How do we construct this figure?

Question

Homi · Accepted Answer

The figure is an 8-petal flower with rotational symmetry.

Step 1 — Draw the base circle and center

Let us draw a point. Let us call this point O. This is the center of our flower.

Let us use a compass. Let us open the compass to a convenient radius. Let us call this radius R.

Let us draw a circle with center O and radius R. This circle defines the outer tips of our petals.

Diagram 1

Step 2 — Divide the circle into 8 equal parts

Let us draw a horizontal diameter through O. Let its endpoints on the circle be $P_1$ and $P_5$ .

Let us draw a vertical diameter through O. This diameter must be perpendicular to the first one. Let its endpoints on the circle be $P_3$ and $P_7$ .

We now have 4 points on the circle: $P_1, P_3, P_5, P_7$ . They are equally spaced.

Let us bisect the angles between these diameters. For example, to bisect the angle $\angle P_1 O P_3$ :

Place the compass at $P_1$ . Draw an arc.

Place the compass at $P_3$ . Draw an arc with the same radius.

The arcs intersect at two points. Draw a line from O through one of these intersection points. This line is an angle bisector.

Extend this line to meet the circle. This gives two new points on the circle. Let us call them $P_2$ and $P_6$ .

Repeat this for the other angles (e.g., $\angle P_3 O P_5$ ). This gives two more points, $P_4$ and $P_8$ .

We now have 8 equally spaced points on the circle: $P_1, P_2, P_3, P_4, P_5, P_6, P_7, P_8$ . These will be the tips of our 8 petals.

Diagram 2

Step 3 — Construct one petal

Let us construct the petal that has $P_1$ as its tip. This petal starts at O and ends at $P_1$ .

Its two sides are curved arcs. These arcs bulge outwards.

Let us find the center for the left arc of this petal. This arc goes from O to $P_1$ .

The center of this arc must be equidistant from O and $P_1$ . So, it lies on the perpendicular bisector of the line segment $OP_1$ .

Let us draw the perpendicular bisector of $OP_1$ .

The center of the arc also lies on the radial line $OP_2$ . This is the radial line adjacent to $OP_1$ in the counter-clockwise direction.

Let us find the point where the perpendicular bisector of $OP_1$ intersects the line $OP_2$ . Let us call this point $C_1$ .

Now, place the compass at $C_1$ . Set the compass radius to $C_1O$ .

Draw an arc from O to $P_1$ . This forms the left side of the first petal.

Next, let us find the center for the right arc of this petal. This arc also goes from O to $P_1$ .

This center must also be equidistant from O and $P_1$ . So, it lies on the perpendicular bisector of $OP_1$ .

The center of this arc also lies on the radial line $OP_8$ . This is the radial line adjacent to $OP_1$ in the clockwise direction.

Let us find the point where the perpendicular bisector of $OP_1$ intersects the line $OP_8$ . Let us call this point $C_2$ .

Now, place the compass at $C_2$ . Set the compass radius to $C_2O$ .

Draw an arc from O to $P_1$ . This forms the right side of the first petal.

We have now constructed one complete petal.

Diagram 3

Step 4 — Complete the remaining petals

Let us repeat Step 3 for all the other 7 petals.

For each point $P_i$ (from $P_2$ to $P_8$ ):

Draw the perpendicular bisector of the line segment $OP_i$ .

Find the intersection of this bisector with the radial line $OP_{i+1}$ (or $OP_1$ if $i=8$ ). Let this be $C_{i, \text{left}}$ .

Find the intersection of this bisector with the radial line $OP_{i-1}$ (or $OP_8$ if $i=1$ ). Let this be $C_{i, \text{right}}$ .

With $C_{i, \text{left}}$ as center and radius $C_{i, \text{left}}O$ , draw an arc from O to $P_i$ .

With $C_{i, \text{right}}$ as center and radius $C_{i, \text{right}}O$ , draw an arc from O to $P_i$ .

After drawing all 16 arcs (two for each of the 8 petals), the figure will be complete.

Diagram 4

Question 11

More questions in IT