Interactive Concepts for Mathematical Problems

Prompt

Create a full-page, responsive, and visually appealing interactive web application to visualize the Collatz Conjecture. Use HTML, CSS (Tailwind CSS), and JavaScript. The page should have a dark theme. **1. UI Components:** - A main title: "The Collatz Conjecture Visualizer". - A brief explanation of the conjecture. - An input field for the user to enter a positive integer. It should only accept numbers. - A "Visualize Sequence" button. - A toggle switch or button to change between "Sequence View" and "Forest View". - A main canvas area where the visualization will be rendered (use p5.js or a similar library for drawing). - An area to display the sequence of numbers as text (e.g., "6 → 3 → 10 → ... → 1"). **2. "Sequence View" Functionality (Default):** - When the user enters a number (e.g., 6) and clicks "Visualize", clear the canvas. - Calculate the full sequence until it reaches 1. - Animate the drawing of the sequence. For each step, draw a point for the number and a line connecting it to the previous point. The y-position of the point should correspond to the number's value, creating a zig-zagging path downwards. - As each number is drawn, display the operation (e.g., "x3 + 1") next to the connecting line. - Display the full sequence as text below the canvas. - Handle invalid inputs (non-integers, numbers less than 1) with a user-friendly error message. **3. "Forest View" Functionality:** - When the user switches to this view, the input field is replaced by a slider to select a range (e.g., "Visualize numbers from 1 to N", where N is set by the slider). - When visualized, calculate the Collatz sequence for every number in the selected range. - Draw all these paths on the canvas simultaneously. The paths should be color-coded based on their starting number or length. - The final visualization should resemble a tree or a network of rivers, where all paths converge to the number 1. This view should be optimized for performance, as it can involve many calculations. The Collatz Conjecture Problem Description: Start with any positive integer n. If n is even, the next term is n/2. If n is odd, the next term is 3n + 1. The conjecture states that no matter what value of n you start with, the sequence will always eventually reach 1. Visualization Concept: The Hailstone Sequence Animator The most effective visualization is an interactive animator that shows the path of any given number. This directly illustrates the core process of the conjecture. Interaction: The user enters a number. The application then animates the sequence, showing each number and the operation performed (/2 or *3+1) to get to the next. The path is drawn as a series of connected points, creating a "hailstone" graph that zig-zags downwards until it hits 1. Advanced View: A "Forest View" button that shows the paths of many numbers (e.g., 1 to 100) simultaneously. This creates a beautiful, tree-like structure where all branches eventually merge and flow into the single root, which is 1. This powerfully visualizes the "all paths lead to one" idea.

A system prompt was added to support web rendering