## Orthogonality is harder to achieve as the number of explanatory factors increases

One of my favorite posts is “New Perspective on Unification” where I discuss the “horizontal” and “vertical” attributes.  Horizontal attributes are associated with collectivity and multiplicity. Vertical attributes are about individuality, individual histories and individual characteristics. In that post I also claimed that all entities are mixtures of vertical and horizontal attributes. All entities exhibit individuality but they also belong to various groups.

Coupling disturbs orthogonality

Horizontal and vertical attributes are supposed to be orthogonal. But, in reality, they are not perfectly orthogonal. Horizontal/vertical attribute pairs cannot be perfectly independent. Loss of independence is sometimes expressed with the term “coupling” which disturbs orthogonality by definition.

The minimum number of explanatory factors is 2

In the pictures of horizontal/vertical pairs above we have 2 dimensions: 1) collectivity 2) individuality. The number 2 is the minimum number of explanatory factors (dimensions).

We sometimes refer to the source of all  – God – as the ultimate explanatory factor but it is clear that 1 explanatory factor cannot be explanatory. Unity – the perspective of unity – must be a spiritual experience. In the realm of the mind we have to have at least 2 factors to create the illusion of explanation. But even with 2 explanatory factors (dimensions) it is very difficult to assume independence between those 2 factors. It is clear that we play this mental game of “explanation” but deep down we know that our 2 factors are not really independent. Do you see what I mean? We use Purusha/Prakrti (also known as Shiva/Shakti) or Yang/Yin pairs to explain things but we are not really satisfied by these mental explanations. We yearn for that spiritual experience known as the perspective of unity.

What happens if I increase the number of explanatory factors to 3

So, instead of explaining everything with 2 explanatory factors we can use the famous 3 explanatory factors  known as 3 gunas…right?

Yes, we can do that but my intuition tells me that these 3 fundamental explanatory factors are coupled pair-wise. In other words, those 3 fundamental explanatory factors (3 gunas) as they function in the physical universe are not totally independent. They may be independent in the stages prior to the physical universe. Spiritual philosophy claims that 3 gunas were independent in the “Nirguna” (attribute-free) phase of Consciousness.

In the physical universe there is a conservation law of 3 gunas.

$\mathbf{S R T = S^2 R^2 + R^2 T^2 + T^2 S^2}$

where $\mathbf{S}$ is the Sattvaguna, $\mathbf{R}$ is the Rajoguna, $\mathbf{T}$ is the Tamoguna. This equation makes the pair-wise coupling of gunas explicit.

Assuming S, R, T are continuous variables, the geometrical shape implied by the equation $\mathbf{S R T = S^2 R^2 + R^2 T^2 + T^2 S^2}$ is known as the Steiner’s Roman surface (the image shown below). It is called Roman surface because Jacob Steiner was visiting Rome when he came up with the idea of this geometrical shape.

Orthogonality is harder to achieve as the number of dimensions increases

The “explanation” game gets more and more difficult as we increase the number of explanatory factors (dimensions). I strongly recommend that you take a look at the post titled “What is Dimension?” at this point. In that post, I tried to explain that the terms “dimension” “explanatory factor” and “degree of freedom” are synonymous. I also briefly discussed dimension as space extension, dimension as elapsed time, fractal dimensions, and statistical factors as dimensions.

Statistical explanatory factors can be found in almost all models used in finance, economics, politics, sociology and many other fields such as climatology and artificial intelligence. In the statistical models we have multiple explanatory factors. We assume that those explanatory factors are independent of each other. Modelers (scientists) try very hard to come up with statistically independent explanatory factors.

It is common knowledge among the developers of statistical models that having more explanatory factors does not improve the predictive power of the model. Having more explanatory factors typically increases the R-square of your model which gives the appearance of more explanatory power but in reality you lose predictive power. The reason for this dilemma is the main point of this post: orthogonality of the explanatory factors diminishes as we add more explanatory factors. This means that some of those “new” explanatory factors are not really “new.” It is really difficult to find more and more independent explanatory factors. If you force the situation you end up with explanatory factors that are correlated with each other. You may then have the illusion of a better explanation because the R-square of your model is  higher but you should know that your predictive power will certainly be lower. This dilemma is known as “over-fitting.”

If there are additional spatial dimensions they will certainly be coupled

Scientists do not pay attention to my “3 fundamental variables are coupled pair-wise” arguments of course but they end up with similar results for the shape of their additional spatial dimensions. It becomes more and more difficult for them to find a unique shape because orthogonality is harder to achieve as dimensionality increases.

Here’s some of the infinite number of possible shapes

These shapes are known as Calabi-Yau manifolds of extremely tiny ($10^{-33}$ cm) curled-up extra spatial dimensions.