In an earlier blog, I wrote about a universe in which light did not move and everything else was moving. In our measured universe, light has zero mass and scale invariance. Light, as it happens, has many of the interesting properties of unparticles in what has been called the unparticle world by Professor Howard Georgi of Harvard University. Georgi's ideas may help cosmic model builders propose an explanation for some of the real mysteries afflicting the explanatory power of modern physics.
What is interesting about the speculations concerning unparticles is that the unparticles proposed are scale invariant. The unparticles would be matter, but they would behave under very different rules from what we experience in our particle intensive universe.
If we search for such matter, we have a problem detecting it because it does not appear as particles. However, unparticles may influence particles and it is possible that we may detect this influence by means of powerful Large Hadron Colliders, and the like. Physicists are puzzled by their inability to detect the source of much of the energy and mass in the universe.
Scale invariance is an attribute that relates to laws governing the length or energy of an entity under different scales. It has different connotations depending on what aspect of physics is under review.
Curves can display scale invariance and one that is often found in nature is that of the logarithmic spiral. If the spiral has very specific and unusual properties, it is a golden spiral. The properties of the golden spiral are not that common in nature and don't add any particular advantage.
The mathematics of fractals has scale invariance as an important contribution to our understanding of natural phenomena.