#mathart

[ozachou]

Computational Drawing CLXXX

[mathhombre]

[regolo54]

[xyymath]

Why the Number Zero Changed Everything

Zero: a concept so foundational to modern mathematics, science, and technology that we often forget it wasn’t always there. Its presence in our world today seems obvious, but its journey from controversial abstraction to indispensable tool has shaped entire civilizations.

1. The Birth of Zero: A Revolutionary Idea

The concept of zero didn't exist in many ancient cultures. For example, the Greeks, despite their advancements in geometry and number theory, rejected the idea of a placeholder for nothingness. The Babylonians had a placeholder symbol (a space or two slashes) for zero, but they didn't treat it as a number. It wasn't until Indian mathematicians in the 5th century, like Brahmagupta, that zero was truly conceptualized and treated as a number with its own properties.

Zero was initially used as a place-holder in the decimal system, but soon evolved into a full-fledged number with mathematical properties, marking a huge leap in human cognition.

2. The Birth of Algebra

Imagine trying to solve equations like x + 5 = 0 without zero. With zero, algebra becomes solvable, opening up entire fields of study. Before zero’s arrival, solving equations involving unknowns was rudimentary, relying on geometric methods. The Indian mathematician Brahmagupta (again) was one of the first to establish rules for zero in algebraic operations, such as:

x + 0 = x (additive identity)

x × 0 = 0 (multiplicative property)

These properties allowed algebra to evolve into a system of abstract thought rather than just arithmetic, transforming the ways we understand equations, functions, and polynomials.

3. Calculus and Zero: A Relationship Built on Limits

Without zero, the foundation of calculus—limits, derivatives, and integrals—wouldn’t exist. The limit concept is intrinsically tied to approaching zero as a boundary. In differentiation, the derivative of a function f(x) is defined as:

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

This limit process hinges on the ability to manipulate and conceptualize zero in infinitesimal quantities. Similarly, integrals, which form the backbone of area under curves and summation of continuous data, rely on summing infinitely small quantities—essentially working with zero.

Without the concept of zero, we wouldn’t have the means to rigorously define rates of change or accumulation, effectively stalling physics, engineering, and economics.

4. Zero and the Concept of Nothingness: The Philosophical Impact

Zero is more than just a number; it’s an idea that forces us to confront nothingness. Its acceptance was met with philosophical resistance in ancient times. How could "nothing" be real? How could nothing be useful in equations? But once mathematicians recognized zero as a number in its own right, it transformed entire philosophical discussions. It even challenged ideas in theology (e.g., the nature of creation and void).

In set theory, zero is the size of the empty set—the set that contains no elements. But without zero, there would be no way to express or manipulate sets of nothing. Thus, zero's philosophical acceptance paved the way for advanced theories in logic and mathematical foundations.

5. The Computing Revolution: Zero as a Binary Foundation

Fast forward to today. Every piece of digital technology—from computers to smartphones—relies on binary systems: sequences of 1s and 0s. These two digits are the fundamental building blocks of computer operations. The idea of Boolean algebra, where values are either true (1) or false (0), is deeply rooted in zero’s ability to represent "nothing" or "off."

The computational world relies on logical gates, where zero is interpreted as false, allowing us to build anything from a basic calculator to the complex AI systems that drive modern technology. Zero, in this context, is as important as one—and it's been essential in shaping the digital age.

6. Zero and Its Role in Modern Fields

In modern fields like physics and economics, zero plays a crucial role in explaining natural phenomena and building theories. For instance:

In physics, zero-point energy (the lowest possible energy state) describes phenomena in quantum mechanics and cosmology.

In economics, zero is the reference point for economic equilibrium, and the concept of "breaking even" relies on zero profit/loss.

Zero allows us to make sense of the world, whether we’re measuring the empty vacuum of space or examining the marginal cost of producing one more unit in economics.

7. The Mathematical Utility of Zero

Zero is essential in defining negative numbers. Without zero as the boundary between positive and negative values, our number system would collapse. The number line itself relies on zero as the anchor point, dividing positive and negative values. Vector spaces, a fundamental structure in linear algebra, depend on the concept of a zero vector as the additive identity.

The coordinate system and graphs we use to model data in statistics, geometry, and trigonometry would not function as we know them today. Without zero, there could be no Cartesian plane, and concepts like distance, midpoint, and slope would be incoherent.

[elkement]

On my mission to combine traditional art, digital paintings, and code art in new ways!

I am using my code-generated art (real and imaginary part of 1/z) as a reference image!

First ever test with acrylics on canvas 🙈

Work in progress!

[daily-public-domain]

Day 257: When Claude met Vincent

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–These images are part of the public domain, meaning you can do anything you want with them! (you could even sell them as a shirt, poster or whatever, no need to credit them!)–

[eccentricstylist]

Winter Daylight Mandala

[olbaid-st]

Deep Mandelbrot Set #158 Mag=9.72e+1774

Magnification = 9.72e+1774 = 2^5896.382 Max iteration = 1,132,860

[kaleidoscopevisualart]

[coffeenuts]

[ozachou]

Computational Drawing CLXX

[mathhombre]

Loop de Loop Closed Curves

From this GeoGebra applet.

[regolo54]

[xyymath]

The Math of Social Networks: How Social Media Algorithms Work

In the digital age, social media platforms like Instagram, Facebook, and TikTok are fueled by complex mathematical algorithms that determine what you see in your feed, who you follow, and what content "goes viral." These algorithms rely heavily on graph theory, matrix operations, and probabilistic models to connect billions of users, influencers, and posts in increasingly intricate webs of relationships.

Graph Theory: The Backbone of Social Networks

Social media platforms can be visualized as graphs, where each user is a node and each connection (whether it’s a "follow," "like," or "comment") is an edge. The structure of these graphs is far from random. In fact, they follow certain mathematical properties that can be analyzed using graph theory.

For example, cliques (a subset of users where everyone is connected to each other) are common in influencer networks. These clusters of interconnected users help drive trends by amplifying each other’s content. The degree of a node (a user’s number of direct connections) is a key factor in visibility, influencing how posts spread across the platform.

Additionally, the famous Six Degrees of Separation theory, which posits that any two people are connected by no more than six intermediaries, can be modeled using small-world networks. In these networks, most users are not directly connected to each other, but the distance between any two users (in terms of number of connections) is surprisingly short. This is the mathematical magic behind viral content, as a post can be shared through a small network of highly connected individuals and reach millions of users.

Matrix Operations: Modeling Connections and Relevance

When social media platforms recommend posts, they often rely on matrix operations to model relationships between users and content. This process can be broken down into several steps:

User-Content Matrix: A matrix is created where each row represents a user and each column represents a piece of content (post, video, etc.). Each cell in this matrix could hold values indicating the user’s interactions with the content (e.g., likes, comments, shares).

Matrix Factorization: To make recommendations, platforms use matrix factorization techniques such as singular value decomposition (SVD). This helps reduce the complexity of the data by identifying latent factors that explain user preferences, enabling platforms to predict what content a user is likely to engage with next.

Personalization: This factorization results in a model that can predict a user’s preferences even for content they’ve never seen before, creating a personalized feed. The goal is to minimize the error matrix, where the predicted interactions match the actual interactions as closely as possible.

Influence and Virality: The Power of Centrality and Weighted Graphs

Not all users are equal when it comes to influencing the network. The concept of centrality measures the importance of a node within a graph, and in social media, this directly correlates with a user’s ability to shape trends and drive engagement. Common types of centrality include:

Degree centrality: Simply the number of direct connections a user has. Highly connected users (like influencers) are often at the core of viral content propagation.

Betweenness centrality: This measures how often a user acts as a bridge along the shortest path between two other users. A user with high betweenness centrality can facilitate the spread of information across different parts of the network.

Eigenvector centrality: A more sophisticated measure that not only considers the number of connections but also the quality of those connections. A user with high eigenvector centrality is well-connected to other important users, enhancing their influence.

Algorithms and Machine Learning: Predicting What You See

The most sophisticated social media platforms integrate machine learning algorithms to predict which posts will generate the most engagement. These models are often trained on vast amounts of user data (likes, shares, comments, time spent on content, etc.) to determine the factors that influence user interaction.

The ranking algorithms take these factors into account to assign each post a “score” based on its predicted engagement. For example:

Collaborative Filtering: This technique relies on past interactions to predict future preferences, where the behavior of similar users is used to recommend content.

Content-Based Filtering: This involves analyzing the content itself, such as keywords, images, or video length, to recommend similar content to users.

Hybrid Methods: These combine collaborative filtering and content-based filtering to improve accuracy.

Ethics and the Filter Bubble

While the mathematical models behind social media algorithms are powerful, they also come with ethical considerations. Filter bubbles, where users are only exposed to content they agree with or are already familiar with, can be created due to biased algorithms. This can limit exposure to diverse perspectives and create echo chambers, reinforcing existing beliefs rather than fostering healthy debate.

Furthermore, algorithmic fairness and the prevention of algorithmic bias are growing areas of research, as biased recommendations can disproportionately affect marginalized groups. For instance, if an algorithm is trained on biased data (say, excluding certain demographics), it can unfairly influence the content shown to users.

[elkement]

Orbital Bloom

A mini-series: Spherical harmonic function Y_33, envisaged via "metallic" lines of constant angle.

The four images show the same "virtual sculpture" from different viewpoints. Colors are inspired by the festive season and not related to physics :-)

Created with hand-crafted JavaScript code using the framework threejs, no AI.

[daily-public-domain]

Day 261: "hot chaos contained"

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–This image is part of the public domain, meaning you can do anything you want with it! (you could even sell it as a shirt, poster or whatever, no need to credit it!)–

Reminds me a lot of PK Fire from Earthbound

(Obviously, this gif is NOT in the public domain)