#1. We are asked to fold a series of isoceles right angled triangles. Each one in the series has double the area of the one before. We are shown a number of patterns made from this series of triangles:
I choose to work out the ratios of the side lengths created by the touching vertices. I think there is a reasonably elegant solution. A friend is intrigued by the same image, but in the loci of the touching vertices.
How do mathematicians, and teachers of mathematics, develop an intuition for the interesting? How might we develop this intuition in the students we teach?
#2. Fran asks: What might we learn from paper folding problems?
I wonder: is it possible to know at the time of doing what we are learning? Perhaps we can only know what we have learned at a later time, when some action comes to mind.
Gattegno described an awareness as, ‘That which enables action’, which is related to this comment on 'locating action' by John Mason (in reply to this blog):
[An] action... initiates an action, and what is vital is access to useful actions to follow it... what about "where did we make a choice and what was the basis for that choice?" ...or, "what would you do in a similar situation? and what would make a situation similar to this one?"
This is how it is possible to expand and enrich one's repertoire AND remember to invoke an action next time!
#3. I hold a skeletal octahedron in my hands (below). I am amazed at its rigidity. By what mechanism did these unconnected, flat sheets of paper form this 3-dimensional object that I can toss around? From where does this rigidity originate? I conjecture it is from closed inter-locking cycles of inter-dependent foldings and tuckings, perhaps across dimensions. The folds in some pieces of paper hold other pieces in place, reducing possible movement (degrees of freedom), relative to each other.
How do origamists design these rigid constructions? From intuition, or through mathematical design? Some of the answers might be in the mathematics of linkages.
#4. For the first half of the session, I become engrossed in the problem described in #1. While working on it, the person next to me asks: ‘Are you struggling?’ A while later, she asks, ‘Are you still struggling?’
I want to say, 'No, I'm not struggling, I'm just doing some maths...'
#5. Look at image of a piece of folded paper below. Can you describe what you see? Now imagine repeatedly unfolding and folding the paper. Can you visualise what folds were performed on the original flat sheet of paper?
If we press the object in this image flat, the border becomes a ‘line’. I marvel at its unfolding, through 3-dimensional space, into a square. I feel a sense of - I really don’t know how to describe this - the manifold in which we exist/perceive.
#6. On my way home, passing through Kings Cross station, I notice a man with a white stick walking towards the escalator. His stick comes to a halt on a raised metal plate on the floor. He turns round and walks away. I pause to see where he is going, suspecting he might need help; he walks towards a dead end suggesting that he might be lost.
I walk towards him and ask if I can help. He says he is looking for the, ‘First capital connect ticket barrier.’ I say I will take him to it, and we walk together back through the station. I cannot see any signs for the barrier, so I say I’ll take him to someone who will be able to help. He says that his train is coming in 6 minutes. We might not have time to ask for help, so I decide to describe what I can see: ‘There’s no first capital connect, I can only see signs for Eurostar, East Midlands trains and Thameslink,,,’ ‘Ah, that’s it, Thameslink!’ he replies. I lead him through the station. He does not need me to take his arm as long as I stay close; he is blind in one eye, and can see very little with the other. I take him to the barriers; he now knows where he is going.
#7. Watch this video of the session. What do you notice?