## Puzzle Board

A circular rosette is made by rotating circles about a fixed point and angle. Circular rosettes have a long history, going back at least 6000 years, and were popular in the Egyptian, Babylonian, Assyrian, and Greek cultures, amongst others.

To begin, explore the following relationships:

$r$: radius of each rotating circle.

$d$: distance between the point of rotation and the circle centers.

- If $r=d$, then the rotating circles all meet at the centre of the rosette.

- If $r <d$, then there will be a smaller hole inside the original circle.

- What happens when $r>d$?

The angle of rotation of the circles determines the number of petals in the rosette.

- If the angle of rotation is 45 degrees, how many petals will the rosetta have?
- If a circular rosetta has 12 petals, what is the angle of rotation?

In the given 4-petals rosette where $d=r$, what is the ratio of perimeter of the rosette to the circumference of the purple circle? In 6 petals? In 20 petals?

## Hints

Let $R$ be the radius of the original circle and $r$ be the radius of the rotating circles. $R=2r$

Examing the rotation angle for the 4-petal rosette.

- Draw the segments $OA$ and $OB$. if the angle between them is $90$ degrees, what can you tell about the arc-length of $AB$?
- How many of the same AB arc does the 4 -petal rosette have?

## Solution

## More on

Explore the relation between the circumference of the original circle and the perimeter of the rosettes when $r<d$ and $r>d$.