Rotational symmetry of order n means the object can be rotated about the point of symmetry by 360/n degrees and the result is the same as original in geometry. Circle has a rotational symmetry of infinite order with respect to its centre. That is, rotation of the circle about its centre by any angle does not change its geometry. Two diametrically opposite points on the circumference divide the circumference into two arcs of equal lengths. Thirty equally spaced points on the circumference make 30 equal arcs out of the circumference. From the point (7) one travels 15 arcs (half the circumference) to point (22) which is half way around. Therefore points (7) and (22) are diametrically opposite to each other.
The process to the solution is based on these two
facts about the circle:
1)
Circle has rotational symmetry of infinite order.
2)
The same angles at the centre correspond to the arcs of the same length.
An extended puzzle: A farmer has fencing material of length 600 meters. The farmer mistakenly assume that circumference of a circle of radius r is 6r. He starts building the circular fence with radius 100 meters, thinking that he would get a circular enclosure. After laying just over 300 meters of fencing material, he realizes that he is wrong. He manages to close the fence by laying the last part straight. Does the farmer get the maximum enclosed area?
The key to this question is to know what region on the flat plane has maximum area to perimeter ratio. The circle has the maximum area to perimeter ratio. The farmer finishes the enclosed area of the following shape.
What
he gets is not circular region. If he correctly calculates the radius for the
circle with 600 m perimeter, then he would get
With this radius he would get a circular area larger than the one in above diagram.
Regarding the solution: In our life when
we face with problems some possibilities are:
1)
There is no solution.
2)
There is one solution.
3)
There are finitely many solutions.
4)
There are infinitely many solutions.
5)
We know there is a solution. However, we are unable to compute exact solution
due to our insufficient computing ability (capacity) or due to resource
limitation. We can find approximate solution(s).
6)
We can simplify the problem to a simpler form. However we cannot get better for
now.
7)
We do not know if the problem is solvable (need more knowledge).
The
number of solutions (including zero) can be due to the constraints. For example,
in the problem: x+y = 5 with both x and y are real integers of at least three, there is
no solution. But if we can take x and y any positive integers of at least two, then
there are two solutions:
(x,y) = (2,3),(3,2). If x and y can be any integers, then there are infinitely
many solutions. Students should be allowed to experience a variety
of solution-situations.
Chloe, thank you for your very mathematical solution, fascinating extension problem (love this!!) and extremely interesting list of the possible solution spaces for any math problems. This is a super interesting tour-de-force!!
ReplyDeleteNow I'd like you to think in your teacherly way as well. If you were teaching, say, a Grade 10 math class, how might you explain this to them? You might want to help them get initiated into university-level mathematical language, but they will certainly not be there at the start. I would love to see you bringing your mathematical thinking skills to share with your students in ways that they can access!