Question 42
Can you try drawing the other tetris shapes on isometric paper?
- Isometric paper uses a grid of equilateral triangles (dots forming axes at , , and ) that allows us to draw three-dimensional solids with accurate proportions.
- A tetromino (Tetris shape) is a 3D geometric figure formed by joining unit cubes face-to-face.
- There are standard tetrominoes: I, O, T, L, J, S, and Z.
- Each shape is constructed by first drawing a single unit cube on the isometric dots and then attaching subsequent cubes along the grid lines.
Step 1 · Draw a Unit Cube on Isometric Paper
To draw a unit cube of side :
- Choose a starting dot as the front-bottom-left corner.
- Draw segments of length from this dot along the grid axes: vertically upward (), up-right (), and up-left ().
- Complete the top rhombus face by drawing two parallel segments of length .
- Draw vertical segments from the remaining visible corners and join them to form the side faces.
Step 2 · Draw the I-Tetromino (Straight)
- The I-tetromino consists of unit cubes in a straight line.
- Draw the first unit cube, then extend more cubes sequentially along the same grid axis, sharing adjoining faces.
Step 3 · Draw the O-Tetromino (Square)
- The O-tetromino is a block of cubes.
- Draw cubes side-by-side along one axis, then draw another pair of cubes directly adjacent to them along the perpendicular isometric axis.
Step 4 · Draw the T-Tetromino
- The T-tetromino consists of a line of cubes with extra cube attached to the middle.
- Draw cubes in a row, then attach the fourth cube extending outward from the center cube's face.
Step 5 · Draw the L-Tetromino
- The L-tetromino has cubes in a line with cube attached at one end.
- Draw a row of cubes, then attach the fourth cube to one endpoint extending in a perpendicular direction.
Step 6 · Draw the J-Tetromino
- The J-tetromino is the mirror image of the L-tetromino.
- Draw a row of cubes, then attach the fourth cube to the opposite end extending in the same direction.
Step 7 · Draw the S-Tetromino
- The S-tetromino has a stepped, zig-zag shape.
- Draw cubes side-by-side, then draw another pair of cubes shifted by one unit to the next level.
Step 8 · Draw the Z-Tetromino
- The Z-tetromino is the mirror image of the S-tetromino.
- Draw cubes side-by-side, then attach the next pair of cubes shifted in the opposite direction.
All standard tetromino shapes (I, O, T, L, J, S, and Z) can be drawn on isometric paper by combining unit cubes along the isometric grid axes.
- Ignoring Isometric Grid Angles: Drawing edges horizontally instead of following the , , and dot lines breaks the 3D perspective.
- Hidden Edges: Drawing interior edges between connected cubes that share a face; only outer, visible boundary edges should be drawn.
- Confusing Chiral Pairs: Confusing mirror shapes like L vs. J or S vs. Z, which differ by the direction in which the offset cubes attach.
More questions in IT
Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Carpet.
By its construction, each step in the sequence has (i) squares of the same size that remain in the figure, and the size of these squares becomes smaller and smaller as the step number increases, and (ii) square holes that are formed by removing square pieces.
Show that by joining the midpoints of an equilateral triangle, we divide it into 4 identical equilateral triangles.
[Hint: Note that the corner triangles are isosceles.]
This fractal is called the Sierpinski Triangle/Gasket.
In previous classes, you've seen solids that are much simpler than an elephant or cat, such as cubes, spheres, cylinders, and cones. What would the profiles of these look like, from different viewpoints?
Can you describe a solid and a viewpoint that would result in each of the following cases? If it helps, you can imagine the solid passing through a wall like Tom did, and leaving a hole of the appropriate shape.
- A solid whose profile has a square outline
- A solid whose profile has a circular outline
- A solid whose profile has a triangular outline
As we saw with the elephant, a given solid might have very different profiles from different viewpoints. Can you visualise solids that have the following contrasting profiles?
Spend some time on this, and if you are finding it difficult to visualise, you may look around and use objects that are around you, or that you will make in the next section. Feel free to consider viewpoints from any direction, including directly above the object.
- A solid with a rectangular profile from one viewpoint and a circular profile from another viewpoint
- A solid with a circular profile from one viewpoint and a triangular one from another viewpoint
- A solid with a rectangular profile from one viewpoint and a triangular one from another viewpoint
- A solid with a trapezium shaped profile from one viewpoint and a circular one from another viewpoint
- A solid with a pentagonal profile from one viewpoint and a rectangular one from another viewpoint
Are there unique solids for each of the conditions, or can you come up with multiple possibilities?
If the congruent polygons of a prism have 10 sides, how many faces, edges and vertices does the prism have? What if the polygons have sides?
If the base of a pyramid has 10 sides, how many faces, edges and vertices does the pyramid have? What if the base is an -sided polygon?
What is a net of a cube?
Visualise how it can be folded to form a cube.
What is a net of a regular tetrahedron? Which of the following are nets of a regular tetrahedron?
Are there any other possible nets?
Draw a net with appropriate measurements that can be folded into a regular tetrahedron. Verify if it works by making an actual cutout.
Draw a net with appropriate measurements that can be folded into a square pyramid. Verify if it works by making an actual cutout.
What is the net of a cylinder?
If the circular faces of a cylinder are unfolded, and if a cut is made along the height of the cylinder, as shown in the figure below, then we get
What are the sidelengths of the rectangle obtained?
How will the net of a cone look?
If the cone is slit open along the line and then unrolled, what will we get?
Observe that all the points on the boundary of the base circle are at equal distances from . So after unrolling the cone, the boundary of the net will be a portion of a circle with centre .
What surface do you construct by using the above net, in which is not the centre of the boundary circle? Make a physical model to help you answer this question!
Draw a net with appropriate measurements that can be folded into a triangular prism. Verify that it works by making an actual cutout.
Taking all the triangles in the net to be equilateral, make a cutout of the net and fold it to form an octahedron.
What is the shortest path for the ant to reach the laddu?
What about in the following case?
If we think that a certain path is the shortest, how can we be sure that it truly is, among all the infinite possibilities?
For example, are either of these the shortest path?
What does this show?
Have we now completely analysed the problem of finding the shortest path between two points on a cuboid?
What is the length of the shortest path between the ant and the laddu?
What happens to the length of a line in its projection?
Can you now compare the lengths and ?
When is the length of the projected line equal to its actual length?
What do you think are the different possible projections of a square that we get based on its orientation?
What do you think is the projection of a parallelogram under different orientations?
Can this ever be a quadrilateral that is not a parallelogram? As a starting point, you could think about the projection of a pair of parallel lines.
What can you say about the projection of an -sided regular polygon?
[Hint: Projection of a polygon is composed of the projections of its sides.]
How would the projections of a cube and a cone look?
See Figures 4.2–4.5. In each case, see if you can visualise another object that gives the same projection.
Find another object that makes the same projection as that of a given cone.
Have you played Tetris? There are five basic shapes in Tetris, corresponding to the different ways of arranging four squares.
Context: In Fig. 4.8, there are five basic shapes in Tetris, corresponding to the different ways of arranging four squares.
Q. Imagine these are cubes, not squares. Draw each of these on your isometric paper (you can find it at the end of the book).
Why is this correspondence between directions on isometric paper and axes of the solid so effective for communicating the shape of the solid?
Can you try drawing the other tetris shapes on isometric paper?