Symmetry: Understanding the Way Things Are

"Although bees have known for ages that hexagons are the most efficient shape for building a honey store, it is only very recently mathematicians have  fully explained the Honeycomb Conjecture: from the infinite choice of different structures that the bees could have built, it is hexagons that use the least wax to create the most cells." ~Marcus du Sautoy

As I read this I think again of how amazing our natural world is. There are ways of seeing in which everything falls into patterns that dazzle the mind.

Near the end of the book Rocket Men, this is one of the effects the astronauts experienced, a new awe at the structure and order of the universe, man's smallness and the masterful combination of complexity, immensity and simplicity of the universe and its design.

“Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.” ― Chen Ning Yang

This is no new insight. The laws of symmetry are vast, and everywhere present.

“Since the beginning of physics, symmetry considerations have provided us with an extremely powerful and useful tool in our effort to understand nature. Gradually they have become the backbone of our theoretical formulation of physical laws.” ― Tsung-Dao Lee

Or as Paul Valery observed, "The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect."

The trigger for this blog post was the Marcus Du Sautoy quote about bees. It reminded me of Buckminster Fuller's work with Geodesic Domes. I heard a presentation on Fuller when I was young and it made an impression on me.

The harmonics between external structure and order and the symmetrical resonance with the internal phenomenon of our minds is endlessly fascinating. This was especially so for Du Sautoy. "Mathematics has beauty and romance," he wrote. "It's not a boring place to be, the mathematical world. It's an extraordinary place; it's worth spending time there... The reason why we do math is because it's like poetry. It's about patterns, and that really turned me on. It made me feel that math was in tune with the other things I liked doing."

The applications are endless. Fuller applied it to architecture and design; Da Vinci and Dali to art. Mozart to music. "I'm obviously attuned to pick up mathematics whenever I can see it. But in Mozart there is a lot of conscious use of mathematical symbolism and numbers..." Most masterfully, the same occurs in the music of Johann Sebastian Bach.

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"But actually a code is a language for translating one thing into another. And mathematics is the language of science. My big thesis is that although the world looks messy and chaotic, if you translate it into the world of numbers and shapes, patterns emerge and you start to understand why things are the way they are." ~Marcus Du Sautoy

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Symmetry is more than just the natural world. This desire for balance reaches into the realm of ethics and morality. Doesn't our desire for justice, and our sense that injustice must be rectified, stem from a sense that wrongs must be righted, or someone must pay when a wrong is done? Where does this sense of a need for moral symmetry come from? Do I dare say it seems innate in the fiber of our very souls?

That, friends, is a much longer discussion and equally profound in its implications. 

Current and Recent Readings: Too Many Books, Too Little Time

There really are so many good books out there. We read them for different reasons. Some serve as simply a diversion. Others provide nutrition for our souls. Still others give our "brain muscles" a workout. Here are a few books that I've been enjoying right now or recently completed.

Symmetry by Marcus du Sautoy
I picked up this gem after recently re-reading my interview with Portuguese artist Margarida Sardinha regarding her 2015 project Symmetry's Portal which led me into revisiting (via books) the remarkable features of The Alhambra.

The Red Book by Carl Jung
The Amazon listing about this volume calls it "the most influential unpublished work in the history of psychology." My friend Dan introduced me to this hefty volume so I could see the illustrations, more than 200 in all. But the substance is Jung's private wrestling with the meaning of Self, consciousness and universal truths about who we are. Four decades ago I read Jung's Memories, Dreams and Reflections and was impressed with his candor.

50 Philosophy Classics by Tom Butler-Bowdon
I finished the audiobook in December. Having found it to be such a valuable resource I purchased the paper version to use as a reference. A great thought-stimulator. Useful tool for stirring up themes to cogitate upon so you can produce the illusion that you're a deep thinker.

The Power of Habit by Charles Duhigg
Exceptionally insightful.  Listened to the audiobook this past month and will do it again. Utilizes insights from the latest research in neuroscience. Compelling stories bring home essential truths. Yes, we're creatures of habit, and when those habits are bad ones we need to apply ourselves to cultivating new ones. We begin by becoming aware of our triggers.

Scientific Advertising by Claude Hopkins
Duhigg's book uses stories from a variety of sources. One of these sources was Claude Hopkins, an influential ad man from the first half of the twentieth century. I'd read Hopkins many years ago because my own advertising guru, David Ogilvy, consider Hopkins his own shining light. Were you aware that it was an ad campaign by Claude Hopkins that prodded a whole nation of people to regularly brush their teeth?

Illustration from The Red Book

The Light on Synanon by Dave Mitchell, Cathy Mitchell and Richard Ofshe
Current bedtime reading. When I read it in the early 1990's it triggered an idea for a story which later became an unproduced screenplay. Still gonna try to resurrect that project if I live long enough.

Rocket Men by Craig Nelson
Currently reading this one as I commute. Absolutely compelling thus far. Reminds us of the context when that first moonshot took place, during the Cold War. The Russians had already embarrassed us with Sputnik and other achievements. A moon landing would be a major PR coup, which really amounted to a "puff our chests out" opportunity to gloat. Had the Russians been first on the moon would that have meant Soviet communism was superior to our democratic capitalism? Rocket Men is an excellent addition to the many other books about the story of NASA. An good follow up to Tom Wolfe's The Right Stuff. (EdNote: A few reviews on Amazon indicate that this book may not be entirely reliable in all its facts, though for now it's been a good read.)

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What are you reading these days?

The Alhambra: An Introduction for North American Readers

I recently received an email regarding a new exhibition, titled Symmetry's Portal, by Lisbon-born artist Margardia Sardinha. The show was exhibited at Centro Cultural Jaime Lobo e Silva in Ericeira and opening this past week at Lisbon's Ismaili Centre, which belongs to the Aga Khan Foundation.

The press release begins with this enticing description:

Symmetry 's Portal is an exhibition organized by the artist Margarida Sardinha that has, over a lengthy period of time, been extensively focused on the concepts of symmetry and optical illusion. The exhibition expands geometrically a wide photographic survey of the Alhambra in Granada documented by the artist. This photographic assemblage is used by Margarida Sardinha to deconstruct symmetry and to generate illusory semblances. Symmetry 's Portal is a body of work of optical illusions where the symmetric and random are diluted in three-dimensional works originated from two-dimensional planes.

In order to properly write about M. Sardinha's current work I found myself challenged by my ignorance regarding the central feature of this work, the Alhambra in Granada on the southern plains of Spain. Since learning of the show last week I downloaded several books about the Alhambra to my Kindle and have been surprised yet again by another wonder of the world.

First surprise: Washington Irving, author of the Legend of Sleepy Hollow and Rip Van Winkle, wrote a book about his journey to the Alhambra in the mid-nineteenth century. Irving, who later completed a five-volume biography of George Washington, had been appointed ambassador to Spain from 1842-45. During this time he took the opportunity to travel through Don Quixote territories to this most unusual palace and fortress complex which had first been established by the Moors when they conquered Spain. Irving, being an author, wrote in great detail and an entertaining manner about the trip to the Alhambra and the marvels he encountered there.

The Alhambra began as an outpost in about the 9th century, was later rebuilt and added onto in the 11th century by the Moorish emir Mohammed ben Al-Ahmar of the Emirate of Granada, who built its current palace and walls. It was converted into a royal palace in 1333 by Yusuf I, Sultan of Granada.

But its history doesn't end there. After the Muslims were removed from Spain, Christian rulers moved in to portions of the Alhambra and added some of their own embellishments. In fact, it was here that in 1492 Christopher Columbus made his presentation to King Ferdinand and Queen Isabel of Spain to obtain funding for that crazy-wild notion of going to India by a direct route West across the Atlantic.

The structure over time fell into disrepair until a much later time it was recognized for the jewel that it was, with restorations initiated in the 19th century.

This history is not, however, the focus of Symmetry's Portal. Rather, it is the remarkable tiling patterns with which this palatial wonder has been decorated. I learned more about this feature of the Alhambra by reading John Jaworski's volume A Mathematician's Guide to the Alhambra.

For those unaware, Islam forbids representation of the human form, hence the "kaleidoscope of colorful tilings, carvings and reliefs" in Islamc architecture. According to Jaworski, the phenomenon that is featured in the Alhambra regarding these designs is that there are examples of all 17 possible tiling patterns here. The designs of the Alhambra thus become a catalog of mathematical forms.

17 is an interesting number. First, as a prime number, it is indivisible. Second, among other things, it is the only known prime that is equal to the sum of digits of its cube (17 to the third power = 4913 and 4 + 9 + 1 + 3 = 17).

To understand  what is going on in the patterns of the Alhambra it is helpful to understand tiling, symmetry and tessellation, the latter being "the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps." Many of us have a measure of familiarity with the concept through the etchings of M.C. Escher, who was himself inspired by the Alhambra.* Not all tiling patterns have symmetry, and according to those who have studied this, there can be no more than 17 distinct ways to repeat these motifs. This standard of 17 exists only in Egyptian temples and the Alhambra.

How did these artisans in Egypt and of the Alhambra discover this? How did this all come about? As this is my first introduction to these notions, I can only say that I look forward to learning more, and aim to share more insights from M. Sardinha's penetrating work, which appears to be an outgrowth of concepts that she has been developing for many years.

"Know oh brother ... the study of sensible geometry evokes skill in all practical arts, while the study of intelligible geometry evokes skill in the intellectual arts because this science is one of the portals through which we move towards the knowledge of the essence of the soul, and this is the root of all knowledge..."
~Rasa'il Brotherhood of Purity, translated by S.H. Nasr

TO BE CONTINUED

* Henri Matisse and David Hockney are additional 20th century artists who found inspiration in the designs of the Alhambra

Photo Credits
Top Right: "Ceiling in Alhambra" by Liam987 - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.
Lower Photo: "Vista de la Alhambra" by bernjan - Flickr. Licensed under CC BY 2.0 via Wikimedia Commons.