Mystic Rose

A Mystic Rose is a beautiful image created by joining together points that are equally spaced around a circle.

Watch the animation below to see how a Mystic Rose can be constructed. You can change the number of points around the circle.

Can you describe how to construct a Mystic Rose?

Alison and Charlie have been working out how many lines are needed to draw a 10 pointed Mystic Rose.

Alison worked out $9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45$.

Charlie worked out $\frac{10 \times 9}{2}$ = 45

Can you explain how each method relates to the construction of a 10 pointed Mystic Rose?

How would Alison work out the number of lines needed for other Mystic Roses?
How would Charlie work them out?

Whose method do you prefer?

How many lines are needed for a 100 pointed Mystic Rose?

Could there be a Mystic Rose with exactly 4851 lines? Or 6214 lines? Or 3655 lines? Or 7626 lines? Or 8656 lines?

How did you decide?

Comments

Mystic rose

How many lines are needed foe a 100 poinyed Mystic Rose ?
Well I calculated this by logic. If you write it down the number of lines needed each time ( 1,2,3,4....98,99,100) you will found out the there are 50 pairs of 101, 100+1 99+2 98+3 ... So in total there are 101*50=5050

Could there be a Mystic Rose with exactly 4851 lines?
Well we use the formula < n/2 * (n+1) >. This means that n**2 + n is equal to 4851*2 which is equal to 9702. Then we solve the quadratic equation and we get the n is equal to 98. You can do this with any number of lines.

Very nice, Sergio. I wonder

Very nice, Sergio. I wonder if you could explain where the formula n/2*(n+1) comes from? Why does this describe the number of lines in a Mystic Rose?

Mystic Rose

How many lines are needed for a 100 pointed Mystic Rose?

Well I calculated this by logic. If you write down the number of lines needed each time (1, 2, 3, 4 .... 98, 99, 100) you will find that there are 50 pairs of 101 (100+1, 99+2, 98+3, ... 51+50).
So in total there are 101x50 = 5050 lines

Could there be a Mystic Rose with exactly 4851 lines?

Well we use the formula $\frac{n}{2}$ x (n+1) = 4851

Mystic Rose

Daniel, can you explain why the number of lines needed will be 1+2+3+4+ ... +98+99+100?

And can you explain how you arrived at the formula?