Fractals and Visualising Solids | FIO

Question 8

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.

Solution

Understand the Question

A net of a cube is a 2D arrangement of 6 connected squares that can be folded along their edges to form a closed 3D cube without overlaps or gaps.
Two nets are considered identical if one can be transformed into the other via rotation or reflection (flip).
Systematically grouping them by the length of their longest straight line of squares reveals all 11 distinct nets:
- 6 nets with a central row of $4$ squares ( $1\text{-}4\text{-}1$ arrangements)
- 4 nets with a central row of $3$ squares ( $1\text{-}3\text{-}2$ or $2\text{-}3\text{-}1$ arrangements)
- 1 net with a step/staircase structure ( $2\text{-}2\text{-}2$ arrangement)

Step 1 · Nets with a Central Row of 4 Squares

These nets have a continuous row of 4 squares, with the remaining 2 squares attached to opposite sides:

1.2.3.4.5.6.

Step 2 · Nets with a Central Row of 3 Squares

These nets have a longest straight row of 3 squares:

7.8.9.

Step 3 · Nets with a Base Block of 2 Squares

These nets contain no continuous row of more than 2 squares:

10.11.

Answer

There are 11 distinct nets of a cube, categorized by their longest continuous row of squares ( $6$ nets with $4$ in a row, $3$ nets with $3$ in a row, and $2$ nets with $2$ in a row).

Common Mistakes

Counting Rotations/Reflections: Counting a rotated or flipped version as a new net; nets related by rigid motion are mathematically identical.
$2 \times 2$ Square Clusters: Any arrangement containing a solid $2 \times 2$ block of $4$ squares cannot fold into a cube because adjacent faces overlap, leaving open sides.
5 or More Squares in a Line: Having $5$ squares in a straight line cannot form a cube because the fifth face overlaps the first.

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.

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?