Fractals and Visualising Solids

Build it in Your Imagination

We will start this section by practising visualisation. For each prompt, feel free to talk to your partner, gesture, draw it in the air — but do not actually draw on paper!

1. Picture your name, then read off the letters backwards. Make sure to do this by sight, not by sound — really see your name! Now try with your friend's name.

Cut off the four corners of an imaginary square, with each cut going between midpoints of adjacent edges. What shape is left over? How can you reassemble the four corners to make another square?

Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What shape do you get?

Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What shape is left?

Net of a sphere? Experiment and see if you can make a paper cutout that can perfectly wrap around a ball without leaving any wrinkles, gaps or overlaps.

Place an object in front of a plane, such as a wall of your room. Shine a torch light on the object in a direction perpendicular to the wall.

What do you see?

Observe what happens to the size of the shadow as you vary the distance between your torch and your object.

Context: Observe what happens to the size of the shadow as you vary the distance between your torch and your object.

Q. Why does this happen?

Construct a model of a cube and use your hands to keep it balanced on one corner vertex. Can you try to understand why all the projected edges have equal length?

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 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

1. 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?

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.

5. A solid whose profile has a square outline
6. A solid whose profile has a circular outline
7. 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.

8. A solid with a rectangular profile from one viewpoint and a circular profile from another viewpoint
9. A solid with a circular profile from one viewpoint and a triangular one from another viewpoint
10. A solid with a rectangular profile from one viewpoint and a triangular one from another viewpoint
11. A solid with a trapezium shaped profile from one viewpoint and a circular one from another viewpoint
12. 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 $n$ 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 $n$-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 $l$ and then unrolled, what will we get?

Observe that all the points on the boundary of the base circle are at equal distances from 'O'. So after unrolling the cone, the boundary of the net will be a portion of a circle with centre O.

What surface do you construct by using the above net, in which O 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 $p$ and $l$?

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 $n$-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?