Simple proofs?
Simple proofs?
OK, imagine for a minute that I know little about mathematics  what simple geometric proofs does origami provide that could be used in a class?

 Posts: 80
 Joined: Tue Dec 06, 2011 11:40 am
Re: Simple proofs?
To show that the internal angles of a triangle always add up to 180°, take a triangle and fold two opposite corners to meet each other, then fold the last corner in. The three corners will meet and form an angle of 180°. I first saw this in an old BOS magazine, but it's probably in books on geometrical paperfolding.
Re: Simple proofs?
That's great  simple, yet satisfying and works for any triangle, as far as I can see. A lot of the maths/geometry I see is just too advanced to be of use with younger children.Edwin Corrie wrote: I first saw this in an old BOS magazine, but it's probably in books on geometrical paperfolding.
I found a video.
Thanks Edwin!
Re: Simple proofs?
Hello!
As a mathematician of sorts I cannot help but feel a bit uneasy about the use of the word 'proof' in contexts such as this. I'm reminded of some correspondence that I had with Paul Jackson about proving that a piece of paper was square.... However, I do agree that there are many things that can be demonstrated or illustrated by practical exercises, and by origami in particular.
In 2009 I wrote a piece about what could be demonstrated by folding a triangle, which started with this demonstration. I shall post the file later, but I should like to refer to an extract from it here. To make this demonstration work, you need to begin by folding the altitude of the triangle from the top vertex, or pinch the foot of this altitude on the base, and fold the vertices to this pinch mark. This demonstration works for any scalene triangle without any problem, but for isosceles triangles two of the altitudes will fall outside the base. In such cases, it is necessary to have the longest edge as the base.
As with a lot of mathematics, there are special cases  in this case the limiting case between scalene and obtuseangled triangles: a rightangled triangle. If the rightangle is on the base there will be just two small triangles to fold over, and the demonstration has to be 'explained'. The remaining two angles add to 90 degrees (on top of the rightangle), so the total is 180 degrees. Of course, if you make the longest edge (hypotenuse) the base, then there is not a problem.
Ian
As a mathematician of sorts I cannot help but feel a bit uneasy about the use of the word 'proof' in contexts such as this. I'm reminded of some correspondence that I had with Paul Jackson about proving that a piece of paper was square.... However, I do agree that there are many things that can be demonstrated or illustrated by practical exercises, and by origami in particular.
In 2009 I wrote a piece about what could be demonstrated by folding a triangle, which started with this demonstration. I shall post the file later, but I should like to refer to an extract from it here. To make this demonstration work, you need to begin by folding the altitude of the triangle from the top vertex, or pinch the foot of this altitude on the base, and fold the vertices to this pinch mark. This demonstration works for any scalene triangle without any problem, but for isosceles triangles two of the altitudes will fall outside the base. In such cases, it is necessary to have the longest edge as the base.
As with a lot of mathematics, there are special cases  in this case the limiting case between scalene and obtuseangled triangles: a rightangled triangle. If the rightangle is on the base there will be just two small triangles to fold over, and the demonstration has to be 'explained'. The remaining two angles add to 90 degrees (on top of the rightangle), so the total is 180 degrees. Of course, if you make the longest edge (hypotenuse) the base, then there is not a problem.
Ian

 Posts: 4
 Joined: Sun Jun 17, 2012 11:03 am
Re: Simple proofs?
There are a couple of proofs that can be demonstrated using origami, but I've found it useful it as a method of teaching HOW to prove something.
In my case I have shown Haga's theorem 'If you fold a bottom corner of a square onto the centre point of the top edge, the bottom edge of the square intersects the side edge at the 1/3 point.
Prove it.'
You can demonstrate the truth of it by simply folding, but using the paper you can direct the students through the process of starting with what you know. (Square, rightangled triangles, top half split in two) and using that to increase what you know (show that the two triangles at the top are 'similar triangles')
Using that knowledge you can then workout the ratio of the triangles, etc. etc. and eventually get to a proof.
At each stage the physical presence of the paper is an aid to the process.
In my case I have shown Haga's theorem 'If you fold a bottom corner of a square onto the centre point of the top edge, the bottom edge of the square intersects the side edge at the 1/3 point.
Prove it.'
You can demonstrate the truth of it by simply folding, but using the paper you can direct the students through the process of starting with what you know. (Square, rightangled triangles, top half split in two) and using that to increase what you know (show that the two triangles at the top are 'similar triangles')
Using that knowledge you can then workout the ratio of the triangles, etc. etc. and eventually get to a proof.
At each stage the physical presence of the paper is an aid to the process.
Re: Simple proofs?
Do you have any visuals of this Dennis? It sounds interestig, but I can't picture it...origamidennis wrote: You can demonstrate the truth of it by simply folding, but using the paper you can direct the students through the process of starting with what you know. (Square, rightangled triangles, top half split in two) and using that to increase what you know (show that the two triangles at the top are 'similar triangles')

 Posts: 4
 Joined: Sun Jun 17, 2012 11:03 am
Re: Simple proofs?
Hi Nick, I don't at the moment, but I could make some.(but it'll be a couple of weeks away before I can get around to it!)Nick wrote:
Do you have any visuals of this Dennis? It sounds interestig, but I can't picture it...