P. Rugby, North Dakota is the geographical center of North America. Amtrak’s Empire Builder train stops there once daily, in the morning, as it passes along highway 2 from Minot, North Dakota toward Montana and Glacier National Park.
The railroad created Rugby. At first it was to be called “Rugby Junction,” after a famous railroad station in England. The ploy was evidently to attract English settlers to the area. The English settlers were not convinced. As of the 2000 census the community still identified 50% German and 40% Norwegian. About three thousand people live in Rugby.
A large cairn in town marks Rugby’s geographical claim to fame. According to the U.S. Geological Survey, it stands about fifteen miles from the actual center of the continent. This is not so many miles, in this part of the country. Rugby is 67 miles from the nearest major city of Minot, North Dakota. It is 223 miles from Fargo, 458 from Minneapolis, 838 from Denver, and 1332 from Seattle.
Much closer is the Canadian border, only 45 miles north. This is the site of the International Peace Garden, a 2,339-acre botanical garden lying along the world’s longest unfortified border. The park can be visited without passing through customs.
Just south of Rugby is another attraction, the Northern Lights Tower. This 88-foot steel sculpture is lit at night to pay homage to the beauty of the aurora borealis. It is a somewhat unusual monument in that it seeks to honor not just a region or political area, but the whole continent at whose center it stands. The tower is currently under construction to improve the lighting.
The train doesn’t stop for long in Rugby. A few passengers disembark, a few board the train, and you roll on toward Montana. As you enter the trees on the west side of town, a sign on the back of the S&S Laundry building reads, “Rugby Says Hi.”
P. As the train pulled out of Rugby I was listening to Horatiu Radulescu’s Fourth String Quartet, which may have contributed to the moment’s atmosphere of subtle menace.
Radulescu, who died in 2008 a fine Romanian mystic composer and impeccable kook, wrote using spectral techniques of composition. This is one of those musical terms almost universally disavowed by those most associated with it. It has to do with basing musical decisions on the physical, harmonic properties of tones. The result is often a shimmering, overwhelming, waving sort of sonic texture.
As a student Radulescu named a piece after the Polynesian god Taaroa. His teachers didn’t like this because they thought it was “mystical” and “imperialistic.” In the early seventies Olivier Messiaen said he was “one of the most original young musicians of our time.” Many of his fellow students disagreed.
My favorite thing about Radulescu is his invention of the “sound icon,” which is a concert grand piano retuned to his own spectral schema, turned on its side, and then bowed. He would write a piece for 16 or 17 of these sound icons. Sometimes there would be also, say, a viola. He once wrote a piece for forty flutes. Often these go on for an hour or so. The effect is immersive, psychedelic. Or rather, I’d assume it is, as like most people I’ve never heard Radulescu’s music live. Most concert presenters are loath indeed to convert their fine pianos even temporarily into sound icons.
So the Fourth String Quartet is actually scored for nine quartets. Typically (if anything regarding Radulescu’s music can be considered typical) it is performed by one live quartet with the remaining eight pre-recorded. This group of eight quartets features instruments retuned dramatically and enthusiastically such that no two strings are tuned to the same pitch. Radulescu imagined them collectively as “an imaginary viola da gamba with 128 strings.” The central, live quartet plays in a traditional tuning adjusted slightly lower than is conventional.
There is a coherent mathematical foundation to all of this, to the intonational decisions being made. When people say music sounds mathematical, the subtext is not generally one of pleasure. We have developed a linguistic fissure between the mathematical and the sensuous that simply doesn’t make sense in music. All music is of course mathematical, as it concerns waves of different frequencies related by different ratios. I suppose what some listeners dislike is when the construction of the music, its formal manifestation, sounds mathematical--or perhaps the more precise word is mechanical. And mechanism is clearly not among Radulescu’s sins. This music as it unfolds is exactly as mechanical as a wolf spider crawling across a window. The music creeps along at a rate it appears to have chosen, stopping when it will, hesitating, looking back, glancing left and right. Its forward motion is not balanced or decisive in a manner that suggests premeditation. It is deeply organic. That word is overused, with vague referent, by composers and critics of new music. I use it to mean that this music sounds distinctly, terrifyingly, alive. Also large. And perhaps completely beyond our control.
Radulescu built the music, sculpted this sonic construction, to reflect individual spiritual questing. The mode is one of ever-growing ascents with intermediate descents. A final, triumphant ascent crowns the fifty-minute form.
He worked on the piece for eleven years before it was premiered in 1987.
The subtitle of the quartet, obliquely enough, is “infinite to be cannot be infinite, infinite anti-be could be infinite.” Radulescu was very much into the writings of Lao Tzu. The subtitle is “hommage á Leonardo da Vinci.” The analogy to the great inventor, a man whose imagination knew no boundary, is an evocative one.
P. The Penrose steps, also known as the impossible staircase, make four ninety-degree turns in two dimensions to loop back onto themselves. They thus climb continuously, such that a person could ascend or descend them forever without getting anywhere at all.
It’s a variation of the same geometric principle exhibited by the Penrose triangle, the classic “impossible object,” which was made famous by the mathematician and physicist Roger Penrose in the 1950s. Its three bars appear to cross behind and back in front of each other. Penrose called the triangle “impossibility in its purest form.” The eye accepts it as a valid construction. It is only when we consider the object, imagine it projected into our own beloved three-dimensional space, that we deem it impossible.
The Penrose stairs were first discussed in a paper Penrose wrote with his father in 1959, but made immortal the following year in M.C. Escher’s lithograph Ascending and Descending, which features faceless figures passing each other as they travel up and down an impossible staircase.
Escher had previously depicted geometrically challenging staircases in House of Stairs and Relativity, both of which featured stairs in multiple directions moving toward and away from the viewer. But the simple looping staircase in Ascending and Descending has a special elegance, paired with the strange mystery of the monkish people utilizing it.
There are twenty-five figures on the steps, walking in both directions. One story below, a lone figure stands on a balcony, gazing up at the stair-climbers.
Two stories below him there is another, sitting pensively on what must be, nominally, the front steps of the house, gazing across the featureless ground upon which the house sits, away off the frame of the drawing. Below the steps, a stairway opens down into a lower level. Near the top of this staircase, a few tiny emblems are carved into the wall.
Escher’s Waterfall, printed in 1961, explores similar territory. Water falls through a wheel and flows, apparently downhill, along a zig-zagged aqueduct reminiscent of two Penrose triangles stitched together--and then reaches again the top of the waterfall. The water always flows back into itself. Escher, like Radulescu, was a “mathematical” artist.
Again my attention is pulled to the interior rooms, the lower balconies. Two figures are represented in Waterfall. One admires the cascade, leaning back on the wall. Another casually hangs clothes from a line. I wonder about these peoples’ lives, about the rooms seen through all those windows. I wonder about the strange whimsical plants below the mill, the trees and ramparts and structures seen in the background. I imagine the fictional engineer who built this city. I wonder who designed the polyhedral capitals that crown the aqueduct’s two grand columns.
In medieval Eurasian churches there is not infrequently seen the symbol of the three hares. They chase each other in a circle, their ears pointing to the center. But each animal shares its ears, one each with its pursuer and its pursued, so that only three ears are actually depicted.
No primary documentation has been found to explain this design’s symbolic meanings. As seen in Christian churches it is generally assumed, perhaps facilely, to be symbolic of the Trinity; yet it originated in sixth century cave temples in China, and was a popular Buddhist emblem before it was adopted by medieval Christians.
The symbol’s meaning seems to shift based on its context.
Emerging from pagan cultures the emblem seems to reflect fertility and the moon cycle, as the hare was traditionally associated with lunar deities. Many cultures have gazed at the moon and seen the shape of a rabbit in its distant hills and craters; and of course, the rabbit is a famously fertile beast. Female hares can carry two distinct litters of offspring at one time. The fertile rabbit, along with its patron Germanic goddess Eostre, gave us the Easter Bunny--still a popular fertility symbol surrounding the Vernal Equinox, although we don’t much culturally recognize it as such.
The Christian significance of the holiday is akin, and resurrection is clearly a powerful symbol for that time of year. At some point it evidently became dangerous for the church to admit that the principle of rebirth applies metaphorically to all of us and to the earth, and not solely literally to Jesus of Nazareth.
The equinox, of course, is the time when day and night are of equal length everywhere on the planet.
The gray edifices of Escher’s lithographs and mathematical purity of Penrose’s formations fail to suggest fertility like the pastel colors of Easter eggs. They do convey, in a forceful, visceral manner, the existence of endless cycles.
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