Possibly, we are the first ones seeing proof of an undiscovered fundamental economics law by our own eyes, but no one realizes it yet.

But before I explain an idea, here is the Mandelbrot set: the most known example of a fractal: a shape, that repeats itself endlessly on any scale:

Fig 1. Mandelbrot set

The mathematical law behind this image was discovered in 1905, but the image itself was published only in 1975, when computers became available. Before computers, it was still possible to calculate several “pixels” of the image manually, but it was not possible to see the whole shape of the set and realize it’s fractal nature. While the math fundamentals were still the same behind this image, they were invisible for a naked eye.

It is known, that trading charts are noisy and unpredictable. But what if there is also a fundamental mathematical law behind any price chart that is invisible for a naked eye due to a noise? Possibly, Bitcoin chart gives us an ability to see this fundamental economical fractal clearest than ever before.

Bitcoin reduces the noises hiding the fundamental fractal shape due to the flowing properties:

Fair exchange. An ability to see any historical charts and buy/sell by a fair market price. The Bitcoin is a unique resource mostly traded this way. The number of people involved in trading. The more people involved, the less is the noise. The number of people involved in Bitcoin trading is huge and grows, so the fractal shape becomes more and more obvious over the time. Stable environment. For example, oil prices could be affected by wars, political decisions or all-electric vehicle’s success corrupting the all-time price chart fractal shape. The whole Bitcoin’s lifetime was relatively stable regarding to major shocks and USD inflation rate. But Bitcoin has an internal major change: the halving. The fractal produced in this conditions is still valid, but it has a unique property: the chart not just represents stable conditions (where the fractal shape could be less obvious and more noise-driven), but represents a response to a periodic sharp step external conditions changes, something never seen before in economics! Bitcoin is also fully speculative resource, it has no volatile price component related to it’s real-world usage or supply sources (like gold).

Let’s look at this chart. This is the price of Bitcoin on a logarithmic scale.

FIg 2. Bitcoin all-time prices, logarithmic scale.

It looks surprisingly periodic, doesn’t it? But what if this is actually the outlines of the shape of an undiscovered economics fractal, like the edge of the Mandelbrot set?

Fig 3. Mandelbrot set edge.

Regardless the early days pricing of Bitcoin was a complete noise due to a low number of traders and an absence of historical price charts, the price chart self-organized into this fractal structure. While sources of noises were gradually reduced over time making the fractal structure more detailed and appeared.

But what about a more thin structure of this fractal, how does it look like? Possibly, we’ve also already seen it for the first time ever. Let’s look at the Bitcoin price full cycle (2017–2019):

Fig 5. Bitcoin price full cycle 2017–2019 outlined.

Do you see this distinctive wave pattern (outlined in purple)? This looks somewhat similar (but not identical, as far as this is a fractal, it’s true shape should be much more complex, it is just a basic shape discovered) to a LC circuit response to a pulse input (or many similar physical phenomena):

Fig 4. Step input response in physics.

But for a Bitcoin, we have the same pulse input: the halving! Since this looks like a full price cycle between halvings, possibly this shape is even more fundamental than all-time chart (Fig. 2), because it is the full cycle instead of a part of it.

There are two zones (marked as “A” and “B” on (Fig. 5)) which seems to break this outlined shape. But it could be explained:

Once the fractal reaches the area where the amplitude of osculations decreases, the fractal becomes more sensitive to noises and “butterfly effects” (the same story with flag-like technical analysis patterns) making a noise-driven transition more likely (the area “A”). But the area “B” is a reaction to an area “A” changes, and it is exactly the same fractal, as the outlined area. The attractor is there on any scale, and the system is trying to return to it on any scale after any noise-driven changes.

Let’s try to outline the Mandelbrot set edge in a similar way (this shape is perfectly detailed because there are no noises at all, just a pure math):

Fig 5. Mandelbrot set edge outlined.

So there could be also much more details in the true price chart fractal shape than outlined on the (Fig. 5), but we already can see the basic principles and building blocks.

Let’s take a look the same 2017 -2019 Bitcoin chart on a logarithmic scale:

Fig 6. Bitcoin price cycle 2017–2019, logarithmic scale.

Doesn’t it look like that the all-time chart (Fig. 2) is just a part of this full cycle halving? This is making the fractal assumption complete: the same shape both on the high and the low level. The shapes similar to (Fig. 5) are known in technical analysis as flags, wedges: so even in noisy real-life trading, special conditions frequently appear to make this fundamental shape visible on a chart.

So the final assumption is, that in ideal conditions, with a periodical changes of a part of environmental conditions, the price chart of any resource is a perfect fractal. The shape of the fractal depends on:

The properties of stable conditions. The properties of conditions that change periodically.

As an example, there are two images of Julia Set:

Fig 7. Julia set A.

Fig 8. Julia set B.

These fractals looks different, but they are both true fractals and share the same mathematical foundation, just having partially different environmental conditions. In a similar way, Bitcoin price chart fractal could be different in a “parallel universe” with different having settings or with different world economics conditions. Possibly, “flags” and “wedges” found on detailed chart prices, are just fractals with different stable conditions.

So, in real-life conditions: