This is an entire post about the number 3, and I am OK with that...
Here are several fascinating, and in several cases unique, properties of the number 3:
First off, 3 is an odd number, a prime number, the integer component of Pi, and it is the first non-trivial triangular number.
More interesting still is that, in two or three dimensional space, one can arbitrarily define any 3 points and, provided they are not all collinear (i.e. as long as they are not in a straight line), they can be treated as the vertices (the points) of a triangle, and can thus be used to define a plane.
Because of this, we can reproduce, or at least approximate, almost any polygon in 2D or 3D space just using triangles. In other words we can decompose almost any shape into triangles.
Here is where things start to get really, really intriguing (at least to me)...
3 is the only number which satisfies the rule:
N(N-1) / 2 = N
What this means is that in any set of 3 objects, there are always exactly 3 unique unordered pairs.
Imagine we have objects A, B, and C. The set of unique combinations is:
(A,B), (B,C), (C,A)
Which is equivalent to (B,A), (C,B), (A,C).
Amongst this new set, there are also exactly 3 possible unique combinations:
((A,B),(B,C)), ((B,C),(C,A)), ((C,A),(A,B))
This recursive structure maintains exactly 3 pairwise relationships at each level. No matter how many times we repeat the process, we always form exactly 3 new pairs from the prior 3 elements — a fixed point under pairwise combination, unique to the number 3, which we can state as:
N(N-1)/2 = N only for N = 3.
This is borne out in the Sierpiński Gasket, which provides a visual proof of this recursive rule.
Now this next point is slightly more complex. Imagine we have two objects, X and Y, floating in a void. One, or both, of these objects are in motion, but the only information we have is the distance between them over time: d(xy)(t)
From this information alone, we can draw some conclusions about their motion (we might see oscillations, collisions, etc) but we cannot understand the distribution of that motion between the two. We could normalise to X (treat X as static) and then attribute all motion to Y, or vice versa, or else we could split the difference and assume that X and Y are both equally responsible for the motion, and thus assume half of the motion is X and half is Y. Regardless of which option we choose, we are making an assumption since we have no way of establishing asymmetry of motion distribution using only d(x,y)(t).
The only way to establish any asymmetry in the motion of X and Y is to introduce a third object, Z, and normalise the system to Z. We now have the set:
d(x,y)(t), d(x,z)(t) and d(y,z)(t)
(three unique pairs, as above).
Armed with this information we can now triangulate to determine the differences in motion between X and Y, enabling a complete description of d(x,y)t.
This means that 3 is the minimum number of objects necessary in order to identify asymmetric motion between any two objects using distance over time - another intriguing result.
There are some fascinating implications to this. Consider the phenomena of hearing and vision. In both cases we have a pair of sensors, of known, fixed distance, and we use them to pinpoint the location and motion of some source stimulus. We cannot, for example, look at two things at once - instead we form a triangle between our two sensors and the subject, and from those three points we can determine direction and motion. Now, vision and hearing are of course far more complex than merely triangulating between three points, but it is still very important - particularly for depth perception or auditory location.
This notion might even echo in quantum physics. In interpretations like Copenhagen, particles in superposition only take on definite states when measured — that is, when a third system (the observer) interacts with them. Much like how asymmetric motion between two bodies only becomes meaningful when framed by a third reference, quantum outcomes seem to require an external frame to resolve uncertainty. While the connection is conceptual rather than formal, the structural resemblance is striking — and it may hint at deeper principles worth exploring. Don't worry if this last point is a bit unclear, though - future posts will address this connection more fully!
Finally, there are interesting implications for why we conceive of space as 3 dinensional at all. Is it merely a computational simplification - arrived at because of nature's tendency toward efficiency? Is the fact that our own motion all evolved relative to one fixed constant (earth's gravity) perhaps a relevant factor?
Anyway, thank you for reading, and I hope I have given you something to think about - even if it is merely that complexity sometimes hides within the apparently mundane.
