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Tree (M.C. Escher, 1919 woodcut print)

Tree (M.C. Escher, 1919 woodcut print)

Here are two early woodcuts from the Dutch graphic artist M.C. Escher. In the course of time, Escher would become extremely famous for intricate black and white prints which picture the paradoxical juxtaposition and interplay of seemingly irreconcilable moral, aesthetic, or mathematical concepts. These two works, however, date from 1919 when the artist was only twenty years old and was still finding his artistic path. World War I had just ended (as had the Spanish flu epidemic) and a dark pall seemed to still hover over humankind. Escher had been a sickly child who failed to excel at any particular course of studies in secondary school. He was studying (and failing) architecture at Haarlem School of Architecture and Decorative Arts. In a few years he would reconcile himself with the artistic life and set off for Italy, but in these works some of the gloom of the war, and of his unhappy youth seems to linger in the solemn simple lines of the huge enigmatic trees.


The Borger Oak (M.C. Escher, 1919, Linocut print)

The Borger Oak (M.C. Escher, 1919, Linocut print)

The two woodcuts here both show fractal trees against the cosmic backdrop of a black sky with a single luminous star burning in the heavens. In Tree, a tiny benighted human figure drops to his knees in front of the great tree which seems to hold a burning star within its interwoven branches. The Borger Oak is even starker: the boughs of the tree are becoming a simple recursive pattern against white hills. A glowing celestial body fitfully illuminates the scene. Already the themes which would dominate Escher’s life work are apparent: the recursive patterns of mathematic sequences are apparent in the prints (albeit not with the vertiginous intricacy which would characterize later works). Both works are simple and beautiful microcosms. The trees represent life, science, and even the entire universe itself (like Yggdrasil, the world tree of ancient northern myth). Living things and the laws of space are both part of an overarching pattern.

One of life’s disappointments is the dearth of fine art concerning outer space.  Outer space is vast beyond imagining: it contains everything known. Indeed, we live in space (albeit on a little blue planet hurtling around an obscure yellow star)–but cosmic wonders do not seem to have called out to the greatest artists of the past as much as religious or earthly subjects. There are of course many commercial illustrations featuring the elements of science fiction: starships, ringed planets, exploding suns, and tentacled aliens (all of which I like) and there are also didactic scientific illustrations, which attempt to show binary stars, ring galaxies, quasars and other celestial subjects.  Yet only rarely does a fine artist turn his eyes towards the heavens, and it is even less frequent that such a work captures the magnificence and enormity of astronomy.

Fortunately the Dutch artist MC Escher was such an artist.  His space-themed engravings utilize religious, architectural, and biological elements in order to give a sense of scale and mystery.  The familiar architecture and subjects are transcended and eclipsed by the enormity of the cosmic subjects.  Here are two of his woodcuts which directly concern outer space.

The Dream (Mantis Religiosa) (M.C. Escher, 1935, wood engraving)

The first print is a wood engraving entitled The Dream (Mantis Religiosa) shows a fallen bishop stretched on a catafalque as a huge otherworldly praying mantis stands on his chest (the whole work is a sort of pun on the mantis’ taxonomical name Mantis religiosa “the religious mantis”.  The buildings arround the bishop and the bug are dissipating to reveal the wonders of the night sky. The bishop’s world of religious mysteries and social control are vanishing in the face of his death.  Greater mysteries are coming to life and beckoning the anxious viewer.

Another World (M. C. Escher, 1947, colored woodcut)

The colored woodcut “Other World” shows a simurgh standing above, below and in front of the viewer in a spatially impossible gazebo on an alien world.  The simurgh is a mythical animal from ancient Persian literature and art which combines human and avian elements.  Sufi mystics sometimes utilize the simurgh as a metaphor for the unknowable nature of divinity.  Yet here the simurgh is dwarfed by the craters beneath him and by the planetary rings filling up the sky above.  A strange horn hangs above, below, and to the side of the viewer.  Perhaps it is a shofar from ancient Judea or a cornucopia from the great goat Amalthea.  Whatever the case, the viewer has become unfixed in mathematical space and is simultaneously looking at the world from many different vantage points.  A galaxy hangs in the sky above as a reminder of the viewer’s insignificance.

Above all it is Escher’s manipulation of spatial constructs within his art that makes the viewer realize the mathematical mysteries which we are daily enmeshed in.  The multidimensional geometric oddities rendered by Escher’s steady hand in two dimensions characterize a universe which contains both order and mystery.  Giant bugs and bird/human hybrids are only symbols of our quest to learn the underpinnings of the firmament. Escher’s art is one of the few places where science and art go together hand in hand as partners. This synthesis gives a lasting greatness to his artwork, which are undiminished by popularity and mass reproduction.

Today, Ferrebeekeeper ventures far far beyond my comfort zone into that most esoteric and pure realm of thought, mathematics.  But don’t worry, we are concentrating on topology and geometry only for long enough to introduce a beautiful, intriguing shape, the torus, and then it is straight back to the real world for us…  Well, hopefully that will prove to be the case–the torus is anything but straight.  It is, in fact, very circular indeed, and, as we all know, it has at least one big hole in it….

Wikipedia defines a torus as “a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle.” That’s hard for me to wrap by head around but the meaning becomes much more comprehensible in the following illustration.

So a torus is a circle wrapped around in a circular path.  You can find a variety of other ways of mathematically representing the torus here, but the simple definition suits our purpose.

One ring toroid them all....

I admire torus shapes because I think they are very beautiful.  Just as the golden ratio and the Fibonacci sequence are aesthetically appealing, there is a pleasure to merely beholding or touching a torus: ask anyone who has contemplated a ring, a golden diadem, or a cinnamon donut.  Talk to an indolent adolescent sprawled on an inner-tube bobbing on the surf, and you will immediately grasp the hold that toroids have on humankind.

And beyond humankind....

In addition to its obvious aesthetic merits, however, there is a mysterious aspect to the torus, a hint at hidden dimensions, negative space, and infinity.  Kindly contemplate the old eighties video game Asteroid (you can play the game here if you are too young to remember: amusingly, your ship is an “A” which reminds me of Petrus Christus’ enigmatic painting).  If you pilot your ship to the far left of the screen you emerge on the right side of the screen: the flat screen represents a cylinder. However, to quote Bryan Clare from Strange Horizons, “the bottom of the screen is connected to the top as well. This has the same effect as if the screen were rolled into a cylinder, and then bent again to glue the two circular ends together, forming the familiar donut shape.” So, when you play asteroids you are trapped in a miniature toroid universe which appears 2-dimensional. Try to blast your way out of that!

The following famous math problem further illustrates the nature of the torus.  Three utility companies need to connect their respective lines (gas, water, and electric) to three different houses without ever crossing the lines.

Connect each utility to each house. Don't cross the lines.

The problem is impossible on a two-dimensional Euclidean plain and even on a sphere, however topologists realized that if you poke a hole through the plane or the sphere (thereby making it a torus) the lines can be connected.

To finish the article here is a movie of a torus being punctured and turned inside out. The result is a torus of the same dimensions but with reversed latitude and longitude.  It’s hard not to love such a funny shape.  But it is hard for me to wrap my mind around the larger implications.  I think I’m going to stop trying and head off for some donuts.

Ye Olde Ferrebeekeeper Archives

July 2020