How do we construct this figure?

Question

Homi · Accepted Answer

Understand the Question

The figure is a symmetric 8-petal flower inscribed inside a circle.
To construct it:
1. Draw a base circle with center $O$ and divide it into 8 equal angular sectors ( $45^\circ$ each) to get petal tips $P_1, P_2, \dots, P_8$ .
2. Construct outward-curving circular arcs connecting center $O$ to each tip $P_i$ using arc centers found at the intersection of the perpendicular bisector of $OP_i$ and the adjacent radial lines.
3. Repeat symmetrically for all 8 petals.

Step 1 · Draw the Base Circle and Center

Mark a center point $O$ . Using a compass set to a convenient radius $R$ , draw a circle with center $O$ . This circle defines the outer tips of all petals. Diagram 1

Step 2 · Divide the Circle into 8 Equal Parts

Diagram 2

Draw a horizontal diameter through $O$ meeting the circle at $P_1$ and $P_5$ .
Draw a vertical diameter through $O$ perpendicular to the first, meeting the circle at $P_3$ and $P_7$ .
Bisect the four $90^\circ$ angles (e.g., $\angle P_1OP_3$ and $\angle P_3OP_5$ ) by drawing intersecting arcs from adjacent points.
Extend the bisectors through center $O$ to meet the circle at $P_2, P_4, P_6,$ and $P_8$ .

This gives 8 equally spaced points on the circumference at $45^\circ$ intervals.

Step 3 · Construct the First Petal

Diagram 3

For the petal with tip $P_1$ :

Draw the perpendicular bisector of segment $OP_1$ .
Let the bisector intersect radial line $OP_2$ at center $C_1$ , and radial line $OP_8$ at center $C_2$ .
With center $C_1$ and radius $C_1O$ , draw an arc from $O$ to $P_1$ (left side of petal).
With center $C_2$ and radius $C_2O$ , draw an arc from $O$ to $P_1$ (right side of petal).

Step 4 · Construct the Remaining Petals

Diagram 4

Repeat the process for each tip $P_i$ (for $i = 2, 3, \dots, 8$ ):

Draw the perpendicular bisector of $OP_i$ .
Find its intersections with adjacent radial lines $OP_{i+1}$ and $OP_{i-1}$ to locate the arc centers $C_{i, \text{left}}$ and $C_{i, \text{right}}$ .
Draw the two symmetric arcs connecting $O$ to $P_i$ .

Drawing all 16 arcs completes the 8-petal flower.

Answer

The 8-petal symmetrical flower design is constructed by dividing the circle into 8 equal parts of $45^\circ$ each and drawing symmetric circular arcs from center $O$ to each vertex $P_i$ .

Common Mistakes

Inaccurate Angle Bisection: Imprecise angle bisectors lead to unequal petal widths and broken rotational symmetry.
Incorrect Arc Centers: Forgetting to use the perpendicular bisector of $OP_i$ when locating centers $C_1$ and $C_2$ , resulting in arcs that do not smoothly meet at both $O$ and $P_i$ .
Radius Inconsistency: Changing the compass width between drawing the left and right arcs of a petal, causing asymmetric petals.

Question 11

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