Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Triangle.

Question

Homi · Accepted Answer

The Sierpinski Triangle is a fractal made by repeatedly removing the middle part of triangles.

Step 1 — Initial Shape

We start with a basic shape.

Let us take one solid equilateral triangle.

This is our first step, called Step 0.

Diagram 1

Step 2 — First Transformation

Now, let us change our triangle.

We find the middle point of each side.

Connect these midpoints with lines.

This creates four smaller triangles inside.

One central triangle points downwards.

We remove this central triangle.

Three smaller solid triangles remain.

These are at the corners.

This is Step 1 of the process.

Diagram 2

Step 3 — Second Transformation

We repeat this for each remaining triangle.

Take each of the three solid triangles from Step 1.

For each, find the midpoints of its sides.

Connect these midpoints to form a central triangle.

Remove this central triangle from each.

This leaves nine even smaller solid triangles.

These are at the corners of the previous three.

This is Step 2 of the construction.

Diagram 3

Step 4 — Repeating the Pattern

We can continue this process forever.

Each time, we take every solid triangle.

We divide it into four smaller ones.

Then, we remove the central part.

Repeating this creates the Sierpinski Triangle.

It is a beautiful fractal pattern.

Answer

(i) Step 0: Start with a single solid equilateral triangle. (ii) Step 1: Divide the triangle into four smaller equilateral triangles by connecting the midpoints of its sides. Remove the central inverted triangle. (iii) Step 2: For each of the three remaining solid triangles, repeat the process: divide it into four smaller triangles and remove its central inverted triangle. (iv) Step 3: Continue this process of dividing each solid triangle and removing its central part.

Question 1

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