March 24, 2015

triangulation

By In musings, yoga

trikonasana is the pose i come home to in my practice. until now for reasons i haven’t really tried to tease out, other than it feels pretty natural to slip into when your quads are fatigued and your hamstrings have been engaged long enough to need a counter pose. i also find it to be particularly easy pose to enter without engagement or complete awareness and still feel a, perhaps, superficial  stretch. geometrically and architecturally  triangles are phenomenally well understood and soundly stable. a fact understood to be true regardless of the characterization or classification of the triangle, in essence independent of whether it is right, oblique, acute, or obtuse. it goes without saying that the proportional length of the sides has an impact, but this is a fact over which we have little control given the legs of our triangles are our legs. we can however control the angles between our feet and the floor  the distance between them, dictating the length of the third side. given this architectural fact the shape we create is strong with or without similar angle measures to our teachers or Yoga Journal models, the strength is inherent in the shape itself and its inability to be reshaped by lateral forces and pressure. unlike, for example, a square or a rectangle which are both easily turned into parallelograms when force is applied to any of the sides, revealing the importance of angles in maintaining these four sided figures.  for a triangle to be misshapen the sides or the joints must be broken, for it is the simplest shape that cannot be deformed when just the length of the sides are fixed.  yet another yogic reminder to heed our joints, of which in the triangle we create there are five;

ours knees cut halfway between the sides and wouldn’t considered a joint by Euclid, but are nonetheless essential to the longevity of our triangle, so, starting at the foundation push down firmly with the ball of your front foot maybe even   lifting and spreading the toes, draw the knee cap up by engaging all the muscles around it, contract the right quadricep to relax the targeted hamstring and in the back, root down with the outer edge of your back foot helping you to engage all the supportive muscles on the inside of the back knee.

there’s more to this than strength though, the triangle is a historical and religious icon as well as the basic symbol for so many things up, down, male, female, solar, and lunar among others. across cultures the triangle has come to represent diametrically opposing directions and entities. this becomes increasingly appropriate given that the sum of its interior angles is 180 degrees, a turn to the diametrically opposite side. the circle is often touted as the strongest shape, and there is certainly beauty to everything coming full circle but 180 degrees inevitably happens first, at some point the other side must be at least glimpsed, all the possible actions of the opposite. sometimes doing something that takes you back to where you are or where you started is unimaginable and wouldn’t suit you or your future. triangles represent our ability to embrace and embody polar opposites and opposing forces, and find strength in doing so. in trikonasana the line of spine is supplementary, forming two complements to the peak angle created at the hips by the thighs. no matter the measure the angle created by the line of spine further embodies this 180 degrees, it is the line over which the triangle flips to denote the opposite meaning.

this angle can be more clearly articulated in your body with an external rotation of the front quad (drawing down of the outer hip), a drawing up and into the pelvis of this femur and, for the back leg a slight hinge forward at the hip and pushing back and drawing into the socket of this thigh. This creates a structure that allows more freedom for axial extension of the spine as it hinges forward the hips aiming toward parallel with the floor.

i don’t mean to detract from circles, i’m obsessed, and someday there will be an even longer post on the topic but my dhanurasana (bow pose) should probably look somewhat circular first. but, triangles are the more subtle foundation of our world, the scaffolding of your house and any construction site can tell you so. they are geometrically defined by any three unique non-collinear points, a simple geometric fact that helps us socially and geographically locate ourselves. the emotional triangle is thought to be the smallest strongest working subsystem of family and community. we triangulate relationships to ease stress on our “dyad” relationships something i am undeniably guilty of with my dog, as is everyone that has ever borrowed him for an OK Cupid date. geographically we can triangulate where we are, or the distance to a point by establishing a known baseline (segment) and determining the angles from its ends to any given point, and then back-calculating the length of the lines that intersect at the point of interest or desired destination. additionally, in higher math, like topology, triangulation is used to determine aspects of spaces (like the homology – or number of holes in shape) that are too arduous to determine without the theories available in a triangulated, or simplicial, space. defining a shape by its holes is a whole other story. triangles can be used to direct us to points otherwise unknown and can also transport, through transformation, us to spaces where the geometry is better known. the full expression of trikonasana is not an internally oriented or particularly introspective pose, the hand not stabilizing the torso is reaching up and out toward the sky, the chest swiveling in that direction with the gaze following; but as a shape a beautiful characteristic of triangles – Morley’s trisector theorem – proved that a trisection of all the interior angles of any triangle yielded an equilateral triangle at the cross-section points of the angle trisectors, with further trisection yielding more equilateral triangles ad infinitum. yogic philosophy is underscored by a belief in every person’s internal divinity around which we deviate in our day-to-day lives by losing our alignment around center. Morley’s triangle can be a manifestation of this, whatever beautiful trinity you’re drawn to, or just an idea to ponder while in the pose, if you’re like me and over-analyzing every anatomical aspect in an effort toward an expression for lack of a better word perfect; per Morley, regardless of your expression a more stable balanced version of the triangle you create is inevitably built within, a payoff just for having done the work to form one with your limbs.

Morleys

unfortunately, Morley’s theorem, isn’t easily generalized to different spaces, spaces such as spherical or hyperbolic space. while widely accepted and proven in Euclidean space, in Non-Euclidean where the parallel postulate and the distance formula are defunct everything we know and assume about triangles drawn on graph paper doesn’t necessarily hold, suddenly the sum of the internal measure of triangles can be greater than or less than 180 degrees and there is no internal equilateral triangle, all these qualities are dependent upon the space in or upon which the triangle was created, a reminder that we must first find a space that supports what we are trying to create before we alone can support it.

my triangle was created in Cape Town, South Africa atop Lion’s Head, a spiritual heartland that always brings me back to the center i deviate from.

Morley’s triangle image came from mathworld.wolfram.com, Thanks 🙂

Leave a Comment