Fractals and Visualising Solids | FIO

Question 20

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.]

Question diagram 1
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Solution

We will draw each 3D figure on an isometric grid, using unit cubes.

Step 1 — Drawing the L-shaped block

Let us start by drawing the vertical part of the L-shape. This part is a tower: 1 unit wide, 1 unit deep, 3 units high.

  1. First, we draw the base of this tower. Pick a point on the grid. From this point, draw a 1-unit line along the 'length' direction. From the same point, draw another 1-unit line along the 'width' direction. Complete the 1×11 \times 1 square base with two parallel lines.
  2. From each base corner, draw vertical lines 3 units long.
  3. Connect the top ends to form the top 1×11 \times 1 square. This completes the vertical tower.
  4. Next, we add the horizontal part. This part extends from the base of the tower. From the tower's front-right base corner, extend 2 units along the 'length' direction.
  5. From the tower's back-right base corner, draw a parallel 2-unit line.
  6. Connect these line ends to form the horizontal part's base.
  7. From the two new outer base corners, draw vertical lines 1 unit long.
  8. Connect the top ends of these new vertical lines.
  9. Connect the tower's top-front-right corner to the adjacent vertical line's top.
  10. Do the same for the top-back-right corner. This completes the L-shape.

Diagram 1

Step 2 — Drawing the T-shaped block

Let us draw the T-shaped block. It is a horizontal base block with a vertical block centered on top.

  1. First, we draw the base block. This block is 3 units long, 1 unit deep, 1 unit high. Pick a starting point. Draw a line 3 units long along the 'length' direction. From the start point, draw a 1-unit line along the 'width' direction. Complete the 3×13 \times 1 rectangular base.
  2. From each base corner, draw vertical lines 1 unit long.
  3. Connect the top ends to form the top 3×13 \times 1 rectangle. This completes the base block.
  4. Now, we add the vertical part. This part is 1 unit wide, 1 unit deep, 3 units high total. It sits on the middle of the base block.
  5. Find the 1×11 \times 1 square at the center of the base block's top.
  6. From each corner of this central square, draw 2-unit vertical lines.
  7. Connect the top ends to form the top 1×11 \times 1 square. This completes the T-shape.

Diagram 2

Step 3 — Drawing the Staircase block

Let us draw the staircase block, which has three steps. Each step rises by 1 unit and extends by 1 unit.

  1. First, we draw the lowest step. This step is 3 units long, 1 unit deep, 1 unit high. Pick a starting point. Draw a line 3 units long along the 'length' direction. From the start point, draw a 1-unit line along the 'width' direction. Complete the 3×13 \times 1 rectangular base.
  2. From each base corner, draw vertical lines 1 unit long.
  3. Connect the top ends to form the top 3×13 \times 1 rectangle. This completes the first step.
  4. Next, we draw the second step. This step is 2 units long, 1 unit deep, and 1 unit higher. It sits on one side of the first step's top surface.
  5. Find the 2×12 \times 1 area on the first step's top. Place it closer to the 'back' or 'left' side.
  6. From these four corners, draw vertical lines 1 unit long.
  7. Connect the top ends to form the top 2×12 \times 1 rectangle. This completes the second step.
  8. Finally, we draw the third step. This step is 1 unit long, 1 unit deep, and 1 unit higher. It sits on one side of the second step's top surface.
  9. Find the 1×11 \times 1 area on the second step's top.
  10. From these four corners, draw vertical lines 1 unit long.
  11. Connect the top ends to form the top 1×11 \times 1 square. This completes the staircase.

Diagram 3

Answer

(i) The L-shaped block is drawn on the isometric grid by combining a 1×1×31 \times 1 \times 3 vertical block and a 1×2×11 \times 2 \times 1 horizontal block. (ii) The T-shaped block is drawn on the isometric grid by placing a 1×1×31 \times 1 \times 3 vertical block centrally on top of a 3×1×13 \times 1 \times 1 horizontal base block. (iii) The staircase block is drawn on the isometric grid by stacking three rectangular blocks: a 3×1×13 \times 1 \times 1 base, a 2×1×12 \times 1 \times 1 middle step, and a 1×1×11 \times 1 \times 1 top step, each offset to create the stair effect.

More questions in FIO

Q1

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

Q2

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

Q3

Find the area of the region remaining at the nnth 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.

Q4

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

Q5

Find the number of sides in the nth step of the shape sequence that leads to the Koch Snowflake.

Q6

Find the perimeter of the shape at the nth step of the sequence. Take the starting equilateral triangle to have a sidelength of 1 unit.

Q7

Figure it Out

  1. Which of the following are the nets of a cube? First, try to answer by visualisation. Then, you may use cutouts and try.
Q8

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.

Q9

Draw a net of a cuboid having sidelengths:

(i) 5 cm, 3 cm, and 1 cm

(ii) 6 cm, 3 cm, and 2 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?

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