I literally just found this article and it’s another fascinating take on Bitcoins behaviour over the last 15 years.

It gets into Fractals and examples of Scale Invariance in nature before a discussion of Bitcoin included below:

“In the context of the Bitcoin Power Law, scale invariance suggests that the mathematical relationships governing Bitcoin's market dynamics remain consistent regardless of the time frame or market size.

Since its inception, Bitcoin's price has increased by eight orders of magnitude. Initially, Bitcoin was virtually worthless, with early transactions valuing it at less than $0.01 per Bitcoin.

For example, in 2010, the famous "Bitcoin pizza" transaction valued 10,000 BTC at approximately $25, equating to a price of $0.0025 per Bitcoin.

In contrast, Bitcoin's price has reached highs above $73,000 in 2024, an incredible growth highlights the cryptocurrency's rise, aligning with the principles of power law behavior.

In the graph below Giovanni applies the principle (pedagogically) to the prediction Harold Christopher Burger made 5 years ago using the Power Law (blue dots). 5 years later the prediction turned out to be correct (red dots).

(Bitcoin Power Law corridor predicted in 2019)

You can see one could have used scale invariance to make the prediction (he did so indirectly assuming the path continued).

Scale invariance is used all the time in science to make predictions.

This means that the patterns and behaviors observed in Bitcoin's price and network metrics can be analyzed and understood across different scales, making it possible to predict long-term trends based on historical data. “