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Rosette

Puzzle Board

polypad.org/3RXrF00MQgd5xQ

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:

rr: radius of each rotating circle.

dd: distance between the point of rotation and the circle centers.

  • If r=dr=d, then the rotating circles all meet at the centre of the rosette.
  • If r<dr <d, then there will be a smaller hole inside the original circle.
  • What happens when r>dr>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=rd=r, what is the ratio of perimeter of the rosette to the circumference of the original circle? In 6 petals? In 20 petals?

Hints

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

Examing the rotation angle for the 4-petal rosette.

  • Draw the segments OAOA and OBOB. if the angle between them is 9090 degrees, what can you tell about the arc-length of ABAB?
  • How many of the same AB arc does the 4 -petal rosette have?

Solution

polypad.org/8NZyZ3uKRayOg#probability

More on

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

polypad.org/Evcby3oM6elpkA