Ad Triangulum (And maybe some Ad Quadratum too)

I spent a long time debating whether to create a new post or add this to the daisy wheel topic. I decided to make a new post, because this isn't really daisy wheel geometry although it is related, and that thread is quite long already.



First off, Here is a link to photographs of me going through a simple Ad Triangulum construction (or at least my own interpretation of this system) for a small building. I did this to explore the process, and took lots of pictures to document all the various steps.

At the end of the album are some pictures of a plan yielded by the use of a matrix system, also Ad Triangulum. This was created by using the points and lines of a simple geometric matrix consisting mostly of equilateral triangles. My primary rule for this construction was that only those lines may be drawn which will yield equilateral triangles.

After experimenting with this system, I have decided that the pure constructive method, without the creation of the matrix, is better (in my opinion) and both can yield essentially the same results. The straightforward construction is a lot faster, and by using it you make only those lines that you need for your plan, nothing more. The disadvantage is that you have to have a very clear picture in your mind of just what it is you want before you start, whereas the matrix system can essentially plan the structure for you. You also really have to know how to use the system if you are going to go with the straight construction method.

It should be noted that the steel rulers are used only as straightedges, and not as rulers.

I tried to explain everything as best as I could in the photo descriptions, so make sure and pay attention to those. Feel free to ask any other questions you may have. I did assume a basic understanding of geometry, the daisy wheel, and the use of dividers so if you don't have that I recommend acquiring it BEFORE viewing the pictures and asking questions. (please)

I also recommend reading up on Euclid, and familiarizing yourself with Euclid's first three postulates:

1 A straight line segment can be drawn joining any two points.

2 Any straight line segment can be extended indefinitely, to form a straight line.

3 Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

These are the foundations of these geometric constructions, and this is often referred to as Euclidean Geometry because of this. Ad Triangulum is simply an application of these 3 postulates for the purpose of designing buildings or creating decorations (and possibly painting as well)

Welcome to the High and Liberal Art of Architectural Geometry

Ad Quadratum geometry to follow at a later date. I have to think of something appropriate to design with it. Which is hard because it was used to design Cathedrals, and great structures like the Hagia Sophia.

Holy moly! If I had to follow all these rules of making lines and circles, I would never have time to cut joinery. To me, Timber Framing is a form of art which flows freely from my imagination.
But alas... there are many perspectives of this craft so fill your boots.

You have to follow some set of rules to design any structure. It takes me maybe about a half hour to lay out every single timber on an elevation like this if I have any kind of idea in my head of what I want beforehand.

This is a system of designing the structure as a whole. This is a system that may have been used by the ancient Greeks, at least the Romans, and some evidence suggests that the Egyptians used this or something very similar, and it fell out of favor and eventually died out altogether during the Renaissance.

The rules in this system exist because you have to have rules. If I gave you a pen and a piece of paper and told you to draw me and accurate layout of a building, one accurate to be used to find all the measurements, would you be able to do it? Would you be able to make it square, and make it exactly the right width, height, etc. or even close? Would you be able to do that with JUST a pen? The answer to all of those questions is no.
Therefore you have to have some other tools. To keep matters simple, this tradition has inserted just 2 other tools, a straightedge and a compass. The rules exist so that you can know how to use these tools to get desired results. If I handed you a compass and a straightedge and told you I wanted a building, would you be able to do it? Not without some kind of a system to use these things. And that's all these rules are.

These 'rules' are realities that are inherent in the system. one 'rule' is that I must set the compass radius to some dimension that is found on the diagram. This sounds like a limitation, but in reality it's absolutely necessary. Otherwise you would have totally arbitrary dimensions with no relationship to each other whatsoever. That's a recipe for disaster. And if you can't find a dimension that fits the bill, either plot some other parts until you do, or else use your compass and straightedge to create some new lines and circles. piece of cake. Think of these not as rules, but as instructions on how to arrive at a desired result.

Really this system is not complicated at all, it is amazingly simple.

What do I mean?
Well consider the alternatives

If I want to lay out a building by modern methods, then I use a ruler to establish known measurements, sure (or tell a computer to make a line of a given length) Then I make more measurements and so on, and then something strange happens. I have to sit down and do some figuring. That's right, mathematics. That's complicated. Especially when you get into roof design and truss layout and all that good stuff. Pretty soon you either need a good computer program or else some sound knowledge of trigonometry. That's complicated.

I can do the same thing with my compass, believe it or not, without once writing down a single number or opening up my calculator. That's simple. I can design a great cathedral with such accuracy, and geometric relations that defy computers and modern architecture. I can make an infinite array or fantastic structures, from the Barns at Cressing to the great Cathedrals of Europe, to the awe inspiring Hagia Sophia of Constantinople (it will always be Constantinople to me, Istanbul just doesn't quite make it I'm afraid) and the list goes on to just about anything built from the reign of Rome to the Renaissance.

This all is not to make you feel bad or wrong, it's to show that these 'rules' are inherent to any system, and quite necessary if you want to do anything good. Any way you design a building is set with a certain set of rules, otherwise you really can't do anything much at all. You have a set of rules that tell you how to determine where to place your floor joists or bents or rafters or.... So do I, I just have a different set, that's all. Or do we place one rafter here, 2 feet from the end wall at this pitch, then place the next one 5 and a half feet away from it at another pitch? Of course not, we follow the rules or the structure will suffer.

Imagination is all well and good, but it too has its own set of rules it must follow, although those rules are different for each person.

This is a beautiful system. Where can I learn more?

The Guild will be teaching a course is a version of this system in October in Colorado. Details are emerging on tfguild.org.

Here's a preview: http://tfguild.org/workshops/geo/

If the software asks for a password, just ignore it!

another great topic and a fasciating design system DL
lignarius, I found Rober Lawlor's -Sacred Geometry very informative, you can d/l on scribd
http://www.scribd.com/doc/8320/Robert-Lawlor-Sacred-Geometry-Philosophy-and-Practice-1982

The best way to find out more is to just sit down and try it for yourself. That's the point of the walkthrough pictures really, is so that some of you would be inspired to just sit down and try it out. Like I said earlier, the walkthrough is just pictures of me doing just that, sitting down and putting theory into practice.

For me, a wonderful resource has been http://medievalarchitecturalgeometry.com/Index.htm
This site deals with the geometries of 2 Medieval English Cathedrals and dabbles a bit with some other buildings also (for example, a few diagrams of the Hagia Sophia can be found)

The best page on this site is this (in my opinion): http://medievalarchitecturalgeometry.com/A3%20The%20Geometry.htm which is a good primer in the methods of Euclidean geometry, although it stops short of telling you how to actually use it to create a building plan or elevation. To do that you really just need to try it out.

Hi DL,

I find geometry fascinating as well. Remember Geometry Proofs? Simplicity of arcs and lines does make beautiful buildings, strong too.

Your methods you are describing, make good sense. Here is where I see that. The method provides a rule for scribing. With a couple of trammel points and chalkline (or string and stakes) you can lay-out full scale for building with scribe techniques. Hell, This seems more fun than taking 6, 8, 10 with two tape measures with pencils "attached to the ends". Ya'll know what I'm talking about right?

Here are a couple that I have drawn with credit due to Mr. Hobson.

1. Making Perpendicular Lines.
2. Finding the Center of a Circle.

Some time I will post some pictures of the Ad Quadratum design of the floor plan of my building.

But for now I have something else to talk about...

There is another method, even more ancient that these, of using geometry to lay out a building or to survey land, and this method has the good fortune of having actually survived into the present day, even in the West.

This method is mentioned by Herodotus as likely being the origin of geometry, it is the Egyptian practice of the Rope Stretchers. It was used by Egyptian officials to mark off land boundaries after the annual Nile floods (for taxation mostly, but it probably also prevented a lot of disputes) and they probably used it to build their buildings as well.

This system is very simple. There are 2 tools used to mark off or measure, a rope with knots at regular intervals and a plumb bob.

If the rope has 13 knots spaced at an equal distance, then it can be used in a number of ways. Perhaps most importantly it can be used to mark off perfect right angles by use of a 3-4-5 right triangle.

I have not experimented with this system, mostly because you need 3 people to do it right, but I imagine it would be much more limited than the other systems.

However, I can certainly see the use of a knotted rope as a simple method of transferring measurements from a drawing to a full size ground plan.

And Mo, you have the makings of a regular pentagon there in the second construction, it would appear.

5 - 12 - 13 can also form a right triangle.

True, but a 31 knotted rope would be unwieldy, not?

Here are some Medieval measurements from various regions:
English:
Digit: 3/4"
barleycorn, about 1/3", originally based on the length of a grain of barley
ynch, or inch: 3 barleycorn
nail: 3 digits
palm: 3 inches
hand: 4 inches
span (width of outstrentched hand from tip of thumb to tip of little finger): 3 palms
foot: originally 13 inches, shortened to 12 after 1066 to base it on the Roman foot.
cubit: 18 inches
ell: 20 nails or 45 inches
yard (after 1066): 3 feet
fathom (distance from one fingertip to the other on outstretched arms): 6 feet
rod: uncertain original length, defined as 16 1/2 feet after 1066

French:

pied: 12 pounce
pounce: 12 ligne
ligne: about 2.2558 mm

German:

Rute: varied, 0, 12, 14, 15, 18 or 20 feet
Klafter: Fathom, usually 6 feet, although in can range to as high as 3 meters in Switzerland
Elle: Distance between elbow and finger tip. In the North often 2 feet, In Prussia 17 / 8 feet, in the South often 2 1/2 feet.
Fuss: foot, varied by region
Zoll: inch, 1/12 of a foot, sometimes 1/10th of a foot.
Linie: 1/12 or 1/10 of a foot

Norwegian:

alen: Forearm, varies locally
fot: foot, 1/2 alen
kvarter: quarter, 1/4 alen
tomme: thumb, inch. 1/12 foot
linje: line, 1/12 tomme
steinkast: stone's throw. approximate measure, perhaps 25 favner
stang: rod, 5 alen
skrupel: 1/12 linje

Back to Egypt, the following image link shows a surveying party at work, note the man holding a triangle, what do you think is the ratio of the sides?

http://mathcs.slu.edu/history-of-math/images/history/3/32/Menna-rope-right.jpg

I don't know...

On another subject however,

So far we have only looked at applying geometry to a small piece of paper. That's good for making plans and so forth, but these methods are not practical or even very accurate should you wish to lay out a full sized building.

So how do we do this?

More importantly, how did they do this 500, 1000, or even 2000 years ago.

How did the Egyptians do this to create things as accurate as the pyramids?

How did the Romans do this to create their great works, and to make their system of roads?

To transfer the geometry of the paper to reality, we need a number of things.

We need some way of laying out a straight line over a very long distance, some sort of surveying device.

We need a way to make things plumb. Plumb lines and plumb bobs do the trick here.

We need a way to make things level. And that's the tricky part. Once again, the plumb line can be our friend.

For the first task, laying out long straight lines, there is an ancient device that the Romans called the groma
The groma consists of a long pole driven into the ground, on the top of which two beams crossing each other at right angles (making a + shape) are mounted offset from the beam on a swiveling arm. Under the center of the + there is a plumb line to be lined up with a datum point. Each of the four arms has a plumb line as well. The whole assembly looks like this:


to establish a straight line, one man sights down 3 of the plumb lines (the North-South lines and the center line) while another marks off stakes according to the directions of the surveyor. And so incredibility accurate lines can be made. In addition, the groma allows us to make a line at a right angle to the first line using the east-west points.

For the second need, a plumb line will tell us the perfect vertical, however we will need a way of applying this vertical to a post, for example. We cannot simply place the plumb line against the side of the post and expect an accurate reading. Instead, we mount the plumb line to a board with a straight line marked parallel to its edges, and use it to set the post to plumb.

So how about level?
The level as we have it today is the result of millenia of development. The simplest level is a long piece of wood with a chanel cut into its center that is filled with water. This is not the most accurate method, but works for approximations. It can be accurate on a large scale, level foundations can be established by filling the foundation ditch with water and marking the water line to use as a reference point.

[img]

A better method for smaller things is to use the plumb bob again. To do this, we exploit the nature of the isosceles triangle. A plumb line is attached at the point of an isosceles triangle, and when the line lines up with the center of the bottom side then the bottom side of the triangle is level. This might be placed against a level line on a timber, for example.



Some variations of the plumb level

what would appear to be a French variation of an ancient Roman leveling device.

So with these tools we can create straight lines to create our building plan, and then we can erect our frames true and square.

The groma can be used in conjunction with a rope, stakes, or other methods of establishing the straight line. when marking off distances with a large compass or trammel it can be used to ensure you stay in line. For the first line of a structure it is unnecessary, but from then on its use makes the whole task much easier. It eliminates the need to do extensive geometric operations to establish right angle and parallel lines.

The groma is believed to have been invented by the Egyptians, along with the plumb line, plumb level, water level, measuring rope, and other such tools. It survived in various forms for thousands of years.

The plumb bob was the ultimate tool of the ancient builder and surveyor. The Egyptians discovered ways to use the plumb line in about every operation from establishing a level surface to making straight lines and complex angles. The plumb line is a reference point that can always be trusted to be exactly true. When combined with the geometry of triangles, the plumb line can be used to establish a wide variety of angles, creating the first protractor. A graduated disk with two sight points attached to the top of a groma can be used to find angles off of the straight or perpendicular lines as well.

I agree, Plumb and level. 2 axis at any location, you can build most anything. Until it comes the time to build it.

Something for Nothing

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Buckyballs.....string.....Egypt......

http://www.youtube.com/watch?v=o5DlXFzj0SE&feature=sub

Tim