Exploring Some Geometric ThemesClass 8 Mathematics Case Study Questions

A digital artist is creating a background pattern based on the Sierpinski Carpet. She starts with a solid black square of side length $27$ cm. In Step 1, she divides it into 9 smaller equal squares and removes the central one, leaving a white hole. In Step 2, she repeats this process for each of the remaining 8 black squares. This iterative process creates a detailed, self-similar fractal pattern. The artist wants to calculate the properties of her design after a few steps to plan her composition.

A park is being enclosed by a unique fence designed using the initial steps of a Koch Snowflake. The design starts with a large equilateral triangle (Step 0) with each side measuring $81$ meters. To create Step 1, each of the 3 sides is divided into three equal segments. The middle segment is removed and replaced by two sides of an equilateral triangle pointing outwards. This process is repeated on every straight segment to get more intricate designs. The park management needs to calculate the fence's length for budgeting.

An architect is designing a modern sculpture for a museum courtyard using identical cubic blocks. The final structure is assembled as follows: The base is a 2x2 square. The back-left pillar is 3 blocks high. The back-right pillar is 1 block high. The front-left pillar is 2 blocks high, and the front-right pillar is also 1 block high. To create the construction blueprints, the architect needs to draw the orthographic projections: the front view, the top view, and the side view (from the right). The direction you are looking from to see the front-left and front-right pillars is the 'Front' view.