Question 19

Step 1 · Construct an Inflexed Arc

1. Draw a horizontal base line segment $AB$.
2. Draw vertical lines upwards from $A$ and $B$. Mark points $C$ on the vertical line at $A$ and $D$ on the vertical line at $B$ such that $AC = BD$.
3. Find the midpoint $M$ of segment $CD$.
4. Draw a vertical centerline upwards through $M$.
5. Set the compass radius to $CM$.
6. Place the compass at center $C$ and draw an arc curving towards the centerline.
7. With the same radius ($DM = CM$), place the compass at center $D$ and draw an arc curving towards the centerline.
8. The two arcs intersect at point $E$ on the vertical centerline, completing the figure formed by vertical segments $AC$, $BD$ and arcs $CE$, $DE$.

Step 1 · Construct a Flower-like Shape

1. Draw a central circle with center $O$ and radius $R$.
2. Mark any point $A$ on the circumference of the central circle.
3. With center $A$ and radius $R$, draw a circle passing through $O$ and intersecting the central circle at two points; label one point $B$.
4. With center $B$ and radius $R$, draw another circle intersecting the central circle at a new point $C$.
5. Repeat this process around the circumference using each new intersection point as the center until 6 surrounding circles are drawn.
6. The overlapping arcs inside the central circle form the 6 petals.

Step 1 · Construct a Regular Hexagon Inscribed in a Circle

1. Draw a circle with center $O$ and radius $R$.
2. Mark a point $A$ on the circumference.
3. With center $A$ and compass set to radius $R$, draw an arc to cut the circle at point $B$.
4. With center $B$ and radius $R$, draw an arc to cut the circle at point $C$.
5. Continue marking points around the circumference with radius $R$ until 6 points ($A, B, C, D, E, F$) are marked.
6. Connect consecutive vertices with straight lines ($AB, BC, CD, DE, EF, FA$) to form the regular hexagon.

Step 1 · Construct Seven Intersecting Circles

1. Draw a central circle with center $O$ and radius $R$.
2. Choose a point $A$ on the circumference and draw a circle of radius $R$ centered at $A$.
3. Locate the intersection point $B$ of this circle with the central circle.
4. Draw the next circle of radius $R$ centered at $B$.
5. Continue moving to each successive intersection on the central circle until all 6 surrounding circles are drawn.
6. Keep all 7 full circles visible to complete the design.

Step 1 · Construct a Hexagonal Tiling Pattern

1. Draw a circle with center $O$ and radius $R$, and construct an inscribed regular hexagon with vertices $V_1, V_2, V_3, V_4, V_5, V_6$.
2. Draw segments from center $O$ to each vertex $V_1, V_2, \dots, V_6$, dividing the hexagon into 6 large equilateral triangles (e.g., $\Delta OV_1V_2$).
3. In each equilateral triangle:
   • Divide each side ($OV_1, OV_2, V_1V_2$) into 3 equal segments.
   • Connect corresponding division points with lines parallel to the sides of the triangle, subdividing each large triangle into 9 smaller equilateral triangles.
4. Repeat for all 6 large equilateral triangles to reveal the complete geometric tiling pattern of stars and smaller hexagons.