If we are asked to connect the dots that are randomly distributed on a piece of paper, the great number of us would draw it the same way. The result would most likely look very close to the picture shown below. Why do we all reflexively choose to connect the dots this similarly, almost identically, is a puzzling question. Even though it is not mentioned to form non-penetrating triangles, we take special care, almost habitually, draw them as non-intersecting as well. Why ?
All things in the universe, living and stationary, obey, act and even evolve thriving for a simple topology, and that is symmetricity. Our DNA sequence is symmetric with pairs of nucleic acids. The crystal structures of matter favor symmetric formations. Balance requires symmetric forces. We try to balance our budget; if we spend more, we have to save as much. If we drink too much, we stay sober the next day. Nature likes symmetric shapes. Even the most random looking objects are formed by the replication of simple patterns. The shape of coastal lines, tree branches, leaves, etc. can all be modeled by applying a simple pattern numerous times over itself. The underlying simplicity of complex looking formations is so extraordinary that you almost feel tempted to mimic creation of these complex shapes by hand, which in some cases you indeed can. For example, if you replace a straight line with a simple pattern and apply it to each of its new straight lines many times, and with some patience, you'd see the evolution of the shape resembling a snow-flake pretty soon discovered by a clever Swedish mathematician, Mr. Koch in 1904.
We also like symmetric faces. Almost all celebrities have symmetric facial features. Symmetric faces seem beautiful to us. Perhaps it is in our DNA to feel it that way. Whatever it is, and whatever scientific explanation we could find for it, it is undeniably true that we adore symmetricity, period.
Some of us indeed tried to find scientific reasons for why we all draw almost identical triangles out of randomly scattered points. Бори́с Никола́евич Делоне́ (Boris Nikolaevich Delone), a pioneering Russian mountaineer and a descendant of a POW from Napoleon's army general, discovered the intrinsic property of triangulation, which we call today as Delaunay property (a fake-french name) that there is no points in the circumcircle of any triangle other than its own three vertices. It was 1934 when the discovery was announced on French and German written scientific journals. However, it took almost another half a century to cast this idea into its algorithmic form that would enable the tedious repetitive task performed by a computer. During early 1980s, the computer hardware, its memory and speed power was at its infancy and the ability to connect thousand points was one of the most advanced computing achievements.
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The virtual modeling of symmetric reality became possible only when the computing hardware was barely enough to do the animations of triangulated scenes. Even though it was only mid 1980s when the color VGA monitors were introduced to PCs, the creative minds of Holywood quickly saw the opportunity to use virtual reality to resurrect the dinosaurs of Jurassic period in less than a decade. Spielberg with his studio crew at Dreamworks spent hundreds of hours to put together the animation scenes using the mighty SGI workstations. It was still too early to create virtual reality in real time in the year 1994. The real time means that the objects in the scene moves as instructed by the script using the power of computers by translating or rotating the object coordinates as it is being recorded to the film. In real time animations, objects could also be deformed due to the motion of another like T-Rex is crushing the roof of 4WD SUV almost as real as how it actually happens in reality. These were perhaps good enough for the entertainment sector and to this day we still use the same technology in recent movies only with more finesse and texture quality attached to the triangles of the scene.
However, when it comes to building the war machines, fighters, passenger aircrafts, automobiles, those engineering design processes have to be super accurate and it does not have be visually appealing. Almost all of us probably know about the simple formula of Newton's law of motion (F=ma). This tells us how much force is needed to move a body at a certain rate. However, this simple formula is lumped and coming from a complicated set of equations. They are known as momentum conservation equations or Navier Stokes (NS). They dictate how things move and interact in the world accurately. The problem is the difficulty in solving NS equations. If only they could be solved analytically, it would be easy to predict motion. No worries though, triangles come in to the rescue again.
The old tactic of divide and conquer is applied to the governing equations as well as to the geometry. Geometry is divided into triangles and the momentum equation is conserved around the simple boundary of the triangles. The overall contribution from all the triangles is summed up and the resulting system is solved using numerical techniques. The bridges, aircrafts, cars are built first in the computer this way and then after making sure that numerical simulation models work, the actual process of manufacturing is waged saving billions of dollars with more safety and efficiency in the final product.
Nowadays, not only we build engineering marvels such as bridges, ships, aircrafts, but we use the same simulation techniques in designing chairs, tooth-implants, the chips in our computers, basically for all the items we as humans use from A to Z. The final product design is created by optimizing the computer model through numerical simulation experiments to achieve the best outcome, least CO2 release, lowest drag and fuel consumption, largest pay-load, without building physical prototypes.
Triangles and symmetry, from an acute perspective, they are taking us to the path of deep space exploration whatever there is for us to find out there. My guess is more of the same self replicating simple processes that evolve to complex organic creatures like us or ugly animals like roaches. But most of us believe that we had to be created by a special act since we think so much of ourselves out of self-admiration. We would not even try to understand and accept this simple story, a marvelous journey that took us from dots to space. All I wish is that we could connect the dots to see the obvious reality and rid ourselves from the vague notion of a universal "creator".
Triangles and symmetry, from an acute perspective, they are taking us to the path of deep space exploration whatever there is for us to find out there. My guess is more of the same self replicating simple processes that evolve to complex organic creatures like us or ugly animals like roaches. But most of us believe that we had to be created by a special act since we think so much of ourselves out of self-admiration. We would not even try to understand and accept this simple story, a marvelous journey that took us from dots to space. All I wish is that we could connect the dots to see the obvious reality and rid ourselves from the vague notion of a universal "creator".
