The general online perception of fractal cosmology appears to be misguided. A case in point is a photograph captured by Hubble, which depicts one of the oldest star clusters in the universe, this globular cluster has been around for 12 billion years, almost as old as the universe itself.

Over the years, during my online browsing, I frequently encounter various discussions in which people share images related to fractals. Typically, these discussions involve asking simple(ish) questions regarding those fractally patterns.

"Does this image display fractal-like geometry?"

## The Problem

Frequently, I observe a considerable number of people misinterpreting the question. A Google search for "is the universe a fractal" often yields responses asserting that it is not, which is true. However, the distinction lies in our focus – we are not discussing whether the universe is a true fractal, but rather if it exhibits fractal-like qualities. It's crucial to note the explicit wording of the question: "Does this image display *fractal-like *geometry?"

## What is the correct response?

So, the source of confusion lies in the common assumption that the question pertains to true fractals. Consequently, responses frequently categorize celestial objects as non-fractals. And this is why the online perception of fractal cosmology and fractals in general is misguided.

So, the image does include fractal-like properties such as the following:

**Limited Self-Similarity:** Star clusters often exhibit self-similarity over a limited range of scales. If you zoom into portions of the image, you may find that the distribution patterns of the stars resemble the overall cluster, but this similarity won't continue indefinitely as you zoom in further. A limited range of self similarity is key here.

**Statistical fractals****:** And so, fractals that show limited self-similarity are known as statistical fractals. The distribution of stars might not form a perfect geometric fractal, but it can be considered a statistical fractal. This means that the statistical properties of the distribution, such as the density and spacing of stars, remain somewhat consistent at different scales.

**Hierarchical Structures:** Star clusters and galaxies often form hierarchical structures that can be reminiscent of fractal patterns. Smaller clusters may be grouped within larger clusters, and those in turn can be part of even larger structures, as evidenced by our globular cluster, Messier 15.

In contemplating the universe exhibiting genuine fractal properties, the absurdity and amusement arise from the potential bizarre implications. As mentioned earlier, the inherent nature of fractal-like structures offers limited self-similarity rather than infinite self-similarity. It's amusing to envision a scenario where, instead of zooming from galaxy to stars to planets to atoms and subatomic particles, there would be galaxies within galaxies, solar systems within solar systems, and so forth ad infinitum.

## The Evidence

Physicists have analysed data from the Sloan Digital Sky Survey (a massive mapping effort of galaxies). Their analysis suggests that the universe might exhibit fractal distribution of matter up to scales around 100 million light-years. This implies that the way galaxies cluster may have a self-similar pattern when observed at different scales of zoom. Notice how the research mentions up to scales of around 100 million light years, so can be classed as limited self-similarity.

### Finite, Not Infinite

So, the research suggests fractal-like patterns in galaxy distribution of up to a scale of 100 million light-years, implying that beyond that scale, the self-similarity might break down, or a different structure emerges.

This limited range of scales is why we classify it as 'limited self-similarity'. The universe might be approximately fractal-like for a certain range, but not across every possible scale.

Now that we have pondered this, we can explore true and fractal-like geometry in more detail.

## True fractal vs Fractal-like.

An example of a fractal-like structure is a coastline. When observed at various scales, the complex details and patterns of the coastline exhibit self-similarity. However, this self-similarity is limited and doesn't continue indefinitely. The complicated details and irregularities of a coastline contribute to its limited self-similarity, making it a notable example of this type of fractal structure.

In other words, in simpler terms, the overall "roughness" or irregularity of a coastline remains statistically similar whether you zoom in on a small section or consider its full length. This doesn't mean every little inlet replicates perfectly at different scales, but the general level of complexity is similar.

Although coastlines show this fractal-like behaviour, it occurs within a certain range of scales. Huge continental outlines don't look exactly like the shape of a tiny cove when zoomed out, hence fractal-like patterns can only be observed at a limited range of scales, so it must be classified as having limited self-similarity.

**As a result of this fractal nature, there is a paradox called the coastline paradox**, in which the length of a coastline varies depending on the scale of measurement. With finer details captured, the coastline appears longer as the measuring scale decreases, i.e., when viewed from different perspectives, such as from a bird's eye perspective or on foot.

**Trees are also a great example of approximate fractals. **Much like the branching pattern of a tree, where the trunk divides into large branches, which in turn split into smaller branches, extending down to twigs—a classic example of a fractal-like structure—similar principles are hypothesized to manifest within the concept of eternal inflation.

Imagine commencing with a single universe, akin to the tree's trunk, and envision a process of self-replication where each universe gives rise to twelve more. With each iteration, this cascade grows, expanding to 144 mini-universes in the first round, and then to 1,728 mini-universes in the next round, and so and so forth. Sorry, digressing here.

**Self-Similarity (But not exact copies):**

Miniature Trees: Each major branch, with its own subdivisions, looks like a miniature version of the entire tree. While not every branch is identical, there's a visually recognizable similarity in how the branching pattern repeats across multiple scales.

## True Fractal: The Mandelbrot Set

On the other hand, the Mandelbrot Set is, of course, an iconic example of a true fractal. As you zoom into different regions of the Mandelbrot Set, the intricate patterns and shapes you encounter echo at varying scales infinitely. This characteristic is known as quasi-self-similarity.

**Another true fractal is the Koch snowflake.**

The Koch snowflake begins with an equilateral triangle. In each step, you divide every straight line segment into thirds, and build a smaller outward-pointing equilateral triangle on the middle third. Then you erase the base of that new triangle. Repeating this process infinitely creates the Koch snowflake.

Perfect fractals cannot exist in the physical world, hence why we must categorise structures in nature as fractal-like.

**Strict Self Similarity:** This construction leads to self-similarity. No matter how much you 'zoom in', you see the same bumpy, triangular pattern repeating at smaller scales. The Koch snowflake is a classic example of a strictly self-similar fractal. Compare this to a natural example like a coastline. Even though coastlines exhibit a degree of roughness that repeats at different scales, it's not perfect. Zooming in reveals details and variations that aren't identical to what you see at a larger scale. This is "limited self-similarity".

The Koch snowflake showcases the distinction between fractal-like structures, which exhibit limited self-similarity, and true fractals, which display strict self-similarity at infinitely smaller scales.

The key difference lies in whether the copies are perfect or approximate. It's a spectrum, and some fractals might fall somewhere in between, but understanding the terms 'strict' and 'limited/approximate' helps classify the fractal's behaviour.

## Closing thoughts

Misunderstandings about fractals are all too common in online discussions, often leaving people wondering if the universe actually displays fractal-like patterns. But, let's face it, we don't blame them because fractals, well, they can be quite complex.

Pondering these distinctions, however, will significantly improve our understanding of fractal geometry as well as its manifestations in the cosmos, especially in the field of fractal cosmology, which Space Ponder advocates.

By pondering the subtleties of self-similarity and scale, we deepen our appreciation for the complex patterns that pervade both natural and cosmic phenomena. With this essay/article, our aim is to help dispel some of the misconceptions prevalent online. We appreciate your time here.

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