Fragments from:

Wakara Way Clearing:
A Contemplation of a Triangle

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I noticed the clouds breaking over the Oquirrh Mountains while driving aimlessly on an overcast afternoon, the scant light drawing a sickly yellow streak across the sky. I turned east and drove up the hillside to the clearing, a pathetic opening between dense brush in an anonymous woods below Red Butte Canyon. When I arrived the light had already thinned as the sun passed behind the ridgeline.  A thick flurry of snow burst from between the looming Wasatch Mountains.

The light falling, I take account of the dense colors in the clearing, visible despite the dimness. Against the luminent blue tint of stale snow, hardened like shimmering, damp sand, a pallet emerges: smears of aubergine; shoots of marigold, maroon; vivid taupes and streaking puces; umbers gleaming in the muted light.

Yet as fresh snow starts falling and blankets the ground, the starkness of the open space grows. Its saccharine whiteness agitates like infantry the muddled line of brush. The sudden contrast in the clearing seizes the loping demarcations between the dark brush and light snow in rigid form: the clearing becomes a triangle in the dark.

Walking back to my truck, I look up at the clouded night sky. Beyond the clouds lie stars, between which the ancients drew lines between and told stories with the constellations. Is there a story to tell in a shape drawn by the clearing in the skint shade of dusk? As lines emerged, the visual constraints of the clearing’s borders sharpened. Perhaps I’m superimposing geometrical abstractions to make sense in the darkness. Or does the clearing allude to some deeper meaning in that brief glimpse of structure?

The triangle is a simple shape: three lines of any length that intersect as angles.  The name describes precisely, with a graceful reflexivity, the triangle itself. The name “triangle” has performed this reflexive function since Latin — triangulus — then triangle in Old French and eventually English. Since antiquity, one need not see a triangle to understand that it is three angles. It is possible to imagine someone living and dying knowing what a triangle is yet never seeing one.

This curious idiosyncrasy lends the triangle a poetic distinction. It functions like a pictograph, simultaneously representing the object, the word and the concept; it is simultaneously a thing and a sensation and an idea.

The Chinese character system evolved in a similar way. In their earliest form, known as Oracle Bones, there is a primitive simplicity to the penmanship of Chinese characters. Used for divination in ancient China, polished animal bones were burned and the resulting cracks were recorded in a simplistic pictographic script that mimicked the crack’s shape.

In the American Southwest, there are hundreds of pictographs and petroglyphs drawn on cliffs and in caves. Done with mineral pigments and primitive tools, these images depict  prehistoric cosmologies, ancient beasts and human figures. The anatomy of the human figures is unnatural. While the essential comprehension of the human body exists in these ancient drawings, they disregard realism to exaggerate the triangular form of the torso. The figure recedes into a triangle, and, suddenly consumed by geometric corruption, becomes more symbolic than representational.

Whereas the Oracle Bones mimicked cracks and thus found the triangle, the prehistoric Southwest cultures veered from perceived reality to a symbolic meaning. Is the triangle an organic geometry, born from burning bones like Galatea from Pygmalion’s marble? Or, is the triangle gesturing towards a primordial profundity? How can a shape emerge from the aether seeping hints of higher order, bearing some arcane significance within its form?

Plato in The Timaeus conceived of the world as fundamentally geometric. The Platonic cosmology elucidated the perceptible world as a projection of shadows from a realm of ideal forms; a floating world masked by reality where the intrinsic truth of things existed in transcendent geometry. The fundamental shape of this realm and our own universe was the triangle.

Plato fashioned a hypothetical craftsman who conjured ideal forms from the noumenal substance of this alternate realm. His material was the triangle, and other forms were made from triangles to create the elements. Fire, water, and air: each, in varying proportion and combination made up all reality. The mutable properties of each — hot, cold; bright, dark; light, heavy — are effects of triangular arrangements. While some triangles made fire, others made faces, trees, snow; even a clearing. Each object, experience and sensation was a triangle composed in perfection and refracted into our reality.

The earlier Greek philosopher Thales of Miletus used the triangle for measuring distance and height. His triangle was less ideal form than manifestation of a theorem that organized relative geometries: a triangle was truly three angles, and Thales used this to practical ends for measurements of great distance.

Seeing an enemy ship at sea from the shore, Thales drew a line in the sand and measured the angle of the ship. He moved to a further point down the shore on the other side of the ship and again measured the angle. These two points were then followed to the point where they intersected on land. By flipping this triangle, Thales could measure exactly how far the ship was from the shore.

A similar deduction determined the height of an Egyptian pyramid using a staff and its shadow. Holding his staff so the shadow matched the pyramid’s shadow, Thales deduced the pyramid’s height proportionally relative to his staff. This was far from the Platonic forms: Thales’ application of triangles shows that the triangle functions not just as a philosophical form but also a tool for comprehending physical space. Thales could not measure directly the distance or height: his subjects were beyond the scale for which a human can accurately account. Rather than constructing an imperceptible realm, Thales’s triangle makes accessible immediate reality with imposition of a simple form.

Beneath the fragile pastel of spring, I’m driving up the hillside. The snow in the woods has persisted until today’s warmth. Arriving, I notice the woods have subtly shifted in color. The haggard veil of slush has vanished and the umbers beneath blur the woods into a wash of earth. The sheen of old snow’s tainted pearlescence has yielded viscous mud and feeble shoots of celadon grasses.

Entering the clearing, I’m unsettled: the stark contrasts have melted with the snow. What form was here that now is not? The triangle has gone, its contours only wispy serrations where once was geometry.

Perhaps the triangle was imagined. Despite how vivid it’s consolidation in the midwinter twilight, it has abandoned my senses with seasons’ change. Miletus’ triangles constrained great distance within the triangle’s enclosure. I had constrained the clearing within his frame, yet that fleeting form now dissipated. Had it slipped my senses and transcended my impositions for Plato’s realm, be that the winter of memory or deeper recesses of my mind?

For now, the light again fading, the sun settling on the Oquirrhs, I scratch some words and sketches in my notebook and hope to retain some essence of that ancient form. Maybe my glimpse of the triangle in the clearing can be remembered in fragments: just as cracks in bones, pigment on cliffsides, or lines between stars in the sky.

 ~ Nicholas Apodaca (Spring 2019)

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For further reading:

Ross, Frank, and Michael Goodman. Oracle Bones, Stars, and Wheelbarrows: Ancient Chinese Science and Technology. Boston: Houghton Mifflin, 2003.

Castleton, Kenneth Bitner. Petroglyphs and Pictographs of Utah. Salt Lake City: Utah Museum of Natural History, 1987.

Plato, and Francis Macdonald Cornford. Platos Cosmology: The Timaeus of Plato. Mansfield Centre, CT: Martino Publishing, 2014.

Redlin, Lothar, Ngo Viet, and Saleem Watson. “Thales Shadow.” Mathematics Magazine 73, no. 5 (2000): 347.