A Bit of N-Dimensional Geometry Applied to Gardening

Disclaimer

The model I am using below is, of course, unacceptably simplified, somewhere at the level of a “medium-weighted spherical horse in a vacuum.” Nevertheless, even an unacceptable level of simplification sometimes helps to understand the complex processes that actually occur in nature, and from this point of view, this model is quite suitable as an illustration.
Also. I apologize in advance, but to understand further, a couple of semesters of higher mathematics under your belt wouldn’t hurt.

So, let’s start by estimating how many parameters need to be optimized to create ideal conditions for a plant.
At the very least, we will need:

1. Average minimum temperature in the coldest month
2. Average summer temperature
3. Daily temperature fluctuation in summer
4. Precipitation levels for the 4 seasons
5. Light levels in the warm season
6. Soil acidity
7. Soil’s ability to retain moisture
8. Content of at least the basic 5 elements in the soil
9. Air humidity

So, taking into account the multiplicity of points 4 and 8, we end up with no less than 16. This will be the dimension of our “configuration space.” For simplicity, let’s normalize the axes so that the entire range of life-suitable conditions, along any of the axes, fits within the range from 0 to 1. Let the metric be the standard Euclidean (spherical horse in a vacuum, as promised).
Now, again for simplicity, let’s assume we need to find a planting spot for a cactus, whose life optimum is at the origin, the “green” zone not further than 0.3 from the origin, the red zone further than 0.6, and the yellow zone in between. As you have already understood, we have three 16-dimensional spheres in front of us.
Let’s say we are very lucky, and in the garden, we found a spot where all the listed parameters are in the green zone: from 0.15 to 0.25. On average, 0.2. Where will our cactus end up with these parameters?
In other words, at what distance from the origin will the point in our 16-dimensional space with coordinates 0.2 along all axes be located?
The correct answer is √16×0.2² = 0.8 (check it yourself). In other words, far into the red zone. At first glance, N-dimensional spaces with Euclidean metric may seem very similar to our familiar 3D. In many ways, that’s true. But there are far from obvious nuances. In particular, as the dimensionality increases, the main volume of the sphere rapidly shifts towards the periphery. For a sphere of radius 2, for example, with N=2, 1/4 of the volume lies at a distance of less than 1 from the center.
For N=3, it’s already 1/8, for N=10 — less than 0.1%!!
And for N=16, the ratio is extreme.
This is what the article is about.
Note that with a random choice of initial parameters, the probability of hitting the zone of optimal growth with N>10 is practically zero. So a professional who makes a conscious choice will quickly outperform an amateur who essentially rolls the dice when making decisions.

Moral

When the efficiency of a system depends on many parameters, even a relatively small deterioration in each of them can throw the system far into the red zone relative to the optimum.
This happens quite often with plants.
It seems like everything is good in terms of parameters, soil, water, fertilizers, but the end result is a solid C.

By the way, the same goes for celebrities. A clone may look very similar to the original, but slight deviations immediately throw it into the red zone of fans’ interest.

Moral 2

Very often, and perhaps always, it is easier to match the optimal plant to a specific location than to try to adjust the microclimate and soil parameters to an already planted one. Think twice before planting something you are not 100% sure about.
And use microclimate sensors, precise knowledge greatly facilitates such decisions.