This release is dedicated to the Riemann zeta function and accompanies my “Meridian of Primes” series on the Riemann Hypothesis.

Back in 2004, while I was still in high school, I attended a guest lecture on RSA by a visiting mathematics professor from Palacký University in Olomouc. I don’t remember his name, but the lecture was so good that it pulled me deeper into prime numbers, cryptography, and the strange question of how mathematics can describe the distribution of primes.

What fascinated me at that point was that even after so many years, we still do not fully understand the pattern behind them (Riemann postulated the hypothesis in 1859). I filled a few journal pages with notes on the topic back then, and apparently never quite let it go.

At the same time, I was learning about complex numbers in school. They felt bizarre at first. Strange, abstract, almost artificial. And then I discovered that these weird imaginary constructions are deeply connected to the very real, very classical world of prime numbers.

So I decided to connect the topic of primes and complex numbers and start a new series on the Riemann Hypothesis. Not because I plan to solve it, but because I want to explore what it is, why it matters, and how the thinking around it works.

And because Synaptory Fractal Traveler had already grown beyond fractals, this update finally pushed me to rename it. It’s now called Synaptory Chaos Explorer, but the name changes... chaotically :)

Live App

The Riemann Zeta Mode

With version 2.2, Synaptory Chaos Explorer now includes a new mode for exploring the Riemann zeta function (ζ). You can now explore:

• A critical line overlay at Re(s) = 1/2

• The zeta path — the famous visual trajectory of ζ(1/2 + it)

• An analytic continuation for exploring zeta on the full complex plane

And most importantly, the release includes a guided audio-visual tour through 29 curated mathematical points of interest, including:

• Trivial zeros at the negative even integers

• Non-trivial zeros on the critical line, including Gateway Zero, Hardy’s Milestone, or Lehmer’s Phenomenon

• Special values such as the pole, the Basel problem, Apéry’s constant, or Ramanujan summation1.

• Historical landmarks, including Titchmarsh’s last zero or Turing’s last zero

Yes, 29 is a prime number.

Live App

What Else Is New in 2.2

Although this release belongs primarily to the Riemann zeta function, it also pushes the explorer further beyond its original fractal boundaries.

I have added a beta implementation of the Rössler attractor as an early preview of one of my future essays on chaos. It is still unoptimized and experimental, but already stable enough to explore if your machine can keep up. I have also upgraded the Mandelbrot engine with a new series approximation shader, which improves deep-zoom rendering by making extreme magnifications faster, cleaner, and numerically more stable.

Beyond that, version 2.2 adds a new Chaos Gallery for curated captures, smoother tours, richer overlays, expanded keyboard controls, better mobile interaction, internal rendering and architecture improvements, and a substantial set of bug fixes.

So while the zeta function is the headline, this release also marks a broader transition: the project is becoming less a single-purpose fractal viewer and more a growing laboratory for visual mathematics, chaos, and complex systems.

You can find full release notes here.

Give It a Try

• Live app: fractal.brnka.com/#zeta

• Source code & Wiki: github.com/rbrnka/fractal-traveler

Enjoy!

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On the strange atlas of the complex plane, where numbers become geometry and geometry becomes chaos, a unique meridian appears. It marks a boundary between what mathematics has proved and what it has only tested to extraordinary depth.

This essay is a guided walk toward that meridian. Just enough mathematics to see its outline, just enough philosophy to appreciate why it feels so uncanny, and just enough restraint not to pretend we have solved the mystery.

The line is simple, but the claim attached to it remains unproven.

It appears in the Riemann Hypothesis, one of the most famous unsolved problems in mathematics1. The claim is simple to state and extraordinarily hard to prove: all non-trivial zeros of the analytically continued Riemann zeta function lie on that line.

Behind that sentence sits a deeper conceptual shift. To even understand what the claim means, we have to leave behind the naive idea that a formula is only what its original series says it is. We have to talk about convergence, divergence, and one of the strangest moves in mathematics: analytic continuation.

If mathematics has something like an “event horizon,” this problem is one of its best candidates. Yet its proof remains elusive. Proving or disproving it would reshape parts of number theory and would echo far beyond it, including areas tied to prime distribution, computation, and mathematical physics.

You’ll need curiosity more than credentials. Some algebra helps. Comfort with the idea that math can be strange and still be rigorous helps more.

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The Complex Plane

Most people have heard of complex numbers. But what are they really?

We make a mathematically radical move. We say that there is a solution to this simple equation

by introducing a symbol that is not a real number, but whose square is −1. We call it i, define it as

and ask what algebra it forces into existence. The answer is the system of complex numbers, an algebraically closed field, richer than anything that had come before, extending real numbers this way:

Taming Infinity

The most beautiful atlas in the history of mathematics to visualize the extended complex plane2

is the Riemann sphere.

Imagine it as a sphere resting on the origin of the infinite complex plane, at the point zero. We connect every point in the plane to the sphere’s North Pole with a straight line. Where that line pierces the sphere’s surface, the image of the number is born.

Points near zero cluster at the South Pole. But the further you venture into the plane, the higher you climb upon the sphere. All paths to infinity, regardless of their direction, eventually converge at a single point: the North Pole.

This way, Riemann transformed an infinite, ungraspable expanse into a compact, closed object3. Infinity is no longer an abstract abyss; it is a concrete point on a map. And it is within this enclosed world, where algebra meets geometry, that the zeta function begins to reveal its deeper meaning.

Rituals of the Infinite Sum

Before we touch the zeta function, we need to talk about a kind of social contract mathematicians make with infinite series.

An infinite series is a promise that says: “If you keep adding, forever, you approach something.” Sometimes this is true in a way that feels natural. If the terms shrink fast enough, the sum settles down to a number. We say it converges. Sometimes it’s false in the weirdest way possible because the sum simply refuses to be a number, and the series diverges—explodes to infinity or keeps oscillating.

Let me begin with one of the simplest examples:

This is a geometric series S, and it only converges in the narrow interval between -1 and 1 described as:

At x=−1, the partial sums oscillate between 1 and 0. At x=1 they grow without bound. Have a look:

How to find its sum within the convergent interval? Instead of trying to add infinitely many terms one by one, we can manipulate it algebraically. Let’s multiply both sides by x:

and subtract from the original sum:

Everything cancels except the leading 1. Now, to get the sum to one side of the equation and the rest to the other:

Summoning the Right Spirit

We have found a closed-form expression for the sum, but only on the interval where the original series converges. The formula itself, however, defines a function on a much larger domain. Outside the convergence interval, that function still has values, but those values are not the sums of the original series:

How is this possible?

We don’t always treat divergence as failure. Sometimes we treat it as a coordinate change: a hint that we’re summoning the right spirit with the wrong ritual.

Let’s dive in and express the new formula as a function of a real variable x:

We intuitively feel that the function g(x) is somehow extending the S because while equivalent, it works on a broader domain. Let’s plot them both:

We can see that g(x) agrees with the series on the interval where the series converges, but remains defined on a wider domain4.

This is the basic idea behind analytic continuation: the series defines a function on one region, and that function can sometimes be extended beyond it.

Over the real numbers, many different functions can agree on the same interval and then diverge elsewhere. In the complex plane, analytic functions are much more rigid: if an analytic continuation exists on a connected domain, it is unique.

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Beyond Convergence

Let’s play with the g(x). For x=1/2, we get the same result as S. As expected, both give the same value inside the interval of convergence:

Let’s try some of the values outside of the S domain.

As we already know from the graph above, S diverges at +1 while g(x) has a pole there5. How about -1?

For x=−1, the series becomes 1−1+1−1+… Its partial sums oscillate between 1 and 0, so the series does not converge in the usual sense. The function g(−1), on the other hand, gives 1/2, which can be viewed as a generalized value associated with that oscillation6.

Here’s where things get uncomfortable. Let’s try a number 2.

At x=2, the S diverges. The continuation doesn’t. They’re built from the same definition, yet one refuses to answer and the other hands you a number.

This is where the distinction stops feeling like a technicality and starts feeling philosophically strange. The continuation isn’t lying. It’s speaking a different dialect of the same truth. Whether those dialects are equally “real” is a question mathematicians and philosophers still argue about over coffee.

Let’s plot them both:

We can see a very different behavior outside the domain of S. Where S does not converge to a sum, its analytic continuation yields a result (except at +1). The distinction between a function and the series that defines it locally is one of the conceptual keys to the Riemann hypothesis.

But what is the semantics of the number we’re getting? It’s surely not a series sum!

What we get is the value of a different object: a function that agrees with the series where the series converges, and continues beyond that region.

Same voice, wider range.

This is the threshold Euler stepped across in the 18th century, when he faced a problem that had resisted solution for nearly a century: the exact sum of the reciprocals of all perfect squares, known as the Basel problem.

If Euler found the heartbeat of these series, Riemann found their DNA.

In the next part, we will see how the Basel problem evolves into the Zeta function, a mathematical prism that refracts the chaotic sequence of primes into our meridian.

Thanks for reading!

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Back in 2001, I got my hands on the first fractal traveler program written in C by Jindrich Novy. There is no mention of this guy on today’s web, but back then, he was my hero. The program was compact, elegant, and worked surprisingly well, allowing very deep zooms on a Celeron CPU running at 500 MHz. You can find it here.

Since then, I attempted to create a similar experience several times but never got far, mostly due to a lack of motivation to dive into the deeper math of perturbation theory required to make it work beyond naive implementations. And sometimes, dreams come true.

So…

Last year, to kick off my chaos theory series, I built the Synaptory Fractal Traveler, a browser-based, accelerated fractal explorer for zooming into the beauty of Mandelbrot/Julia sets. It came with a whole backlog1 of ideas on how to push it further.

This winter, I finally sat down and burned down enough for the next version.

The quality of my notes let me put Claude Code to work this time. I hoped it would help me with some of the math-heavy challenges, but it caused more damage than clarity there. Where it did help a lot was test generation and a big chunk of the front-end work that would otherwise have taken me ages.

The App

The app is still not where I want it long-term. It’s compute-heavy and will happily spin up every fan your machine has, but it can also generate an absurd amount of beauty.

So after ~a year of iteration, I’m releasing a deeper, faster, more colorful, more polished, responsive, browser-ready version of my fractal zoomer for exploration, learning, and pure visual pleasure of mathematical structures.

If you’re into math, chaos, complexity, visual art, or you just like poking at things you don’t fully understand (but that look amazing), take it for a spin. I also dedicated a lot of time to preselecting some of the intricate structures for you and inventing creative names for them.

• Live App: fractal.brnka.com

• Source, Wiki, Release Notes: github.com/rbrnka/fractal-traveler

Coming

On the roadmap for the next stretch of bad weather: Zeta-function experiments and strange attractors. I’ve actually already rendered the Riemann zeta function in this version of the app (including its analytic continuation), but it’s hidden behind a secret hotkey because it’s not ready yet. If you want to see it, DM me.

Don’t forget to check out the original article on chaos paired with the app:

Enjoy!

This release is dedicated to Jindrich Novy, the author of the original Fractal Traveler from the year 2000, who has inspired me for years to build my own!

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When you think, for example, of a living city and inspect its pulse, you can see some regularities, periodicity, or patterns, but not all of them. You miss the context. You have no idea how the city works because it’s a dynamic system of individual trajectories of everybody and everything there. But if you could understand everyone’s path, the chaos would slightly shift toward order. One step closer but still missing information hidden deeper in the minds of every individual to understand the motivations and processes behind their decisions. Then another step. Their brains, the atoms they are made of, go down and down until you ultimately end up at a dead end of physics, with God or some other level of uncertainty. There will always be a little unknown left, but you will see many symmetrical patterns and self-similarities on the way.

This resemblance of chaos starts at the threshold of predictable structure towards unpredictable complexity. It is ultimately the level of order that transcends our simulation capabilities, meaning predicting the future state of such a system is impossible.

The Essence of Deterministic Chaos

I spent several years commuting by train, traveling to and from school for three hours daily. The journey took me through diverse landscapes and lush forests, where I often admired the intricate shapes crafted by nature. As I spent so much time looking out the window, I began to understand that the repetitive complexity of the structures I saw likely stemmed from something fundamentally simple. This reflection led me to uncover the concept of chaos and the fractal essence of nature. Later on, I studied the work of A. Lyapunov, R. Elliot, or E. Lorenz, and applied concepts of chaos theory in my thesis.

Let’s unveil the mathematical theory of chaos1. It describes non-linear systems that are highly sensitive to initial conditions, where even a minor change can lead to drastically different outcomes. It’s the profane butterfly effect.

Thanks to this sensitivity, a system can start behaving in a way that resembles randomness, but it is deterministic in the sense that it is formally defined, does not contain any random parameters, and if you start with the exact same initial conditions, you will get the exactly same behavior every time.

Even deterministic chaos poses a considerable challenge. You might think that since it is deterministic, it should be manageable with numerical methods. One hallmark of chaotic systems is their high dimensionality. That, in combination with a vast number of possible states, means these systems can shift between various stable configurations through what appears to be random fluctuations, significantly complicating simulations.

A good example is one of the early problems of celestial mechanics and the founding pillars of chaos theory. The Poincaré’s three-body problem. Its idea is to calculate and predict trajectories of three bodies interacting through gravitational force (e.g. three-star solar system) using Newton’s law of universal gravitation. Let me demonstrate the trajectories of these three bodies for you2:

All the bodies have the same mass, zero initial velocities, and only marginal differences in initial vector positions. The rest is done by gravity. This chaotic behavior comes out of three second-order differential equations that don’t have a general analytical solution and are entirely computed numerically.

Fabric of Reality

Now, we understand both notions of chaos. One represents overwhelming complexity, limited only by perception, and the other generates complexity out of simplicity. You surely feel they’re intertwined, and perception and computational capabilities are the keys to their common nature.

Chaotic systems are all around us and shape the reality as we know it. At one end of the spectrum lie simple systems with minimal information – for instance, a mathematical pendulum whose motion can be easily described and predicted. But even Earth’s spin will eventually turn the physical pendulum into a chaotic system.

At the other end are systems containing so many variables and interactions that even the most advanced models can only partially describe them or find limited solutions even for their mathematical representations. Climate, population dynamics, economics3, or even rhythms in biological processes may seem unpredictable, but they have underlying chaos principles at their core.

One notably challenging problem due to its complexity is fluid dynamics. You can imagine it as if you were looking out the window during a snowfall. You can see each snowflake floating beautifully on its unique path; some twirl gracefully with the wind, while others pause for a moment, almost as if they’re enjoying the view before being whisked away again. To truly grasp their behavior, we need an intricate model known as the Navier-Stokes equations. While these equations may explain a lot, they remain an unsolved mathematical problem and keep the prize of one million dollars waiting to be collected.4

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Self-Similarity and Expanding Symmetry

Historically, many mathematicians noticed that iterating on simple formulas in a complex plane could quickly generate unprecedented complexity, and when visualized, actual chaos could be seen! But all early attempts to draw this complexity failed. One of the simplest formulas is

Yes, it is the most famous one—the Mandelbrot set prescription. How does it work? The sequence must not escape to infinity for a complex number c to belong to the Mandelbrot set. Let me show you. Let’s set the initial value of c to a proven constant, like

It is a simple real number within the set, and its sequence converges instead of diverging. Now, to render the first few points in this set without a computer, you would go through this manual exercise:

As you can see, you need very precise decimal point arithmetic just to calculate the first four points. During each iteration, you must check whether the complex number c belongs to the set or not by examining the magnitude (absolute value) of zₙ against this condition:

If |zₙ| is greater than 2, the sequence diverges to infinity, meaning the c value is not in the Mandelbrot set M. It belongs to the set if |zₙ| is less than or equal to 2.5 This binary distinction tells us the basic representation of the point when rendered.

And why 2? It would work for any larger number, but 2 determines the minimal guaranteed escape radius and size of the disk surrounding the entire set.

Let’s use asterisks (*) for points in the set and dashes (-) for points outside the set as Benoît Mandelbrot did in the 1980s when he rendered probably the very first Mandelbrot set using a computer6. After hours of patient waiting, the result in the form of a vaguely ass-shaped blob finally appeared:

This grainy, low-resolution render was a doorway to a world of infinite beauty, where math and art collide—a world of fractals.

Let’s Travel Together!

If you’re curious about what the Mandelbrot fractal may look like today, I wrote a web app allowing you to travel the set. It is limited by the threshold of the 32-bit floating-point arithmetic in your GPU’s fragment shader and can’t go very far, but you can still explore some of the intricate structures. It also contains Julia mode to explore Julia fractal.

~ Synaptory Fractal Traveler ~

The pseudocode of the main method is very simple7:

def mandelbrot(c, max_iterations):
    z = 0
    for n in range(max_iterations):
        z = z**2 + c
        if abs(z) > 2:
            # Normalize the escape value for smooth coloring
            return n + 1 - log2(log2(abs(z)))
    return max_iterations

The values are normalized to achieve fancy coloring (faster escape = brighter colors, slower escape = darker colors). The app allows you to randomize the color palette, and I have also preprogrammed a few interesting zoom-ins you can admire. Let me know your thoughts, or share some of your zoom-ins!

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Compare the difference. Medieval mathematicians could only imagine a few points. In his first render, Mandelbrot saw a few hundred points of the set on a print corresponding to the two orders of magnitude. My app can do seven orders of magnitude before it pixelates. The best fractal zoomers can render the set at more than 20,000 orders of magnitude. Computing power allows us to push the limits and delve into previously hidden structures and patterns of chaos.

But Where Is The Chaos, Exactly?

With increasing precision and the number of iterations, we discover sets of points that do not change their behavior very often during the calculation but also sets of points that are very sensitive to changes. The boundaries between them create the fractals. That’s why the Mandelbrot set has a distinctive shape, and the Julia set changes its look rapidly with a small change of the c at certain points. It is always the coastline where fractal shapes appear.

Fractals are, therefore, another example of deterministic chaos. Even a simple equation can generate unpredictable complexity, yet it is always the same. Fractal geometry reveals that even seemingly disordered structures contain recurring patterns. Discovering its existence transformed my previous commuting insights into a genuine passion. Fractal shapes can be found in tree growth, cloud formations, river flows, or Romanesco broccoli, forming the world around us.

World Without Chaos

Although chaos may initially appear negative, you just learned that it is, in fact, an indispensable source of innovation and change. Systems that are overly rigid and orderly can easily become stagnant. Chaos introduces diversity, new possibilities, and unexpected connections. In evolution, chaos drives adaptation and the emergence of new species. Without it, humankind would probably not exist.

Chaos provides a space for new ideas and perspectives. It challenges us to think beyond the confines of established patterns and explore uncharted territories of thought and imagination. Psychedelic hallucinations often present themselves in fractal shapes, reflecting the structure of our brains and our thoughts.

Chaos forces us to confront the nature of our existence. It blurs the boundary between determinism and free will, suggesting that the universe is a dynamic interplay of patterns and unpredictability at the verge of our perception. In chaos, we find a mirror of human life itself: an intricate balance between order and spontaneity.

Thank you for reading! If you want to learn more about how chaos affects the world around us, what role prime numbers play in shaping reality, or more about fractals, subscribe for free to receive new posts and support my work.

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