Question 21
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.]
- Isometric drawings represent three-dimensional objects along three primary axes: Left/Right, Forward/Backward, and Up/Down.
- By tracing the path of the ball along the cube surfaces step by step, we check the transitions between different height levels.
- While rolling down a level under gravity is physically possible, moving vertically upward onto a higher cube without a ramp or external force makes the path physically impossible.
(i) Is there anything strange about the path of this ball?
Step 1 · Trace the Ball's Movement Across Levels
Analyzing the path using the three primary spatial directions (Left/Right, Forward/Backward, and Up/Down):
- The red ball moves horizontally along the top level.
- It transitions downward from a higher cube into the central depression, which is physically possible as a ball can fall or roll off an edge.
- It travels horizontally along the lower level.
- An arrow then indicates the ball moving vertically upward from the lower cube back to the higher cube.
Without a ramp, slope, or lifting mechanism, this upward vertical step is physically impossible for a rolling ball.
(i) Yes, the ball moves vertically upward from a lower level to a higher level without any ramp or lifting mechanism, creating a physically impossible path.
(ii) Recreate it on the isometric grid.
Step 1 · Construct the Structure on an Isometric Grid
Follow these steps to recreate the figure:
- Draw the outer boundary of the cube arrangement along the isometric axes.
- Form the central depression by drawing the lower level of cubes.
- Trace the arrows starting from the red ball on the top level, descending into the depression, moving across the bottom, and looping back up to the blue ball on the top tier.
(ii) The path is recreated on the isometric grid as shown in the diagram above.
- Overlooking Level Differences: Assuming all cube surfaces lie on the same horizontal plane instead of recognizing the central depression as a lower level.
- Missing the Physical Impossibility: Failing to notice that an arrow directly ascends a vertical step without any incline or mechanism.
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 step of the shape sequence that leads to the Koch Snowflake.
Find the perimeter of the shape at the 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) , , and
(ii) , , and
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?