# Rich "isotropic tensor" concept

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• apostolosdt
In summary, the conversation discusses the concept of isotropic tensors in physics and the discovery of the Motzkin sum numbers, which are a sequence of numbers that follow a specific recurrence relation. The conversation also mentions a generating function that produces a polynomial series with coefficients that are aesthetically pleasing. Additionally, the conversation references a documentary on the debate surrounding the use of math to describe nature.
apostolosdt
My field is physics and I'm very cautious about the "math describing the Nature" attitude, but I can't help admiring the deep richness of mathematics!

The other day, I was checking about isotropic tensors. An isotropic tensor keeps its components in all coordinated systems transformed under rotation. Then, unexpectedly, I came across some beautiful remarks on the topic. Here are some of them.

How many isotropic tensors exist? There are only single rank-0, rank-2, and rank-3 tensors, respectively, a scalar, the Kronecker ##\delta^{ij}##, and the permutation symbol ##\epsilon_{ijk}##. There are no rank-1 isotropic tensors, that is, vectors.

Now, if one attempts to enumerate all the isotropic tensors, starting with the ones above and going to higher ranks, one gets the sequence:
$$1, 0, 1, 1, 3, 6, 15, 36, 91, 232, {\rm etc.}$$
These numbers are called ##{\it Motzkin\,sum\,numbers}## and obey a recurrence relation:
$$a_n = {n-1\over n+1}\left(2a_{n-1} + 3a_{n-2}\right)$$
with ##a(0) =1, a(1) = 0##.

There is also a generating function:
$$G(x) = {1\over 2x}\left(1 -\sqrt{1-3x\over 1+x}\right)$$
that produces the following polynomial series:
$$\sum_{n=0}^\infty a_n\, x^n = 1 + x^2 + x^3 + 3x^4 + 6x^5 + 15x^6 + \cdots$$
No need to draw your attention to the coefficients of the powers! Plain beautiful!

Last edited:
jedishrfu
To answer your question of "math describing nature" debate, you might enjoy this documentary featured on NOVA some years ago:

## 1. What is a "rich isotropic tensor" concept?

A "rich isotropic tensor" concept refers to a mathematical concept used in physics and engineering to describe the properties of materials. It is a tensor that is both symmetric and isotropic, meaning that it has the same properties in all directions.

## 2. How is a "rich isotropic tensor" different from a regular tensor?

Unlike a regular tensor, a "rich isotropic tensor" has the same properties in all directions. This means that it is not affected by changes in orientation or direction, making it a useful tool for describing the behavior of materials under various conditions.

## 3. What are some applications of the "rich isotropic tensor" concept?

The "rich isotropic tensor" concept is commonly used in materials science, mechanics, and fluid dynamics. It is often used to describe the properties of materials such as metals, polymers, and fluids, and to model the behavior of these materials under different conditions.

## 4. How is a "rich isotropic tensor" represented mathematically?

A "rich isotropic tensor" is typically represented by a matrix, with each element corresponding to a different property or component of the material. The matrix is symmetric, meaning that the elements are arranged in a symmetrical pattern around the main diagonal.

## 5. What are some limitations of the "rich isotropic tensor" concept?

While the "rich isotropic tensor" concept is useful for describing the properties of many materials, it does have some limitations. It cannot fully capture the behavior of anisotropic materials, which have different properties in different directions. Additionally, it may not accurately describe the behavior of materials under extreme conditions, such as high pressures or temperatures.

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