A proposal for a Fractal Catalog

The catalog proposed here is by my colleague, Vincenzo Manto.

GitHub - VincenzoManto/Fracta: A Python lightweight domain-specific-language to define, plot and export fractals. · GitHub

Abstract: This article proposes the creation of an Online Encyclopedia of Fractals (OEF), a comprehensive, standardized repository for cataloging the infinite diversity of fractal geometries. The OEF would utilize a novel, line-oriented domain-specific language called Fracta to encode fractal generation algorithms in a compact, human-readable format. The article also addresses the challenges of duplicate detection and mathematical mimicry in fractal classification, proposing automated visual fingerprinting and empirical dimension estimation techniques to ensure a clean, non-redundant database.

How we could catalog the infinity of fractals

If you’ve spent any time down mathematical rabbit holes, you are probably intimately familiar with Neil Sloane’s Online Encyclopedia of Integer Sequences (OEIS). It is an absolute treasure trove for discrete numbers. James and I have often published there. Cataloging integer sequences is quite simple from a CS perspective. But when I wondered if there was a similar repository for fractals, the answer was difficult to find. There are a few scattered databases, but they are mostly image-based and lack any standardized way to index the underlying mathematical formulas.

Historically, our approach has been pretty messy. We usually dump lossy PNGs into web archives or write fragmented scripts in Python, C++, or Mathematica that don’t talk to each other. Because the underlying algorithms are hidden behind different coding styles, it is incredibly difficult to tell if two different equations actually map to the exact same geometric shape.

By introducing a standardized, ultra-lean language called Fracta alongside a strict data schema for a possible future online encyclopedia of fractals (OEF?), we might get a clean, plaintext index for infinity.

The first step: define a language to rule them all

The core issue with fractal generation is syntactic bloat. To fix this, I thought to formalize a line-oriented Domain-Specific Language (DSL) to be read easily by both humans and compilers. I called it Fracta.

Instead of forcing developers to write custom memory allocation scripts or complex graphic rendering pipelines, with Fracta I tried to unify the three main pillars of fractal generation under a single ENGINE directive:

• L_SYSTEM: String-rewriting engines that pass commands to a stack-based “turtle” graphics tracker.
• PIXEL: Escape-time formulas evaluated over a continuous complex plane (think classic Mandelbrot or Julia sets).
• IFS (Iterated Function Systems): Stochastic linear transformations used to compute random affine mappings, like the classic Barnsley Fern.

When dealing with L-Systems, Fracta uses an elegant, deterministic state vector tuple, $S=⟨x,y,\theta ,\text{Stack}⟩$, where the turtle’s radial heading $\theta$ initializes at exactly ${90.0}^{\circ }$ (pointing straight up). As the compiler parses tokens like F (move forward and draw), + (rotate right), or [ (push state to stack), it traces exact vector trajectories without any platform overhead.

Here is a practical look at what the syntax looks like for a classic dual-state rewrite system:

# Configuration for the Heighway Dragon Curve
ENGINE L_SYSTEM
AXIOM FX
RULE X -> X+YF+
RULE Y -> -FX-Y
ANGLE 90
ITER 10
RENDER
                                                                     

Fracta definition is not obviously the only way to do this and may not even be the best. It’s my first attempt at a clean, human-readable format that can be easily parsed by a compiler, and it might evolve as and if the OEF project develops.

Solving the “equivalence problem”

If we want to build a true encyclopedia of fractals, we need to solve the problem of duplicates. The basic problem is: How do we know if two different mathematical formulas are actually describing the same geometric shape? This is what I call The Equivalence Problem.

So we need to sort it out.

From my basic knowledge, in fractal geometry, completely disparate mathematical frameworks can converge on identical geometric attractors. You could write an L-System or an Iterated Function System that look entirely different on paper, but produce the exact same spatial point-cloud. If you only index the source code or an image file hash, you will never realize they are actually duplicates.

The OEF registry must thus handle this by passing every submission through a background rendering matrix to extract its physical geometric footprint, testing it via, at least, two automated steps I identified:

1. Visual Fingerprinting via Jaccard IoU

The system should bound-normalize the resulting point-set and map it to a discrete binary grid ($M\in \{0,1{\}}^{64\times 64}$). It then calculates spatial occupancy similarity between a new submission $A$ and an existing entry $B$ using the Intersection over Union (IoU) index:

$IoU(A,B)=\left(\frac{\mathrm{\mid }A\cap B\mathrm{\mid }}{\mathrm{\mid }A\cup B\mathrm{\mid }}\right)\times 100$

If the visual correlation hits $75\mathrm{\%}$ or higher, the system triggers a deeper mathematical comparison. This threshold is a balance between catching true duplicates and allowing for minor variations in rendering or scaling. I tested with a few known fractals and found that this threshold effectively captures identical shapes while minimizing false positives, but it can be adjusted as the database grows.

2. Empirical Hausdorff Dimension Estimation

To check if the shapes truly scale the same way, the compiler estimates the Minkowski-Bouligand (Box-Counting) dimension. It segments the footprint matrix across decreasing box scale-factors ($ϵ$) to find the precise fractional density of the shape:

$D_H={\lim }_{ϵ\to 0}\frac{\log N(ϵ)}{\log (1\mathrm{/}ϵ)}$

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How to handle “mathematical mimicry”

This is where things get wild. What happens if a submission looks exactly like an existing fractal, but its scaling behavior is fundamentally different? This is what I call Mathematical mimicry. It’s the phenomenon where two different mathematical formulas can produce visually identical shapes, but their internal density and scaling properties diverge. I think this is the worst problem for the OEF, because it can lead to a flood of near-duplicates that are technically different but practically indistinguishable.

I tried to develop it using a conditional conflict resolution routine:

Let $\mathrm{\Delta }D=\mathrm{\mid }{D}_H(\text{Submission})-D_H(\text{Existing})\mathrm{\mid }$

• If $IoU\ge 75\mathrm{\%}$ AND $\mathrm{\Delta }D\le 0.15\to$ REJECT (It’s a duplicate or scale variant).
• If $IoU\ge 75\mathrm{\%}$ AND $\mathrm{\Delta }D>0.15\to$ ACCEPT AS ANOMALY.

It should prove that two entirely different mathematical formulas can construct a visually similar spatial footprint, yet exhibit completely divergent internal densities. Finding these anomalies is a massive deal for chaos theory research.

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What’s next?

By reducing complex mathematical systems down to deterministic, compressed text files that can be instantly evaluated, cross-referenced, and queried, the OEF standard is doing for geometry what the OEIS did for number theory. It’s a massive step forward for computational mathematics, plotting hardware, and graphics engines alike.

I’ll be keeping a close eye on the repository as the database populates. Check out the Fracta documentation and take a look at the Repo of Fracta compilers and renderers.

Highlights from Magnifica Humanitas, the first encyclical letter from Pope Leo XIV

 

Highlights from Magnifica Humanitas, the first encyclical letter from Pope Leo XIV - Excerpts for young adults

Below is a collection of excerpts from the encyclical letter. I selected these with the hope of encouraging the reader to return to the encyclical letter in its entirety. The full text can be here: Encyclical Letter of His Holiness Leo XIV Magnifica Humanitas (15 May 2026). Note: numbers appearing before the excerpts indicate the paragraph number of the text (in most cases, only a portion of the paragraph is given). Numbers appearing in brackets, e.g. [123], are links to the references (the letter’s biography includes 224 references). Excerpts chosen by James C. McMahon.

 

INTRODUCTION

4. In recent years, it has become increasingly evident how rapidly and profoundly digitalization, artificial intelligence (AI) and robotics are transforming our world. Technology should not be considered, in itself, as a force antagonistic to humanity. On the contrary, it has formed part of our history since the beginning as “a profoundly human reality, linked to the autonomy and freedom of man.” [5] Over the centuries, technological development has significantly improved the living conditions of humanity.

Two biblical images

7. The story of Babel appears in the Book of Genesis. After settling in a plain in the land of Shinar, the people decided to build a city and a tower “with its top in the heavens” (Gen 11:4). Fearing being scattered across the earth, they sought to guarantee stability and power for themselves, and above all to “make a name” for themselves

8. The Book of Nehemiah - After the Babylonian exile, a portion of the people returned to Jerusalem, but the city was still in ruins, the walls collapsed and the gates burned (cf. Neh 1–2). 

The narrative shows how the city is reborn, not through the initiative of one man, but through the shared responsibility of all: men, women, priests, artisans, heads of households and young people all play a part. It is an undertaking with God at the center, which rebuilds relationships before rebuilding with stones. 

CHAPTER ONE, A DYNAMIC APPROACH FAITHFUL TO THE GOSPEL

Social Doctrine as a shared discernment

25. What matters most is not occupying positions of power or defending cultural strongholds, but initiating good processes and enabling them to mature.  In this way, the truth of the Gospel is not imposed from above, but grows over time within the concrete interweaving of lives, communities and cultures. This is not a truth that fears diversity, but instead welcomes and guides it. It does not eliminate conflicts, but transforms them, reuniting that which history tends to scatter.

CHAPTER TWO, FOUNDATIONS AND PRINCIPLES OF
THE SOCIAL DOCTRINE OF THE CHURCH

The principle of subsidiarity

68. The principle of subsidiarity stems from the very same understanding of the human person that has guided our reflection on dignity and the common good. If every woman and man is called to take ownership of his or her own life and to contribute to the formation of society, then social institutions must also respect and support this responsibility. The Social Doctrine of the Church refers to subsidiarity as the principle according to which the role of individuals, families, local communities and intermediary organizations should not be supplanted by higher-level authorities. Moreover, higher-level institutions must recognize, protect and promote the freedom and creativity of lower-level entities, coordinating their contributions so that they can cooperate effectively for the common good. [91]

CHAPTER THREE, TECHNOLOGY AND DOMINANCE. THE GRANDEUR OF HUMANITY IN LIGHT OF THE PROMISES OF AI

The technocratic paradigm and digital power

95. When such power is concentrated in the hands of a few, it tends to become opaque and evade public oversight, increasing the risk of distorted forms of development that give rise to new dependencies, exclusions, manipulations and inequalities.

Artificial intelligence

99. So-called artificial intelligences do not undergo experiences, do not possess a body, do not feel joy or pain, do not mature through relationships and do not know from within what love, work, friendship or responsibility mean. Nor do they have a moral conscience, since they do not judge good and evil, grasp the ultimate meaning of situations, or bear responsibility for consequences. They may imitate language, behavior and analytical skills, or even simulate empathy and understanding, but they do not understand what they produce, for they lack the affective, relational and spiritual perspective through which human beings grow in wisdom.

The authentic “more than human”: grace and Christian humanism

128. For an algorithm, an error is a flaw to be corrected; for a person, however, an error can be a catalyst for profound change. A person’s future is not calculable, but depends on one’s freedom — elevated by the inexhaustible grace of God — and on the relationships cultivated.

Two cities and two loves

129. Christian humanism does not reject science or technology, but embraces them with gratitude and realism, and grounds them within a higher vocation. 

130. “Two loves have built two cities: the earthly city, the love of self even to the contempt of God; the heavenly city, the love of God even to the contempt of self.” [139] As throughout history, these two loves continue to contend for dominance in our hearts today. The age of AI is no exception: the construction of Babel or the rebuilding of Jerusalem begins within each one of us.

 

CHAPTER FOUR, SAFEGUARDING HUMANITY AT A TIME OF TRANSFORMATION.

TRUTH, WORK, FREEDOM

An educational alliance for the digital age

140. As Plato wrote, the deepest and most important things are learned only after much time and effort, by engaging in discussion with others, “striking upon” ideas and experiences together like flint until the spark of understanding is kindled within us. [147] We must learn, then, how to exercise restraint in the use of AI and to protect our young people from the promise of the perfect machine, from that subtle temptation which renders human thought seemingly superfluous precisely when it is most needed.

The central role of schools

143. School is the place where new generations can learn to seek and love the truth, to reflect on the meaning of life and to recognize the dignity of every person. For this reason, many parents, who want their children to grow in the capacity to form relationships, develop critical thinking skills and embrace solid values, place great expectations on schools as valuable partners in their children’s education.

An economy that values dignity

157. Entrepreneurial initiative can indeed be a true vocation, generating wealth and improving lives, rather than a variable that is dependent only on profit. This is possible when it recognizes that the creation of dignified, valuable jobs are an essential part of its proper service to society. [158]

 

CHAPTER FIVE, THE CULTURE OF POWER AND THE CIVILIZATION OF LOVE

We can all do our part

212. Yet, no one is without responsibility. We all have our own areas for action, and it is precisely there — and nowhere else — that we must choose whether to fuel the mentality of force (even if only through indifference, cynicism, lies or hatred), or to preserve the mindset of peace (with truth, moderation, closeness and care).

213. The twentieth-century Catholic author J.R.R. Tolkien, in the words of a protagonist in one of his novels, described our responsibility in this way: “It is not our part to master all the tides of the world, but to do what is in us for the succour of those years wherein we are set, uprooting the evil in the fields that we know, so that those who live after may have clean earth to till.” [187] 

CONCLUSION

229. “Let each builder choose with care how to build” (1 Cor 3:10). With these words, Saint Paul encouraged the Christians of Corinth to preserve unity.

233. The dignity inscribed in each of us by the Holy Spirit can also be seen in our capacity to reflect critically, choose and love freely, and form authentic relationships. No computational system, however sophisticated, can create a heart that gives itself, or a conscience that discerns good from evil. Even when machines excel in efficiency, a human face that asks to be gazed upon remains the center of our history.

The song of hope: the Magnificat

245. With the same faith as Mary, let us become “weavers of hope” in our world, sharing who we are and what we have, so that the presence of Jesus may grow among us and his Kingdom take shape. In the humble fidelity of daily life, even the era of AI can become a time in which the Holy Spirit brings about the civilization of love in our lives.

Given in Rome, at Saint Peter’s, on 15 May, in the year 2026, the second of my Pontificate.

LEO PP. XIV

 

Optimal Strategy in Dating - Part 4

(Dinning icon by Icons8)

Optimal stopping strategy came up recently in an article in Scientific American. Once, when dinning out, the physicist, Richard Feynman, asked the question if one should order a favorite dish or try something new. The same question can be asked about going to a favorite restaurant or trying a new place to dine. Richard Feynman’s “dining optimization” solution is essentially an optimal stopping rule for the explore‑exploit dilemma: how long to keep trying new dishes (or restaurants) before settling on the best one you’ve found so far.

Feynman derived a declining threshold strategy (one such declining threshold is Tn=n/(n+1), where n is the number of remaining visits; the T starts very high with the first visit and declines to 1/2 on the last visit):

- Each night you compare the best restaurant (or dish) you’ve tried so far to a quality threshold.  

- If your current best exceeds the threshold, you stop exploring and keep returning to it.  

- If not, you try a new restaurant.  

- The threshold starts high and decreases as the number of remaining nights shrinks.  

  - Early in a trip, you should be picky and explore aggressively.  

  - Near the end, you should lower your standards and exploit whatever is “good enough.”

This strategy maximizes the expected total quality of all meals over a fixed number of nights.  

Why the threshold declines

With many nights left, the value of discovering an exceptional restaurant is high, so the threshold is strict. As time runs out, the benefit of exploration drops, so the threshold relaxes. This is mathematically optimal for maximizing cumulative reward.  

In plain English, Feynman’s rule says:

Explore new restaurants until the best one you’ve found so far is “good enough” by a standard that gets easier to satisfy as your remaining time decreases. Then stick with that best one.