Sunday, December 6, 2015

La Sagrada Familia - Gaudi's dreams through geometry

The stunning La Sagrada Familia since 1882 is still "under construction", however it is not an obstacle for estimated 3.2 million people visiting it every year. The first architect of this cathedral was Francisco del Vilar who built the crypt but then had some disagreements about the design with commissioners of the project and the project was passed to Antonio Gaudi. There is a hope that the project will be finished by the centennial of Gaudi's death in 2026, but already there are estimates that to see really everything finished we will have to wait till 2040. When completed, the cathedral will be 170 m high, one meter less than the highest point in Barcelona because Gaudi believed that no man-made object should be higher than something created by God.
There is quite noticeable difference between parts that are completed in various times.

The pictures of La Sagrada Familia I have seen before but it was absolutely jaw dropping moment when we walked in the building and there was this amazing light. We were so lucky to be there in sunny day.
"I am geometer, therefore I synthesize." A.Gaudi
Helicoidal staircases are in every tower.

Magic square as a door knob
Alfa and Omega sign above the door of Passion Facade

Explanation of surfaces used by Gaudi - going through the museum in the basement of La Sagrada Familia is like having aplied geometry lesson :-)
Model of hyperboloid which is developable surface - that is it can be constructed using straight lines.
A mirror under teh model gives an idea how the ceiling was constructed.

Gaudi had a lot of models and plans, because he understood that La Sagrada Familia can not be completed during his life time, he set up everything so that the construction can go on after he passe away. Unfortunately in 1936 during Spanish Civil War leftists set a fire on his workshop and most of Gaudi's plans were burned. Luckily from whatever was left painstakingly new generation is reviving Gaudi dreams - inspired by natural forms, based on geometry.
This is how catenary arcs were constructed.

This magic square is on Passion Facade. Gaudi never revealed why he choose 33 as a magic number for this square. I think he was depicting the age of Jesus.
There is also this labirinth on the outside wall.
Infinity sign

To get a larger perspective of this wonderful building watch the video La Sagrada Familia. It was filmed in  April 2012, when Luis Caldevilla and Alberto Castaño were invited by the Director of the Construction Board, to film in the basilica using time-lapse and drone aerial techniques. BBC Culture article.

We also visited Casa Mila or La Pedrera -

Door knobs in La Pedrera - I was wondering why they are in this strange shape but it apears that they are very comfortable - Gaudi constructed door knobs and furniture based on human figure - for example, door knobs were constructed by taking a piece of clay, holding it, then this imprint was transferred to the metal.
Wish all architects would hide ugly chimneys under such playful forms!
Such fun - to have an art lesson on a roof of La Pedrera!
View to La Sagrada Familia
Construction of catenaries
and realization of catenaries in the attic of La Pedrera
The other house I visited was Casa Batlló 
Nice catenaries also here.
Unfortunately all tickets were sold for the most interesting part of Park Güell we had only a few glimpses from outside. Actually it is not one park but a system of municipal parks on hillside and most of it is freely accessible.

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 ...