Daniel Gray

GLSL 3D Fractals - 3D IFS Fractals Iterated Function Systems (IFS) are a powerful method for generating fractals using affine transformations. In 3D, IFS fractals can create stunning organic and geome...

GLSL 3D Fractals - Distance Estimation Methods Distance estimation is the cornerstone of efficient 3D fractal rendering. Unlike traditional ray tracing, which requires expensive intersection tests, di...

GLSL 3D Fractals - Hybrid Fractals Hybrid fractals combine multiple fractal formulas to create entirely new structures. By blending Mandelbulbs with Julia sets, mixing geometric and algebraic fractals...

GLSL 3D Fractals - Mandelbox The Mandelbox is a fascinating 3D fractal discovered by Tom Lowe in 2010. It combines sphere inversion with box folding and scaling operations, creating intricate geometri...

GLSL 3D Fractals - Mandelbulb The Mandelbulb is the 3D extension of the classic Mandelbrot set, discovered by Daniel White and Paul Nylander in 2009. It creates stunning organic, bulbous structures wi...

GLSL 3D Fractals - Menger Sponge The Menger Sponge is the 3D generalization of the Sierpinski carpet, discovered by Karl Menger in 1926. It's a fascinating geometric fractal with paradoxical propertie...

GLSL 3D Fractals - Quaternion Julia Sets Quaternion Julia sets extend the classic 2D Julia sets into 4D space using quaternion algebra, then project them back into 3D for visualization. The result is ...

GLSL 3D Fractals - Sierpinski Tetrahedron The Sierpinski Tetrahedron is the 3D generalization of the classic Sierpinski triangle. It's created by recursively subdividing a regular tetrahedron, removin...

GLSL 3D Fractals Series 3D fractals represent some of the most visually stunning and mathematically fascinating objects in computer graphics. This series explores how to render these infinite structur...

Mathematics & Learning Mathematics is a powerful tool for understanding patterns, solving problems, and thinking logically. This section focuses on effective techniques and resources for learning math...

Advanced Concepts and Applications This is Part 6 of the Turing Machines series. Here we explore advanced topics including computability theory, complexity theory, variations of Turing machines, and c...

Building Your First Turing Machine This is Part 5 of the Turing Machines series. Here we walk through building Turing machines step by step, from simple examples to more complex problems. We'll use pr...

Complexity Theory: How Efficiently Can Problems Be Solved? Complexity theory is the branch of theoretical computer science that studies how efficiently problems can be solved, not just whether they ca...

Computability Theory: What Can and Cannot Be Computed Computability theory (also called recursion theory) is the branch of theoretical computer science that studies what problems can be solved by algo...

Computer Science Computer Science is the study of abstract mathematical concepts related to information theory, computation, and algorithms. This section explores the theoretical foundations that unde...

Exploring turingmachine.io This is Part 4 of the Turing Machines series. Here we provide a comprehensive hands-on guide to turingmachine.io, an interactive visualization tool that makes Turing machine...

Formal Languages: The Mathematical Study of Languages and Grammars Formal language theory is the mathematical study of languages—sets of strings over an alphabet—and the grammars that generate them. U...

Introduction to Turing Machines This is Part 2 of the Turing Machines series. Here we explore the foundational concepts, historical context, and mathematical structure of Turing machines—one of the mo...

State Machine: Modeling Computation and Behavior A state machine (also called a finite automaton or finite state machine) is a mathematical model of computation that describes a system with a finite n...

Turing Machines: A Complete Guide Welcome to the Turing Machines series! This comprehensive guide explores one of the most fundamental concepts in Computer Science and Discrete Mathematics—the Advance...

Understanding State Diagrams and Transitions This is Part 3 of the Turing Machines series. Here we explore how Turing machines are visually represented through state diagrams and how to read and under...

Discrete Mathematics: The Foundation of Computer Science Discrete mathematics is the branch of mathematics that deals with discrete, countable objects rather than continuous ones. Unlike calculus, whi...