1 – Empty Circles: The Geometry of High Dimensions
In high-dimensional spaces, most of the volume of a sphere lies on its “surface,” not in its interior. As the number of dimensions increases, the “interior” of the circle or sphere becomes insignificant compared to the total volume of the hypersphere. Essentially, circles “empty out”!
This has important implications:
- The average distance between points in high dimensions tends to increase, and nearly all points are far from each other.
- The intuitive concept of “closeness” breaks down. Everything lies on the edge.
2 – Normal Distribution and Volume Tending to Zero
In a multivariate normal distribution, most of the probability volume moves away from the center and concentrates in the “tails” or edges of the dimensions. As the number of dimensions increases, the effective volume of the probability space becomes so dispersed that the total volume approaches zero.
This happens because the “density” decreases exponentially with the number of dimensions, while the space grows disproportionately.
In other words:
- Normality (the concentration around the mean) ceases to be meaningful in high dimensions.
- This phenomenon is known as the curse of dimensionality.
3 – The Philosophy of the Problem
All of this teaches us a lesson: when we increase dimensions without considering their meaning, our conceptual and mathematical tools begin to fail. Pretending that everything will remain “normal” in a complex, high-dimensional space is an illusion.
Practical interpretation: It is crucial to carefully select the dimensions that truly matter, so that models or problems make sense in the real world
4 – Ideas
Something equivalent is sufficient and necessary. But what is sufficient, and what is necessary?
That is where we define efficiency. Without care, it can devolve into greed; but when done properly, it reveals itself as evolutionary efficiency. Nothing is perfect—things are simply done with meaning, as they emerge. Otherwise, a not-polinomial of non-sense arises.
5 – Sources
- Veritasium by Derek Mulller [youtube]
- Stand-Up Maths by Matt Parker [youtube]