Fractals and Visualising Solids | FIO

Question 6

Find the perimeter of the shape at the $n\text{th}$ step of the sequence. Take the starting equilateral triangle to have a sidelength of 1 unit.

Solution

Understand the Question

The sequence generates a Koch snowflake fractal starting from an equilateral triangle of side length $1$ unit (Step $0$ ).
At each subsequent step, the middle third of every line segment is replaced by two segments of equal length forming an outward point. As a result:
- The number of sides multiplies by $4$ .
- The length of each side divides by $3$ (multiplies by $\dfrac{1}{3}$ ).
Therefore, the total perimeter at each step is multiplied by a factor of $\dfrac{4}{3}$ .

Step 1 · Find Perimeter at Step 0

For the starting equilateral triangle:

Initial side length: $L_0 = 1\text{ unit}$
Number of sides: $N_0 = 3$

\begin{aligned} P_0 &= N_0 \times L_0 \\[0.6em] &= 3 \times 1 \\[0.6em] &= 3 \text{ units} \end{aligned}

Step 2 · Find Perimeter at Step 1

Each side is divided into three equal parts and the middle part is replaced with two new segments.

New side length:

\begin{aligned} L_1 &= \dfrac{L_0}{3} \\[0.6em] &= \dfrac{1}{3} \text{ units} \end{aligned}

Number of sides:

\begin{aligned} N_1 &= N_0 \times 4 \\[0.6em] &= 3 \times 4 \\[0.6em] &= 12 \text{ sides} \end{aligned}

Perimeter at Step 1:

\begin{aligned} P_1 &= N_1 \times L_1 \\[0.6em] &= 12 \times \left(\dfrac{1}{3}\right) \\[0.6em] &= 4 \text{ units} \end{aligned}

Notice that $P_1 = P_0 \times \dfrac{4}{3} = 3 \times \dfrac{4}{3} = 4\text{ units}$ .

Step 3 · Find Perimeter at Step 2

Applying the same rule to each segment:

New side length:

\begin{aligned} L_2 &= \dfrac{L_1}{3} \\[0.6em] &= \dfrac{\frac{1}{3}}{3} \\[1.1em] &= \dfrac{1}{9} \text{ units} \end{aligned}

Number of sides:

\begin{aligned} N_2 &= N_1 \times 4 \\[0.6em] &= 12 \times 4 \\[0.6em] &= 48 \text{ sides} \end{aligned}

Perimeter at Step 2:

\begin{aligned} P_2 &= N_2 \times L_2 \\[0.6em] &= 48 \times \left(\dfrac{1}{9}\right) \\[0.6em] &= \dfrac{16}{3} \text{ units} \end{aligned}

Notice that $P_2 = P_1 \times \left(\dfrac{4}{3}\right) = 4 \times \left(\dfrac{4}{3}\right) = \dfrac{16}{3}\text{ units}$ .

Step 4 · Derive Formula for Perimeter at Step $n$

Observing the pattern:

Number of sides at step $n$ : $N_n = 3 \times 4^n$
Length of each side at step $n$ : $L_n = \left(\dfrac{1}{3}\right)^n$
Perimeter multiplies by $\dfrac{4}{3}$ at every step

Starting from $P_0 = 3$ units:

\begin{aligned} P_n &= P_0 \times \left(\dfrac{4}{3}\right)^n \\[0.6em] &= 3 \times \left(\dfrac{4}{3}\right)^n \text{ units} \end{aligned}

Answer

$3 \times \left(\dfrac{4}{3}\right)^n \text{ units}$

Common Mistakes

Confusing Side Length with Total Perimeter: While individual side lengths decrease by a factor of $\dfrac{1}{3}$ at each step, the total number of sides increases by a factor of $4$ , making the overall perimeter grow infinitely as $n \to \infty$ .
Off-by-One Exponent Error: Using $\left(\dfrac{4}{3}\right)^{n-1}$ instead of $\left(\dfrac{4}{3}\right)^n$ . At Step $0$ ( $n = 0$ ), $3 \times \left(\dfrac{4}{3}\right)^0 = 3$ units, which matches the initial triangle's perimeter.

More questions in FIO

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

Find the number of holes, and the triangles that remain at each step of the shape sequence that leads to the Sierpinski Triangle.

Find the area of the region remaining at the $n$ th step in each of the shape sequences that lead to the Sierpinski fractals. Take the area of the starting square/triangle to be 1 sq. unit.

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

Find the number of sides in the $n^{\text{th}}$ step of the shape sequence that leads to the Koch Snowflake.

Find the perimeter of the shape at the $n\text{th}$ step of the sequence. Take the starting equilateral triangle to have a sidelength of 1 unit.

Figure it Out

Which of the following are the nets of a cube? First, try to answer by visualisation. Then, you may use cutouts and try.

A cube has 11 possible net structures in total. In this count, two nets are considered the same if one can be obtained from the other by a rotation or a flip. For example, the following nets are all considered the same —

Find all the 11 nets of a cube.

Draw a net of a cuboid having sidelengths:

(i) $5\text{ cm}$ , $3\text{ cm}$ , and $1\text{ cm}$

(ii) $6\text{ cm}$ , $3\text{ cm}$ , and $2\text{ cm}$

Q10

Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between their lengths?

Q11

Find the front view, top view and side view of each of the following solids, fixing its orientation with respect to the vertical, horizontal and side planes: cube, cuboid, parallelepiped, cylinder, cone, prism, and pyramid. If needed, see the next problem for clues.

Q12

Match each of the following objects with its projections.

Q13

Draw the top view, front view and the side view of each of the following combinations of identical cubes.

Q14

Imagine eight identical cubes, glued together along faces to form the letter 'C'.

Q15

Which solid corresponds to the given top view, front view, and side view?

Solids to choose from:

Q16

Using identical cubes, make a solid that gives the following projections:

Q17

Find the number of cubes in this stack of identical cubes.

Q18

What are the different shapes the projection of a cube can make under different orientations?

Q19

In addition to the 5 ways shown in Fig. 4.8, are there any additional ways of gluing four cubes together along faces? Can you visualise and draw these as well?

Q20

Draw the following figures on the isometric grid.

[Hint: It may be useful to determine whether the edge to be currently drawn — say, along the height — goes from down to up or up to down. Accordingly, draw the line segment on the grid either in the direction of the height axis or opposite to it.]

Q21

Is there anything strange about the path of this ball? Recreate it on the isometric grid.

[Hint: Consider a portion of this figure that is physically realisable and identify the 3 primary directions.]

Q22

Observe this triangle.

(i) Would it be possible to build a model out of actual cubes? What are the front, top, and side profiles of this impossible triangle? (ii) Recreate this on an isometric grid. (iii) Why does the illusion work?