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?
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?
perfect, thanksAndy 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.
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.
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.
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.
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.
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.