Fractals and Visualising Solids | FIO

Question 20

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.]

Solution

Understand the Question

An isometric grid uses isometric axes at $120^\circ$ angles to represent 3D objects on a 2D plane without distorting edge lengths along the three main directions: length, width, and height.
Each figure is constructed cube-by-cube or block-by-block by drawing line segments along these grid directions to form the edges of the solid.

(i) Draw the L-shaped block on the isometric grid.

Step 1 · Draw the L-Shaped Block

Diagram 1

Vertical tower ( $1 \times 1 \times 3$ ):
- Mark a starting point on the grid and draw a $1 \times 1$ rhombus base along the length and width directions.
- From each corner of the base, draw vertical line segments of length $3$ units upwards.
- Connect the top ends to complete the $1 \times 1$ top face of the vertical tower.
Horizontal extension ( $2 \times 1 \times 1$ ):
- Extend the base from the tower by $2$ units along the length direction.
- Draw vertical lines of $1$ unit height at the outer corners.
- Connect the edges to complete the top and side faces of the lower horizontal section.

Answer

(i) The L-shaped block is drawn on the isometric grid by combining a $1 \times 1 \times 3$ vertical column with an attached $2 \times 1 \times 1$ horizontal base.

(ii) Draw the T-shaped block on the isometric grid.

Step 1 · Draw the T-Shaped Block

Diagram 2

Horizontal base ( $3 \times 1 \times 1$ ):
- Draw a $3 \times 1$ base along the length and width directions.
- Draw $1$ -unit vertical lines from each corner and connect their top ends to form the $3 \times 1 \times 1$ base block.
Central vertical column ( $1 \times 1 \times 2$ ):
- Locate the central $1 \times 1$ square on the top face of the base block.
- Draw vertical lines of $2$ units height upwards from each corner of this central square.
- Connect the tops of these vertical segments to complete the $1 \times 1$ top face.

Answer

(ii) The T-shaped block is drawn on the isometric grid by placing a $1 \times 1 \times 2$ vertical block centrally on top of a $3 \times 1 \times 1$ horizontal base block.

(iii) Draw the staircase block on the isometric grid.

Step 1 · Draw the Staircase Block

Diagram 3

First (bottom) step ( $3 \times 1 \times 1$ ):
- Draw a $3 \times 1$ rectangular base on the grid and raise it by $1$ unit vertically to form the lowest platform.
Second (middle) step ( $2 \times 1 \times 1$ ):
- On the rear $2 \times 1$ section of the bottom step's top surface, draw vertical lines of $1$ unit height.
- Connect the top ends to create a $2 \times 1$ horizontal step surface.
Third (top) step ( $1 \times 1 \times 1$ ):
- On the rearmost $1 \times 1$ section of the second step's top surface, draw vertical lines of $1$ unit height.
- Connect the edges to complete the top $1 \times 1$ surface of the staircase.

Answer

(iii) The staircase block is drawn on the isometric grid by stacking three stepped layers of dimensions $3 \times 1 \times 1$ , $2 \times 1 \times 1$ , and $1 \times 1 \times 1$ .

Common Mistakes

Axis Misalignment: Drawing lines parallel to normal horizontal grid lines instead of following the $30^\circ$ isometric axes for length and width.
Counting Grid Units Incorrectly: Miscounting isometric dot spaces for the height or length dimensions of individual unit cubes.
Drawing Hidden Edges: Drawing interior or rear edges with solid lines instead of only drawing visible outer edges.

Question 20

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