Question 19
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?
We need to find all unique shapes formed by connecting four squares edge-to-edge. These shapes are called tetrominoes. We will consider shapes that are reflections of each other as distinct if they cannot be rotated to match.
Step 1 — Understand the problem
The problem asks us to find if there are more ways to arrange four squares (representing cubes) connected along their edges (faces), beyond the five shown in Fig. 4.8. We are also asked to draw these additional shapes. When we consider reflections as distinct, there are more unique arrangements.
Step 2 — Identify the given shapes
The five shapes shown in Fig. 4.8 are: (i) A straight line of four squares. This is called an I-tetromino. (ii) A 2x2 square. This is called an O-tetromino. (iii) An L-shaped arrangement. This is called an L-tetromino. (iv) A Z-shaped arrangement. This is called an S-tetromino. (v) A T-shaped arrangement. This is called a T-tetromino.
Step 3 — Find the additional shapes
There are a total of seven unique arrangements of four squares when reflections are considered distinct. Since five shapes are already shown in Fig. 4.8, there must be two additional shapes. These two additional shapes are the reflections of the L-tetromino (shape (iii)) and the S-tetromino (shape (iv)). The reflection of the L-tetromino is commonly called the J-tetromino. The reflection of the S-tetromino is commonly called the Z-tetromino. These reflected shapes cannot be rotated to perfectly match the original L or S shapes.
Step 4 — Visualise and draw the additional shapes
The L-tetromino (shape (iii)) is: Its reflection, the J-tetromino, is: The S-tetromino (shape (iv)) is: Its reflection, the Z-tetromino, is:
Answer
Yes, there are two additional ways of gluing four cubes together along faces, considering reflections as distinct shapes. These are:
(i) The J-tetromino (a reflected L-shape).
# # ##(ii) The Z-tetromino (a reflected S-shape).
## ##
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 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 nth step of the shape sequence that leads to the Koch Snowflake.
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.
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 cm, 3 cm, and 1 cm
(ii) 6 cm, 3 cm, and 2 cm
Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between their lengths?
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.
Match each of the following objects with its projections.
Draw the top view, front view and the side view of each of the following combinations of identical cubes.
Imagine eight identical cubes, glued together along faces to form the letter 'C'.
Which solid corresponds to the given top view, front view, and side view?
Solids to choose from:
Using identical cubes, make a solid that gives the following projections:
Find the number of cubes in this stack of identical cubes.
What are the different shapes the projection of a cube can make under different orientations?
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?
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.]
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.]
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?