Saturday, November 10, 2012

Connecting the eight points of intersection of the circle and the sides of the square form an octagon.

Roof over the Hall of  Abencerrajes @ Alhambra

Interior view of the Hall of  Abencerrajes @ Alhambra

Just about every medieval cathedral has an octagonal baptismal and the Alhambra in Granada Spain has several octagonal ground plans. The roof over the Hall of the Abencerrajes is unique because the roof actually follows the Ad Quadratum ground plan that is used to develop the octagonal ground plan. Squares rotated in the circle.

Ad Quadratum ground plan geometrically developed for hip and valley rafters.

Developing the roof framing plan is pretty straight forward if you use polygon roof framing geometry. The hip rafter is base on a polygon with 4 sides and the valley rafter is based on a polygon with 8 sides.

Using the Rafter Tools for Android app select the Polygon Rafter Angles calculator and enter 4 for the Number of Sides of Polygon to calculate the angles for the hip rafter and use number of sides = 8 for the valley rafter slope angle. These angles can also be found on the Chappell Master Framing Square.

The adjacent radius can be calculated using the law of sines.

Octagon 8 Sides
2 - Squares Rotated in Circle

Adjacent Radius = ( Main Radius × sin( B ) ) ÷ sin( C)
Adjacent Radius = ( 144 × sin(45 ) ) ÷ sin(112.5)
110.2128 = ( 144 × sin(45 ) ) ÷ sin(112.5)

Dodecagon - 12 sides
3 - Squares Rotated in Circle

Adjacent Radius = ( Main Radius × sin( B ) ) ÷ sin( C)
Adjacent Radius = ( 144 × sin( 45 ) ) ÷ sin( 120)
117.5755 = ( 144 × sin( 45 ) ) ÷ sin( 120)

Hexadecagon - 16 sides
4 - Squares Rotated in Circle
Adjacent Radius = ( Main Radius × sin( B ) ) ÷ sin( C)
Adjacent Radius = ( 144 × sin( 45 ) ) ÷ sin( 124.08620)
122.4619 = ( 144 × sin( 45 ) ) ÷ sin( 124.08620)

I'll post more information on the hip rafter and valley rafter head cuts in the next article on the Ad Quadratum ground plan.

1. This is just fantastic... I've been searching for a long time for information about Medieval construction and your articles and posts and now this blog are so informative! Just as you got interested after seeing "Pillars of the Earth," so did I just this past week on Netfilx. Was able to watch it streaming on my computer here in Guatemala, Central America, and then find all this fascinating information on the web. Thank God for large and small favors!

"The world is made by artists and architects, and art makes life worth living!"

Gracias, Senor. Will spend many happy hours here reading away. Hope to be able to put some of this into practice in the coming years! I've posted a bit of your writing on my own eco-architecture blog with many links back to your writings. Thank you so much!

Catherine Todd,
Panajachel, Lake Atitlan, Guatemala, Central America

1. Catherine,

Yes, the movie "Pillars of the Earth" really peaked my interest in medieval construction/architecture as well. After I saw the movie I read the book and several others on medieval architecture. The history of the medieval craftsman is as interesting as the architecture. It was only through my studies on medieval architecture that I was able to see that the Alhambra Granada was based on an Ad Quadratum Ground Plan that all medieval craftsman would have used regularly.

Thanks for stopping by,
Sim

2. I am trying to subscribe by email...

3. Dear Sim, thank you for responding so quickly! I have had many interesting hours reading through your discussions and information online and have posted some of it on my eco-architecture blog. Can't wait to give a lot of this a try here in Panajachel, Lake Atitlan, Guatemala, Central America where I live now. There's lots of Spanish Colonial Architecture and many old mission churches based in some of the beautiful and inspiring Medieval Gothic techniques.

You wrote:

"The history of the medieval craftsman is as interesting as the architecture."

I bet! Can you recommend some books? My husband will bring them with him when he comes and some I might be able to download via Kindle on Amazon. Watching "The Pillars of the Earth" was quite an eye-opener, as I've always wondered how in the world the Medieval Cathederals were built and I've been collecting books for quite some time. It's the only "time" in the world that seems important to me. I don't know why, but there it is.

Living in France was the highlight of my life, being able to be in the cathedrals on a daily basis and walking through the marketplaces that have changed so little over time... and it's *almost* the same here in Guatemala, especially in la Antigua Gautemala, where over 300 churches, monasteries and convents were built before the capital moved to Guatemala City.

Where can I find out more about " the Alhambra Granada [that] was based on an Ad Quadratum Ground Plan that all medieval craftsman would have used regularly."

This blog entry has a lot of good information, but most of it is beyond me. How would I get actual plans to build something like this here? I'm on a very tight budget (I work with indigenous Mayan Indians doing beautiful beadwork and handicrafts) but am planning on starting to build with bamboo and adobe, materials that are available here free or cheap.

Plus I'll be using recycled materials and sand-filled plastic bottles as "bottle bricks." The arches and roofs you are describing could be used on top of any of the different types of walls we will build. Concrete block is also used here frequently, although I prefer adobe myself.

Do you have any ideas to visit Guatemala or Central America in the future? It's "poor man's Paris," or "Poor Man's Paradise" due to the incredible 75 degree weather year-round and the art and flowers and birds all the time.

Any ideas you have would be most welcome! I am determined to build my own small cathedral one way or the other, as I have been waiting my whole life to do so. Until I found your information, I was gathering books on the subject of Cathedral Construction, but I never understood the details until I've been able to read your entries. I still don't understand much, but the pictures and descriptions really help. Who knew simple Geometry could accomplish so much? And by so many people who could not read or write. I still don't understand how the framing layouts were accomplished or the heights without transit or levels.

1. Off to work, but you can start on this page

Medieval vaulting geometry research, links, notes and keywords

The Alhambra Granada that is based on the Ad Quadratum Ground Plan is something that I discovered. It's not in any books that I'm aware of.

2. Catherine,

I first saw the picture of roof over the Hall of the Abencerrajes at Alhambra Granada in the documentation for the Carpenters Master Framing Square (Chappell Framing Square)about a year ago. Then a couple of months ago I was in the process of studying some polygon roof framing angles and started to draw out the roof the Hall of the Abencerrajes thinking it was based on some type of Octagonal ground plan. That's when I discovered it was based on the two squares rotated in the circle, Ad Quadratum, that all other medieval cathedral baptismal's are based on. It's not a cathedral, unless it has a baptismal based on the sacred cut.
Ad Quadratum,the Sacred Cut, & Roman Architecture
http://www.dartmouth.edu/~matc/math5.geometry/unit7/unit7.h
tml#cut

Stonemasons and carpenters of the medieval ages learned how to draw out Ad Quadratum, Ad Triangulum, Seed of Life (Daisy Wheel), like you I learn our ABC's. The ground plan of any cathedral would start with Ad Quadratum, then they would use Ad Triangulum for the elevation and then use the daisy wheel to transfer angles. And when they got bored or drunk they would use their compass to inscribe the Seed of Life (Daisy Wheel) on stone or timbers.

Sim

4. You wrote:

"The Alhambra Granada that is based on the Ad Quadratum Ground Plan is something that I discovered. It's not in any books that I'm aware of. "

Well, I'll be! That is pretty incredible. I think I saw this image (or one like it) on the Master carpenter square website; did you send it to them? I take it you figured out the geometry posted above. Perhaps your real name is Leonardo!

5. If you don't mind, I've reposted some of your discussions on my eco-architecture blog, with proper credit to you and links. If you don't want them there, let me know and I'll remove them. But I can't tell you how thrilled I am to have found your blog and information! I've been looking for this for years!

Here's my blog: http://catherinetoddarchitecture.blogspot.com/

Thanks, CatherineTodd2 at gmail dot com

1. It's great that you've posted some of the material. I forgot that I even wrote that tongue in cheek article about Tom Builder.

Thanks for sharing,

Sim

6. Dear Sim, thank you so much for granting permission to re-post some of your material! I just watched "Tom the Builder" in action, in "The Pillars of the Earth," and was simply overwhelmed. I wish someone would do a documentary just on the building techniques, and leave out all the drama. That part was scary! I didn't know your article about Tom was "tongue in cheek." I only recognized the name because I had just watched the show, streaming from Netflix here in Guatemala.

I know what you mean about this: "Stonemasons and carpenters of the medieval ages learned how to draw out Ad Quadratum, Ad Triangulum, Seed of Life (Daisy Wheel), like you I learn our ABC's. "

It's true! Here in Guatemala, young students 8 to 12 years old learn geometry and how to make beautiful "drawings" with colored string, all done in geometric patterns. They use tiny nails around pieces of wood and what they come up with is quite incredible. They also learn to make musical instruments out of found and recycled materials. Just goes to show what people can do even if they have "little" to work with. Imagination (which this is all about) goes a long way.

I really had to laugh when you wrote:

"And when they got bored or drunk they would use their compass to inscribe the Seed of Life (Daisy Wheel) on stone or timbers."

This I believe! I think Guatemala would be a great place to re-introduce Medieval Gothic building techniques. We've got plenty of available materials, although not much quarried stone, but surely we can use other materials.

There's a beautiful old Spanish Colonial Catholic Church here in the main part of town that was built of stone and timber. It doesn't have the ceiling arches that the Gothic churches do, and it has a different kind of ceiling (maybe it's called a "tray" ceiling?) but it's very interesting and very rustic and beautiful. Very tall ceilings with gorgeous chandeliers and St. Francis and his wonderful prayer for peace (attributed to St. Francis, although not really written by him).

I took some pictures but forgot to take some outside, although I have a lot from last year. I will post them on the architecture website. I am determined to build something like this on my small building lots which total about one acre.

I saw that fascinating roof on the Chappell Framing Square website, and that's where you saw it and then you realized it was based on "the two squares rotated in the circle, Ad Quadratum, that all other medieval cathedral baptismal's are based on. It's not a cathedral, unless it has a baptismal based on the sacred cut. "

Ad Quadratum,the Sacred Cut, & Roman Architecture
http://www.dartmouth.edu/~matc/math5.geometry/unit7/unit7.h
tml#cut

I had no idea! Heading to your link now. So much to learn and so glad to find out. Isn't the internet a marvelous thing?

I hope my architect here can help with some of these drawings. Do you sell drawings / plans yourself?

7. ... cont'd:

There's a company in the U.S. that is selling arch and ceiling kits for a reasonable price. I'd love to be able to use something like this to finish off my future small cathedral / cloisture / monastery here in Guatemala, right where it belongs. They have a bunch of YouTube videos. Here's one:

Building a plywood arch vs the Universal Arch Kit (\$19.95)

http://archkit.com/
Find it at The Home Depot

Whatever we end up building, it will be a refuge for artisans who do the most beautiful work even today, when many of the older ones cannot read or write. I work with 30 to 40 artisans right now, and seem to be adding more all the time. It will be even more interesting when we get to the construction phase of things. Wells are still hand dug here: two guys and two days or a week until they hit water. May people in Guatemala still live just like our grandparents and great grandparents did in rural America, 50 to 100 years ago. And they work side-by-side with modern, educated Guatemalans who know computers, internet, math, physics, history and more. It's two worlds blended into one at all times. People cooking with wood next door to wealthy people who live in mansions at the lake.

Here in Guatemala, school is only free until third grade (it was never free before) so although the younger generation can read and do simple math, and many have learned how to use the computer, and everyone has a cell phone, the culture still relies on many time-tested and old-time methods of building construction and design. We have modern high-rise concrete construction next door to handmade adobe brick construction. It's quite remarkable. Like living in the present time mixed with the old, 500 to 1,000 years ago, all blended into one.

La Antigua Guatemala was a town built by more than 300 monasteries and churches until the capital was moved to Guatemala City, and now is a historic Nato cultural site (or called something like that). It's gorgeous. All the old convents and cloisters have been turned into hotels and the churches have been restored. Many, but not all. It's a fascinating place to visit.

I live at Lake Atitlan, where people still wear traditional "traje" (hand woven clothing) and work with machetes out in the fields. It's remarkable. (I think I said this a number of times before, but I still can't get over it!)

Anyway, I've taken up too much of your time. Let me know if you have plans to sell so we can do some of these beautiful geometric roofs or other kinds of construction, and I'm heading over to your suggested links right now. Gracias, amigo! More later I hope...

Catherine Todd, AtitlanArts.com
Panajachel, Lake Atitlan, Guatemala, Central America

and Oxford, North Carolina, USA