Sierpinski Triangle Fractal Construction

Animation Description and Purpose

This animation demonstrates the construction of a Sierpinski triangle, a classic fractal, through recursive subdivision. The purpose is to visually illustrate how starting from a single equilateral triangle, the shape evolves into the fractal by repeatedly removing the central triangle from each subdivided triangle. This helps viewers understand the self-similar nature of fractals and the process of infinite subdivision.

Mathematical Elements and Formulas

The Sierpinski triangle is generated by recursively subdividing an equilateral triangle into four smaller equilateral triangles and removing the central one. Mathematically, this can be described using barycentric coordinates or midpoint calculations. For a triangle with vertices $A$, $B$, and $C$, the midpoints are calculated as:

• Midpoint of $AB$: $\left( \frac{A_x + B_x}{2}, \frac{A_y + B_y}{2}\right)$
• Midpoint of $BC$: $\left( \frac{B_x + C_x}{2}, \frac{B_y + C_y}{2}\right)$
• Midpoint of $CA$: $\left( \frac{C_x + A_x}{2}, \frac{C_y + A_y}{2}\right)$

The process repeats recursively, with each iteration reducing the size by half. The fractal dimension is $\frac{\log 3}{\log 2} \approx 1.585$, indicating its space-filling properties.

Visual Elements (Shapes, Colors, Objects)

• Initial Triangle: Start with a large equilateral triangle in blue color, filled with a light shade.
• Subdivision Lines: Use thin black lines to draw the midpoints and connect them to form the subdivided triangles.
• Removed Triangles: The central triangle in each subdivision is highlighted in red before being removed (faded out or erased).
• Recursive Levels: Up to 5 levels of recursion to keep the animation manageable; each level adds more detail without overwhelming the viewer.
• Background: Plain white background for contrast.
• Labels: Optional text labels for each recursion level, e.g., "Level 1", "Level 2", etc., appearing briefly at the start of each phase.

Animation Timing and Transitions

• Total Duration: Approximately 7 seconds, divided into phases for each recursion level.
• Phase Breakdown:
  • 0-1 seconds: Display the initial triangle with a fade-in effect.
  • 1-2.3 seconds: First subdivision – draw midpoints, highlight and remove the central triangle with a smooth fade-out.
  • 2.3-3.7 seconds: Second subdivision – repeat the process on the remaining three triangles, with staggered timing for each.
  • 3.7-5.1 seconds: Third subdivision – continue recursively, adding a slight pause between removals for clarity.
  • 5.1-6.2 seconds: Fourth subdivision – accelerate slightly to show the pattern emerging.
  • 6.2-7 seconds: Fifth and final subdivision, followed by a hold on the complete fractal for 0.8 seconds.
• Transitions: Use smooth morphing or drawing animations for lines and shapes. Fade-ins and fade-outs for additions and removals. No abrupt cuts; ensure fluid progression.

Camera Angles and Perspectives

• Maintain a static 2D view with the triangle centered on the screen. No camera movement is needed, as the fractal builds in place.

Other Relevant Details

• Audio: No audio elements; rely on visual pacing.
• Interactivity: Not applicable for this static animation.
• Resolution and Quality: Ensure high resolution for clear visibility of fine details in later recursions.
• Edge Cases: If recursion exceeds 5 levels, the animation might become too dense; stick to 5 for optimal viewing.