Ch 04 - Exploring Some Geometric Themes8th Ganita Prakash Part 2

• Fractal: A geometric shape or pattern that is strictly self-similar across different scales.
• Self-Similarity: This means that if you zoom into any sub-part of the fractal, the magnified portion mathematically resembles or is an exact identical copy of the entire whole shape.
• Nature Examples: Fern leaves, coastlines, clouds, and snowflakes display approximate fractal properties.

Boundary Conditions / Domain:

• True mathematical fractals exhibit self-similarity infinitely as the scale factor approaches zero ($n \to \infty$). Natural fractals only exhibit this down to a certain physical threshold.

• Indian Architecture: The Kandariya Mahadev Temple in Khajuraho (c. 1025 CE) uses a towering structure made of smaller identical copies, which themselves are made of even smaller copies.
• African Textiles: Traditional Nigerian Fulani wedding blankets feature diamond patterns containing smaller diamond patterns recursively.
• Modern Art: The Dutch artist M.C. Escher utilized strict mathematical fractals in his tessellation work, notably in 'Smaller and Smaller', where lizards repeat at infinitely decreasing scales.

Component Anatomy:

• Step 0: The initial solid shape must strictly be a perfect square.
• Sub-squares: The 9 equal squares formed by dividing the shape using a $3 \times 3$ Cartesian grid.

Boundary Conditions / Domain:

• The initial shape must be a perfect square to ensure isotropic scaling.
• The operation is infinitely recursive ($n \to \infty$).