Geometry Geometry

1,000 Lines of Symmetry


“You’d run out of room in the circle!”

How do kids come to see lines as having no thickness at all? What experiences would support that change?

6 replies on “1,000 Lines of Symmetry”

It’s more troublesome than being in love. That can be experienced, and is of a considerably lower level of abstraction. In any real world situation lines do have a thickness, but it can be very small, a silk fiber, only just visible. All one can hope for is that the abstract, no breadth line is a very useful mathematical fiction (called an abstraction).
I do like the answer of 1000. It is a good real world answer.

Aren’t the ideas of infinite thinness and limits kind of connected? Like, being able to see circles as having infinite, infinitely short sides is related to being able to conceptualize lines in a similarly playful way. There are as many lines of symmetry as their are side on a circle… Because those lines are as skinny as you want them to be!

Isn’t 1000 the largest natural number? If a very young child gave that answer, it would be pretty decent. 🙂

As for the “no thickness” idea, they must be reading their Euclid: “Def. 1.2. A line is a breadthless length.”

Lines and points are some of the things that can’t be defined in math. Perfect shapes only exist mentally, if the world is quantized (not continuous like a real line).
You can’t prove it, but the new geometry without the “breadthless” condition doesn’t seems very fun or different either.

This simply one picture interested me a lot. From the answer that child came up with, I wondered how did the child come up with the idea and exact number with “1,000”. I believe the reason why the child answered as 1,000 is because the largest number that child knows is 1,000. Also after 1,000, the child believes that will be no enough room to draw lines in the circle since the image of circle is too small. From this idea, I could notice abstractions of child that how he thinks and draws circle in his mind. Also, when I was child, I did not understand why circle as infinite lines. Because other shapes have vertex but for circle there were no clear vertex that is shown from the shape. I think this problem set is a good idea for children to promotes critical thinking and reasoning.

I also think that the largest number that the student knows is 1,000. This is the reason why the student specifically wrote 1,000. Of course there are more than 1,000 lines of symmetry like infinity, but even though it is over 1,000, the student may not be able to count more than 1,000. For the second question, the student may not know what “Symmetry” means though from the student’s answer “I can’t really.” If it was triangle or square, it was easier for the child to divide the shape in half, but since circle is hard to measure where is the middle or half, the child may get confused with it. I honestly do not understand what the student’s third answer means. To teach this kind of problem better, teacher should give the students actual circle and ask to fold it in half. In addition, teacher should give clear explanation about what is “Symmetry.”

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