# Fractal Symmetry & Noether's Theorem: Chaos & Conservation

• Michele Zappano
In summary, Steven Strogatz discusses the self similarity of fractals as a form of symmetry in his book. It is important to differentiate between physical and mathematical symmetries. Noether's Theorem states that continuous symmetries in a system lead to conserved quantities. It is unclear if the symmetry in fractals is continuous and physical. However, it may be related to conformal symmetry, which has been linked to phase transitions and the generation of a conserved energy-momentum tensor.

#### Michele Zappano

I was reading a Steven Strogatz book and he said that the self similarity of fractals is a symmetry. Has any conservation law been linked to this type of symmetry using Noether's Theorem?

Michele Zappano said:
I was reading a Steven Strogatz book and he said that the self similarity of fractals is a symmetry. Has any conservation law been linked to this type of symmetry using Noether's Theorem?

One needs to distinguish between physical symmetries and mathematical symmetries.

Noether's Thm can be stated informally as:

If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

I haven't read the Strogatz book, but two questions that come to mind are:

Is the purported symmetry in fractals a continuous symmetry?

Is the purported symmetry in fractals a physical symmetry (really a perfect symmetry in a physical realization)?

A lot of purported symmetries in physical systems are neither exact nor continuous.

• Michele Zappano
Michele Zappano said:
I was reading a Steven Strogatz book and he said that the self similarity of fractals is a symmetry. Has any conservation law been linked to this type of symmetry using Noether's Theorem?

This is at the edge of my understanding: scale invariance (self similarity) is associated with conformal symmetry:

https://en.wikipedia.org/wiki/Conformal_symmetry

These are typically associated with phase transitions, but I think you can also generate a conserved energy-momentum tensor from this.

• Michele Zappano
Andy Resnick said:
This is at the edge of my understanding: scale invariance (self similarity) is associated with conformal symmetry:

https://en.wikipedia.org/wiki/Conformal_symmetry

These are typically associated with phase transitions, but I think you can also generate a conserved energy-momentum tensor from this.
perfect, thanks

## What is fractal symmetry?

Fractal symmetry refers to the self-similarity of patterns at different scales within a fractal. This means that the same pattern or shape can be found at different levels of magnification, creating a visually complex and intricate structure.

## What is Noether's theorem?

Noether's theorem is a fundamental principle in physics that states that for every differentiable symmetry of a physical system, there exists a corresponding conservation law.

## How does chaos relate to fractal symmetry?

The concept of chaos is closely related to fractal symmetry because both involve the repetition of patterns at different scales. In chaotic systems, small changes in initial conditions can lead to vastly different outcomes, just as small changes in the scale of a fractal can result in a completely different pattern.

## What types of systems exhibit fractal symmetry?

Fractal symmetry can be found in a wide range of natural and man-made systems, including coastlines, snowflakes, clouds, trees, and even financial markets. It is also observed in mathematical equations and computer-generated images.

## How does Noether's theorem apply to the conservation of energy?

Noether's theorem shows that the conservation of energy is a result of the symmetry of time in physical systems. This means that the laws of physics remain the same regardless of when an event occurs, leading to the conservation of energy over time.