Monday, November 23, 2015

Alhambra, Granada, Spain

In October our dream finally came true - David and I went to Spain to see two places which we have been mentioning as examples in geometry class how geometry is used in design. (see about math here). Those places are Alhambra and Sagrada Familia (Gaudi), Granada and Barcelona. We actually visited Barcelona first but I want to start with Alhambra because that was a place we always wanted to visit. I just finished to sort through my 1400+ pictures, so here are just a few glimpses.
We arrived in Granada around midnight. Taxi driver took us into Albacin, Granada's Old City, stopped on one-way street, and showed on the left stone paved steps - This is your street, he said. I got out of the car and looked to the right and here it was - Alhambra, lit up at night, and so close.

On our way at night I noticed street lights in interesting geometric shape - cube with cutout pieces. What would be the volume of this solid? 
It turned out that we were too optimistic thinking that the end of October will be a low season and there will be no problem of getting into Alhambra. It was still high season and on-line all possible tickets were sold out until November. We walked up to Alhambra in hope that maybe somehow at the end of the day we can get in but at the gate we were stopped by tired and unkind woman who just said - everything is sold out. With millions visiting this World Heritage site every year, no wonder she did not explain us what we can do. I wish she would. But that is an advantage to travel without tight schedule. If anything else would fail, we could stay until the beginning of November when tickets were available on-line. When we told our hosts how our hopes to visit Alhambra failed, they promised to try to help. And they did - it was arranged that we go to one of the hotels and get there tickets for a guided tour. At that point we did not know that you can go to Alhambra early in the morning, wait in line until ticket office opens (there are even two lines - one for cash, one for credit cards). Every day they sell 1000 tickets there. (We used this to visit Alhambra second time.) 
And now - here we are - in Generalife. Our guide is talking about two types of fountains we can see - Muslim ones are quiet, Christian created ones can be heard from afar. I am trying to capture this nice double row of parabolas but light is not very good, and we have to move further. Each of us has a headphone set through which we are hearing our guide. At times I am so busy with taking pictures that I cannot see anymore where the group went. It is clear that walking through Alhambra with a group I will never have enough time to see and to take pictures - I am not the first one who wishes to capture everything. There is no time to think what geometric constructions I am seeing - take a picture, at home will be able to study on big screen... By the time we reach Nasrid Palaces, I have a plan - I explain to our guide that we both came here to look at geometry and we need  more time and ask can I give the headphones back to him. He understands us and while we are waiting in line to get into Nasrid Palaces, I slip two sets in his bag, and thank him for understanding. 

Now we can move slowly, enjoying every step. 
It does not matter how many times you have seen Alhambra, particularly this iconic image, entering Alhambra you are becoming breathless by the beauty that surround us - exquisite craftsmanship and imagination.
I am trying to take as many pictures as I can, not forgetting to put camera down and letting the beauty surround me.
When we reach Palace of the Lions, I am overwhelmed - I feel tears streaming over my cheeks - it feels so very personal to be here, in the place I was dreaming about so many years. I turn off my camera and walk around trying to imagine how it was when exquisite ceiling ornaments were still having a color which now has left only traces here and there. Mathematics is no more on my mind.

The Alhambra was so called because of its reddish walls (in Arabic, («qa'lat al-Hamra'» means Red Castle). It is located on top of the hill al-Sabika, on the left bank of the river Darro, to the west of the city of Granada and in front of the neighbourhoods of the Albaicin and of the Alcazaba. I saw the documentary The Alhambra:from sultan palace to legend before, but now I watch with different eyes. Thinking about geometry in Alhambra will be later, for now - just sweet memories ...

Friday, June 5, 2015

Geometry in Art, Geometric art or Artful geometry?

The Japanese artist folds light as a paper - very appropriate for  2015 - The Year of the Light

Or how about The Geometry of Hope?
It is a work by Iranian artist Monir Shahroudy Farmanfarmaian (b. Qazvin, Iran, 1924) who at the age of 91 had her first U.S. museum exhibition of mirror works and drawings in Guggenheim museum. Once a master told her that "everything is in geometry" and she followed his advice, exploring geometry through her art. Some more about her art here and here and here and her life story is here.

If light is too visceral - have your Zen moment watching these mesmerizing sand drawing machines:

or think of sand turning into glass and then into segmented glass sculptures by Jiyong Lee:

More pics and a video here.

I know - late following fashion - still enjoy Carolina Herrera's SS 2014 collection Lines and Geometry, which reviewers called "Lessons in Geometry";
Line segments turning into conceptual art in Hamid Naderi Yeganeh art:
More about his art here .

Lines become alive with dancers "bending" the light:

Saturday, February 28, 2015

Creativity of Palestinian math teachers

Yes, I am here - in Palestine, where I am teaching History of Mathematics and Vector Analysis in Palestinian Technical University - Kadoorie. Besides teaching these two courses I am working with students in creating exhibit about mathematics which hopefully will grow into something larger and more permanent. 
Today I was in Nablus and saw this:

It reminded me that I still owe this blog post to all creative and passionate teachers I met on our trip to five schools near and in Tulkarem. We were welcomed very sincerely with traditional Arabic coffee, homemade bread, juice, tea, sweets - something like that in every school. But most importantly - we had a chance to see wonderful exhibits made y teachers and their students in order to make mathematics more colorful and fun. I tried to take as much pictures as I could, not all of them came out well but I hope they captured the creative spirit in teaching mathematics. From five schools we visited, four were girls schools and one small one was co-ed. 
Mathematics becomes a fun when you can play with it. Students who were demonstrating us how to play these learning games were very confident and clearly enjoyed this playful approach to learning.
But these games teaches not only mathematics - they are all created from what would be considered trash - this is all very creative recycling that teaches students that things could and should be re-used. Recently one of our university hosts asked me if there is anything I do not like in his country. And I quickly replied - yes. Perhaps, too quickly because he looked at me quite surprised. I explained - what I do not like here is enormous amount of trash almost everywhere. What I cannot grasp is that people here are so passionate about Palestine at the same time tolerate their beloved land covered in trash. I think that keeping your homeland clean must be a part of patriotism.
I was very happy to see how students and teachers put a lot of efforts to re-use plastic bottles, jars, coffee cups, plastic silverware etc.

Here is a game where you spin two wheels then calculate result and that is where your colored cup will go on the board.
Good use of velcro to demonstrate formula of the area of the circle

This girl was ready to recite for us 400 digits of pi. 
The numbers on this "calculator" are glued on plastic water bottles.
Another use of water bottles to demonstrate how to solve simple equations. (we were visiting mostly elementary schools.)
Some examples of students geometric artwork.
Another way to solve an equation.

Wedding cake? No, illustration of number sets - the bottom is real numbers.
Another way to solve equations.
Greater, smaller or equal?

Which angle is which?
This is larger project students were working on - the school got a little extra piece of land, (which is really good news because most of the schools had almost no yard or very little one); and students are building scale model how they are going to use this land, doing all measurements, planning and calculation. 

No need to explain this ;-)
some  like the math to be illuminated
This is for me - for learning arabic numerals from 1 to 100. Remember - it all starts in bottom right corner...
Another project - building farm from geometric shapes

what is function?
I like to show this magic trick to my audiences but here it was on display.
triangles with the same base and same height have the same area.
of these trig functions - but this is very helpful tool.
I did not know this trick how to remember sin and cos values for these angles!
do you recognize Pascal triangle?

David and I were accompanied by very nice people from Education Department who really care about the education in this area.
Equations and the real life
These girls were playing math game during their break.
Thank you for such warm welcome and great math exhibit! 
The most successful exhibit was were math and art teachers worked together. This school is very lucky to have talented art teacher. Wouldn't it be great to have walls only for murals like these?

I do hope that this wall one day in not so far future will be gone. For now it is what David and I see when we turn towards Mediterranean sea which is only 12 km from Tulkarem if you get through this wall. This wall also has cut off part of university campus which is here seen from the wall side.