art. The “Circle Limit” series is a good example of works that use cyclical tiling with shrinking patterns from the center to the border in a circle. Although a circle is depicted, the patterns theoretically reach an infinitive number of repetitions on the border. In his “Metamorphosis II,” it is also interesting to see that a closed cycle is formed when the two vertical ends of the picture are joined together. Similarly, in his famous woodcut ‘Day and Night,’ which shows black and white birds flying in opposite directions, not only do the birds and landscape complete cycles, but also the print is symmetric with respect to a vertical line (4).

Figure 14: Convex and Concave, 1955
Another famous work by Escher, “Convex and Concave,” exemplifies his “impossible reality” works. The essential idea that governs the entire lithograph is the Islamic tumbling baby block pattern, which appears in the flag on the upper right side. Although the picture, as a whole, seems to portray a normal scene, it actually consists of multi-purpose planar surfaces. Depending on the location upon which the eye focuses, the same planes may serve as ceilings, walls, or floors. Escher’s “Waterfall,” “Ascending and Descending,” “Relativity,” and “House of Stairs” all present similar mind-boggling characteristics (2).

In conclusion, Escher is a world-famous graphic artist, well-known for his impossible structures and transformation prints. He is one of the unique figures appreciated for his ability to apply his mathematical talent in artistic creation. He was strongly influenced by the Islamic patterns in the Alhambra – a fourteenth century palace in Spain. He developed his extraordinary style and mastered his skills after exploring the tessellation techniques Islamic artists used to create the figures in the Alhambra.

Fatih Gelgi has a PhD in computer science. He is currently the computer coordinator of Accord AMSP team in Los Angeles.

Bibliography

1. M.C. Escher, the Official Website. [Online] [Cited: January 10, 2009.] http://www.mcescher.com/.
2. Desoe, Carol D. Marthematics: The Blending of Mathematics and the Art of M.C. Escher. [Online] [Cited: December 10, 2008.] http://caroldesoe.com/IslamicArt/mARThematics.pdf.
3. Broos, C.H.A. Escher: Science and Fiction. In: J.L. Locher. The World of M.C. Escher. New York : Abradale Press, 1988.
4. Locher, G.W. The Work of M.C. Escher. In: J.L. Locher. The World of M.C. Escher. New York : Abradale Press, 1988.
5. Abbas, Jan S. Islamic Patterns: The Spark in Escher’s Genius. In: Doris Schattschneider, Maurits Cornelis Escher and Michele Emmer. M.C. Escher’s Legacy. New York : Springer, 2005.
6. M.C. Escher. Wikipedia. [Online] [Cited: January 10, 2009.] http://en.wikipedia.org/wiki/M.C._Escher.
7. O’Connor, J.J. and Robertson, E.F. Maurits Cornelius Escher. School of Mathematics and Statistics. [Online] May 2000. [Cited: December 21, 2008.] http://www-history.mcs.st-andrews.ac.uk/Biographies/Escher.html.
8. Ernst, Bruno. The Magic Mirror of M.C. Escher. New York : Barns & Noble Inc., 1994.
9. Islamic Patterns and M.C. Escher’s Tessellations. North Texas Institute for Educators on the Visual Arts. [Online] [Cited: December 6, 2008.] http://www.art.unt.edu/ntieva/pages/about/newsletters/vol_14/no_1/.
10. Melikian-Chirvani, A. S. Treasure of Islam. New Jersey : Wellfleet Press, 1985.
11. Islamic Art through the Eyes of M. C. Escher. Desoe, Carol D. Salt Lake City : NCTM Annual Meeting, 2008.
12. Schattschneider, Doris. Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M.C. Escher. New York : W.H. Freeman and Company, 1990.
13. Graber, Oleg. Arts of Islamic Peoples. Encyclopedia Britannica. 1974, Vol. 9.

Seeing the Third Quality of Light Polarization Vision

Louima Cunningham — Thu, 01 May 2008 00:00:00 +0000

Sun is the main source of light for Earth. Without light, there would be no sight for us. Light is nothing but an electromagnetic wave which has three fundamental properties and with naked eyes humans are capable of sensing only two qualities of light – brightness (intensity) and color (frequency). We are essentially blind to the third quality of light. The third property is polarization. The direction in which the electric field oscillates as it propagates is known as polarization. Although unable to naturally sense polarization, we have still been able to measure and analyze polarization in our environment. Polarizing filters are used in photography, certain kinds of sunglasses, digital watches, and laptop screens. Polarization is also used in the entertainment industry to produce and show 3-D movies. We wear polarized sunglasses, for example while fishing, to filter out the glare from polarized light that is reflected off the water’s surface. This makes the water more transparent and thus we can more easily see fish swimming in the water. The knowledge we have accumulated over the years leaded us to use polarization in our daily lives and scientifically our understanding of polarization information is still limited. On the other hand, the ability to analyze polarized light is widespread among animals. Here we will explore how animals make use of polarization information available in light.

Solar radiation is unpolarized before entering the earth’s atmosphere. Unpolarized light is a mixture of photons having randomly oriented electric fields. According to the simplest theory (Rayleigh), when unpolarized sunlight scatters from atmospheric constituents (gases, aerosol particles, water droplets, ice crystals), it becomes partially polarized, depending on the scattering angle – the angle between the incoming (direct solar) and outgoing (skylight) rays. Unpolarized light can also undergo polarization by reflection off of nonmetallic surfaces such as asphalt roadways, soils, racks, snow fields and water. Therefore, there is an abundance of polarized light in natural environments in various forms. Recently, it has become apparent that animals can take advantage of these rich sources of information in the underwater world, on the water surface, and in the terrestrial habitat that are of celestial polarization patterns. They utilize this polarized light prevailing in their visual worlds in various ways associated with their behavioral tasks like navigation, communication, mate recognition, eggs laying, detection of water surfaces, enhancement of visual power (similar to colors), or perhaps even camouflage.

The skylight compass

The best understood use of polarization is the skylight compass of insects. The orientation of the electric field changes with the position of the sun. This can make the sun as a compass usable even when the sun is obscured. In 1949, Nobel laureate Karl von Frisch discovered that when the sun is not visible, honey bees can orient their flights and communication dances by means of the extensive patterns of polarized ultraviolet (UV) skylight¹. For clear sky, these patterns are quite regular and depend so strongly on the position of the sun. It is amazing to see how these little hard-working creatures come programmed to use them to calculate the sun’s location.

Since von Frisch’s pioneering work, several other researchers investigated polarization vision and found that the polarization pattern of the sky offers many other insect species (desert ants, dung beetles, field crickets, and house flies) a reference for visual compass orientation². For example, desert ants were shown to make long and tortuous foraging walks, but use the sky polarization pattern to return to their nest on a straight line³. They are able to continuously compute their present location from their past trajectory and, as a consequence, to return to the starting point by choosing the direct route rather than retracing its outbound trajectory.Moreover, interestingly enough, researchers discovered that one species of dung beetles navigate by using million-time dimmer polarization patterns of moonlight. Dung beetles use it as an orientation guide to leave their food source in a straight line to avoid aggressive fights⁴. To find out how the beetles are able to use the polarized light of the moon to navigate, researchers observed the beetles under the night sky. On nights when the moon was visually clear, the beetles continued to forage and roll their dung balls in a straight line. On moonless or cloudy nights the beetles could not maintain a straight path.

Reflections from water

In nature, important reflections come from water where the polarization distinguishes between water and other reflective surfaces. Horizontally polarized UV light reflected from the surface of water is the main optical cue for habitat finding by insects living in, on, or near water. Weak UV light emitted by a horizontal surface below flying backswimmers can cause the animals to turn their flight paths vertically downward, bringing them to the horizontal surface⁶. Polarization sensitivity has, likewise, been demonstrated in crustaceans, like in the shore-living water flea Daphnia pulex. These animals were shown to swim toward polarized light, which in nature would lead them away from the shore towards deeper water⁷.

Human activity can have overwhelming effects on the natural environment and man-made objects, such as crude or waste oil surfaces, asphalt roads, glass surfaces, or plastic sheets used in agriculture are unfortunately more attractive to water-seeking polarotactic insects than the water surface itself. This effect can be very dangerous for polarotactic insects as these objects function as insect traps. Researchers have observed that every year, in May and June, swarms of mayflies mate, not above lakes and rivers, but above dry asphalt roads and lay their eggs in vain on dry asphalt roads or car-bodies. The horizontally polarized light from these surfaces mimics a highly polarized water surface.⁸.

Ocean animals

For many ocean animals, sensing polarization may be even more important than sensing color. One possible use for polarization in the ocean (and elsewhere) is signaling: communicating with neighbors, rivals, and potential partners. Recent discoveries have shown that stomatopods (Mantis shrimps), a sort of shrimp found on reefs around the world, use special body areas to communicate with polarized light (Fig.6)⁹. Polarized light can also be used to ‘break the camouflage’ of aquatic organisms because, although from most viewing angles they match the color of the water behind them, the nature of the polarization is quite different. Researchers have found that transparency of aquatic organism to avoid detection can be broken with the help of polarization sensitivity¹⁰. In their experiment, they observed that squid detect zooplankton prey under partially linearly polarized lighting 70% greater than those achieved under non-polarized illumination.

In summary, polarization is central to most of the animals’ lives. It is abundant in the nature in various forms. Here, we have given only couple of examples of ways of various animals’ exploitation of polarized-light information. It seems, as researches continue, that the already long list of animals utilizing polarized light will get even longer as we learn more about it. Yet, even these mentioned examples above are enough to help us realize how perfectly these small animals have been created, and how well they are taken care of in their daily lives when they navigate, communicate, recognize a mate, lay eggs, detect water surfaces, or perhaps even break camouflage. Here, it seems necessary to observe that “The tiny body of a fly is connected with most of the elements and causes in the universe; indeed, it is a summary of them. If it is not attributed to the Pre-Eternal and All-Powerful One, it is necessary for those material causes to be themselves present in the immediate vicinity of the fly; rather, for them all to enter into its tiny body; and even for them to enter each of the cells of its eyes, which are minute samples of its body.” We refer the interested reader to Said Nursi’s reputable article of “A Treatise on Nature”¹³ and conclude with his aphorism: “He who created the eye of the mosquito is the one who created the sun.”

Cross Cultural Communication: A Foreign Language Perspective

Louima Cunningham — Mon, 01 Jul 1996 00:00:00 +0000

To be aware of issues in cross-cultural communication is becoming increasingly important. Any of us may travel or meet travellers. We may work with members of other cultural groups. We may learn a foreign language. Such situations inevitably bring us into contact with other ways of speaking, other modes of behaviour and other views of life. In this article we examine how communication across cultures can be affected by participants’ interpretations, assumptions and expectations which largely derive from their own cultural background. We consider some ways in which cross-cultural communication functions at the various levels of words, grammar, pronunciation and at the less obvious levels of discourse patterns, sociolinguistic uses of language and levels involving cultural presuppositions. Finally, we draw some conclusions about the importance of attitudes in cross-cultural situations and of the need to raise awareness and understanding of other cultures. We take a foreign language perspective, asking what kinds of knowledge and understanding about cultures is necessary to learn another language and communicate with speakers from different countries. Cross- cultural communication often involves difficulties but fundamentally it should be viewed as an opportunity for learning and development. The examples that we give relate particularly to students learning English. They are taken from our field- notes and observations made in several countries.

Interpretations, assumptions and expectations

In situations of cross-cultural communication it is not only what happens or what is said that is important, it is how participants interpret the interaction which ultimately counts. It is this interpretation which guides our perception of meaning and our memory of other people. Most of us draw conclusions about others from what they say, or rather from what we think they mean. The gap between what we think others mean and what they intend to say can occur in any communication. This gap is often wider in cross-cultural contexts. This is evident when there is a lack of knowledge of the common language of communication, say English, which may he a second or foreign language to one or both sides. Less obviously the gap is often wider because in intercultural communication participants may not realize that they are using language in different ways which go beyond purely linguistic competence. Our consideration of cross-cultural communication needs to include: discourse competence in which conversations or texts may be structured using different principles; sociolinguistic competence in which language users may draw on differing ideas about who may speak to whom, on what sorts of topics, on what kinds of occasion, in what manner and for what purposes; cultural competence in which cultural norms and beliefs are used to interpret actions and language behaviour and to attribute values and interpretations to interaction. The problem is that our own perception of these aspects of language use is influenced by our own cultural background. It is all too easy to be unconsciously ethnocentric about such matters and to assume that our way is normal, logical or better than those ways used by speakers who come from other cultural backgrounds.

An analogy illustrates this point. In Figure 1 the middle item may be interpreted as a letter B or as a number 13, depending on whether it is read vertically or horizontally. Our interpretation depends on the context of what system we expect to use, in this example an alphabetical or numerical context.

In learning English, students need to be constantly alert for shifts in meaning as participants use varying systems and principles of interpretation. Objectively, the form and shape of the middle item in Figure 1 has not changed. Subjectively, the meaning can he completely different when the figure is seen in an alphabetical or numerical context. Different contexts lead to different expectations which in turn lead to different interpretations of the same object. Similarly, the context of our own culture may lead us to interpret another person’s words, behaviour or attitude quite differently from the way in which that person intends them to be interpreted. We may not be aware of the patterns of interpretation which members of a particular culture use.

In our own culture we can afford to take much communication for granted. Since childhood we have learned what word, normally mean, how and why things are said. Our own culture has provided us with a framework of working principles and systems of interpretation which most of us automatically use every day. We do not need to work out how to use greetings or apologies, how to respond to invitations or compliments, how to take turns or interrupt others, or what silences might mean. In learning to use a foreign language, however, we need to be aware that speakers of the target language may be using quite different assumptions and systems for such ways of using language. We need to become aware of alternatives. We need to expect the unexpected. We need to check our interpretations of what is apparently obvious.

Levels of communication

Language is like an iceberg: some aspects are visible with fairly obvious meaning, but a larger part is hidden or taken for granted.

The greater the foreign language skill in pronunciation, grammar or vocabulary, the greater the danger that the other hidden levels of communication may come into play. Participants on both sides will assume that they mean the same thing by different gestures or patterns of discourse, but in fact they often have quite different interpretations. Hearers often that if a speaker has a reasonable level of skill in the obvious areas of words, grammar and pronunciation then the speaker will he equally skilled in the other kinds of competencies. Often this is not at all the case, especially if the speaker’s foreign language learning has concentrated on language competence. Many learners of English have focussed their main attention on learning words and grammar. In many foreign language classrooms little attention is given to the role of culture and cross-cultural communication.

Words

For many students, learning a foreign language is all about learning words. The students’ aim seems to be largely to acquire a knowledge of a wide vocabulary, concentrating on new and difficult words. These students may not realize the importance of learning new meanings to known words, especially apparently simple words. However, simple words often turn out to have unexpected cultural meanings, as the following dialogue noted in Britain shows.

In this situation, A is an Arabic speaker, a visitor to Britain. She was only expecting a cup of tea and was puzzled by the offer of food. The British hostess (B) was upset that A had already eaten since she had, she thought, specifically invited A for food. The source of the misunderstanding is the word ‘tea’ which in Britain, especially among lower social classes, often means an early evening light meal. Although A speaks excellent English and is, in fact, an experienced university English teacher in her own country, she had not realized that a simple word like ‘tea’ can have different cultural meanings.

Similarly, ‘simple’ common words used in idioms can often catch out learners who are used to concentrating on ‘difficult’ words. The word ‘house’, for instance, takes on a variety of unexpected meanings in such examples as ‘The comedian really set the house on fire’ (the comedian got a good response from the audience, or ‘house’), The drinks are on the house tonight’ (the owner or manager will pay for all the customers’ drinks), ‘After the minister’s speech the House rose at nine’ (the members of the House of Commons, in the British Parliament, went home at nine o’clock).

Grammar

Grammar can often present unexpected difficulties in cross-cultural communication when learners of another language have not worked out the relationship between grammatical form and language function. This happened in the following example in Britain where a British person (B) wants to visit a Chinese student (C) in her room.

The problem ‘here is that the expression ‘do you mind it’ is a polite form of a request which anticipates a negative response, No, I don’t mind…’. C realizes that this is a request but responds only to the function, ‘Yes’ (meaning ‘Yes, do come in’). Since C has not responded with the expected negative grammatical form this leaves B to understand that she is busy (‘Yes, I do mind, I am busy at the moment’) . Fortunately, B did not leave immediately after C’s first response and the misunderstanding was cleared up.

Pronunciation

Clearly when words are mispronounced this can cause problems in cross-cultural communication. This usually happens when speakers have poor pronunciation or confuse words. Less obvious problems can crop up when fluent speakers of English, for instance, are influenced by local varieties of the language. This would be perfectly acceptable in local situations but can cause difficulties when English is used in international contexts. For example, Malaysian speakers of English may stress the second syllables of words like ‘colleague’ or ‘management’ where speakers of other varieties of English expect to hear the stress on the first syllable. Since the difference in stress is also accompanied by changes in the pronunciation of stressed or unstressed vowels (schwa) this can cause momentary confusion. More seriously, hearers’ perceptions of speakers of a language like English are influenced by the fact that stress and intonation commonly convey attitudes. Thus in English a heavy falling intonation can mean definiteness, abruptness or rudeness. Unfortunately, Arabic speakers who learn English have often not been taught this and they transfer Arabic falling intonation patterns to English. One result is that English hearers sometimes perceive the other group (wrongly) as being aggressive or pushy. A solution is to raise the learners’ awareness of the meanings of various intonation patterns in English and the attitudes which might be interpreted from their use.

Body Language

The same gestures or body language may express quite different meanings in different cultures. In Northern Europe yes’ is generally signalled by a downward head movement or up-and- down nodding. In contrast, in Turkey and neighbouring countries a common gesture for ‘no’ is an upward movement of the head, easily mistaken for the European ‘yes’ by those who are unfamiliar with Turkish people. Further scope for misunderstanding arises because the Turkish ‘no’ is often accompanied by a click of the tongue. This noise and the upward head movement means ‘you are stupid’ in Britain! There are cultural differences in the use of space, e.g. how close to others people expect to stand or sit. Many Latin Americans or people from the Middle East prefer to come quite close to their hearers when talking. This shows friendliness and solidarity. North Americans or Northern Europeans, on the other hand, tend to keep more space between themselves and hearers. This shows their awareness of the other person’s individuality and need for personal space.When speakers from the USA and Saudi Arabia, for example, come together they may feel uncomfortable without knowing the reason. Both parties unconsciously try to maintain their own natural polite and friendly distances, The American may feel the Arab is aggressive or pushy when the latter comes close, while the Arab may believe the American is unfriendly or untrustworthy if that person keeps moving away.

Further cross-cultural mismatches can occur in eye contact. Whether and how listeners look at a speaker’s eyes varies from culture to culture. One contrast seems to be that in Britain and the Middle East listeners gaze at a speaker’s eyes to show that they are listening and showing respect whereas in many parts of Africa and Asia this can signify disrespect or anger and be interpreted as insulting. On the other hand, the African or Asian manner of showing politeness, respect and honour to a speaker – by lowering one’s gaze or looking below the other’s eyes – can be interpreted as disinterest, suspicion or guilt by British or Middle Eastern listeners.

Even a smile can cause problems, as an American teacher in Taiwan discovered. One of her students arrived late. He was smiling. She became angry and said, ‘You are late and it’s not funny. Take that smile off your face. He then became very upset because she had publicly become angry with him. Later she realized that a smile is not always a sign of humour – the student was smiling with embarrassment. Such potential sources of difficulty are not likely to be pointed out by participants in cross- cultural situations. Openly drawing attention to misunderstandings may be thought impolite or over-direct unless the speakers are well known to each other.

Discourse Patterns

Speakers from different cultures make use of different discourse patterns in the way they structure information or interpret what others say. Even silence is used to structure discourse: participants know by the length of a pause that a speaker has finished speaking and they can take a turn. However, the exact timing of such turn taking can vary. Among many Greek speakers the pauses between turns are minimal; speakers alternate rapidly and overlaps between one speaker and the next are common and arc accepted as showing solidarity between speakers who understand each other. In contrast, in Scandinavia and Finland such pauses are often one or two seconds longer as members of those cultures show respect and perhaps think carefully about what they want to say. In cross-cultural situations between these two different groups it is very likely that English will be used as a common language of communication. Greek speakers report that they feel there are long silences between themselves and Scandinavians, which leads them to wonder if they have said something wrong or (given that Scandinavians are often highly competent in English) whether they have made a language mistake. As a result the Greeks feel rather insecure (unnecessarily). The Scandinavians meanwhile feel that the Greeks keep interrupting them. They feel (wrongly) that the Greeks are aggressive.

Important cultural differences can emerge when we consider where a speaker puts the main point. Chinese speakers frequently put the main point near the end of what they say. First they establish common ground and give relevant background information before they lead the hearer up to the main point. Sometimes this point only gets a brief mention – after all, it will be clear to the listener familiar with this discourse pattern where the argument is going. This kind of inductive discourse pattern seems to be oriented to the hearer. Many British and American speakers, in contrast, use a more deductive discourse pattern which is more oriented to the speaker in this second pattern the speaker usually gives an early indication of what is to come. Often the main point conies right at the beginning, especially if the speaker is answering a question in a formal situation. The idea seems to be to get to the heart of the matter quickly. Background information or supporting arguments follow. Since the hearer already has a good idea of the main point, it is clear how this background information will be relevant. Each of these contrasting discourse patterns is completely valid and can he taken for granted in its own cultural context. In cross-cultural situations the differences can cause problems. British people listening to Chinese speakers expect the main point to come quickly. Not hearing one, the British may become impatient or lose concentration and miss the point when it finally comes. Some British listeners report that they think the Chinese keep ‘beating around the bush’, they go round and round but don’t seem to get to the point. Chinese listeners expecting the background first often feel they do not get this information from British speakers so they sometimes miss the significance of the main point or do not see the logic behind it. It would help if both sides realized that for the British the background comes from the main point, while for the Chinese the background leads up to the main idea.

Sociolinguistic Uses

Sociolinguistic uses of language relate closely to discourse patterns, but there is greater emphasis on the social context and variation. For instance, to ask a person’s age, how much they earn or whether they are married is acceptable in all cultures, but in very different circumstances. To ask such questions of a stranger is normal in Turkey or China hut quite unexpected in Britain, America or Australia. Western tourists in Turkey or China may not appreciate the friendliness behind such questions. Instead they may think that local people are too curious about what they think are private matters or questions for job interviews. They would prefer to talk about the weather or their jobs (but these may not be such interesting topics in Turkey or China).

Part of the challenge in learning a foreign language is to learn how to manage [he sociolinguistic uses of the language. At a simple level this means understanding how greetings vary across the world. In China, ‘Have you eaten?’ is a greeting, not an indirect invitation to a meal. In Fiji or Malaysia, ‘Where are you going?’ is not always an enquiry about a person’s destination, but again is a greeting. In Botswana, a greeting is ‘How did you wake up?’. Each language also has many informal greetings. An Indonesian student in Britain (I) did not realize this when greeted by a British teacher (B) at a bus stop:

B Hello. How’s it going?

I I’m going home.

This left B puzzled. His greeting had not been returned and he wondered why the student mentioned he was going home. Later he realized the student was about to catch the bus.

The sociolinguistic uses of compliments can cause dilemmas about the nature of the expected response, as this dialogue between a Chinese speaker (C) and a British visitor (B) shows.

C Your Chinese is very good.

B Oh, thank you

This dialogue looks harmless until we consider what each speaker is thinking. C thinks B must be very boastful: B’s Chinese is not, in fact, good and C expected B to say, ‘No, no, it’s very bad’, since in Chinese a compliment should be rejected to show modesty. B is much happier: he has been complimented and he has thanked the Chinese speaker for his kind thoughts; he is not, in fact, immodest but is following the English rule that a response to a compliment often relates to the complimentary, not to the content.

Cultural Presuppositions

In many instances of cross-cultural communication it is important to understand the cultural presuppositions which lie behind speakers’ words and their expectations and interpretations. For instance, a Chinese student (C) asks a British person (B) for help.

C Can you help me?

B I would like to help you….but I’m afraid I can’t because….

When C heard the first words she was very happy, believing she would get help; when she heard the second phrase she was very disappointed. She thought, ‘Why did you raise my hopes and then let me down?’ She concluded that B was hypocritical. It would help if she understood the cultural presuppositions that B is using: first, to show good will and kindness by saying he would like to help, then moving to the main point that he cannot help before explaining why not. A Chinese speaker would probably give the reasons for not helping first before concluding that it was impossible: this would prepare the hearer for the bad news.

Many Chinese and Latin Americans respond to personal invitations by accepting to come, but when the day arrives they may not turn up. This has left many British and North American hosts puzzled, thinking: why did they promise to come, then break their promises? Can they he trusted? But this interpretation misses the Chinese or Latin American cultural presupposition behind their reply: it is better to show good will, by accepting and perhaps not go, than to refuse and bring immediate disappointment to the potential host. This shows regard for the hosts face, and for that of the person invited, who does not have to provide an excuse for refusing the invitation. Thus the Chinese and Latin Americans in this situation base their reply on social values, while the British and North Americans put truth values first. If this is understood, the situation becomes easier on both sides, although there will still be further variation depending on whether the invitation is by telephone, letter, or face to face, on whether it is a group invitation and how well the people know each other.

Fundamentally, relevant cultural presuppositions relate to how members of a culture view the world, how they think about human nature, time, space and society. Also crucial are the balance between individual and social identity, the role of language in social relations and getting things done, and how concepts of politeness and face are realized in interaction. Probably all of these is important in all cultures, hut the nature and emphasis of each may vary.

Some Caveats

When we consider cross-cultural communication it is natural to consider cross-cultural differences but we should first remember that cultures have much in common: we are all members of humanity, there has been extensive interaction between cultural groups for centuries and for most of the time most people get along very well with each other. Differences and problems should not obscure common elements.

Secondly, we need to remind ourselves that generalities about cultural groups do not always apply to individuals. There is always individual variation even in those cultures which emphasize collective thinking and action. Every culture has some balance between unity and diversity, between the individual and the group, between expected conventional responses and freedom of choice.

Thirdly, while general insights are very helpful we also need to bear in mind that different situations elicit different responses even in the same culture. Situational variation is common and much cultural activity is determined by context even in those cultures which stress principles which apparently transcend contextual variation.

Fourthly, in thinking about cross-cultural communication we should avoid a tendency to think about ‘us’ and ‘them’- For example, it is very useful to analyse some cultures (like British, French, German, American) as being individualistic, since they put emphasis on the individual, and other cultures (say Turkish, Chinese, Japanese, Indian) as being collective, since they tend to emphasize the group. But in using such binary categories we should remember that any culture probably has both individualistic and collective tendencies – it is a question of emphasis and relative balance.

Some Conclusions

We have looked at cross-cultural communication from a foreign language perspective. We have emphasized that it is not only cross-cultural language and behaviour which count but also participants interpretations of situations and people, since this interpretation often frames perceived meaning. Our own culture provides us with systems of interpreting language and interaction and in cross-cultural situations we need to be aware of these systems and endeavour to transcend them. Here, we have used a framework of different levels of communication to discuss how the more obvious levels of words, grammar and pronunciation often obscure the crucial role of body language, discourse patterns, sociolinguistic uses of language and cultural presuppositions. In face-to-face communication all these levels usually work simultaneously, in combination,

Speakers’ or hearers’ attitudes can be influenced by their interpretations, which in turn can be influenced by their own cultural systems. There can be a vicious circle here: cultural expectations can lead to different language use, which can lead to miscommunication. This in turn can lead to wrong assessments and stereotypes of participants from other cultures, which cart reinforce or mould cultural expectations, and so on. However, a major way to break such vicious circles is to be aware of possible difficulties, to have some knowledge of other cultures, and to try to develop intercultural skills.

To finish on a positive note, it is worth remembering that most of the time people from different cultures do get on with each other: as members of different cultures we share a common humanity. Good will and a friendly smile can overcome many barriers. Patience, trust and sensitivity are part of an international language of humanity which goes beyond words.