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A River Find


peter johnsson

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Here is a short animation presenting an analysis of the proportions of a sword found in the River Ljubljanica in Slovenia. The study is part of a joint project between the National Museum of Slovenia and Albion Armorers (whom I work for as designer and researcher).

 

As there seems to be some interest in my hypothesis on geometrically based design of swords, I thought you guys might like to see this. It is just one example of how it may have been done.

 

Please enjoy and let me know what you think.

 

<iframe width="960" height="720" src="http://www.youtube.com/embed/HOwEPot6tok?rel=0" frameborder="0" allowfullscreen></iframe>

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Like I said earlier on MyArmoury discussion on this video, I think it is a perfect way to introduce someone new to your theory, Peter. Very nice to see it out there,, thanks for putting all the work into making this happen!

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First, thank you for this video. It's a very clean and easily understood application of your theory to this sword, something I can share with friends and they'll understand where you're coming from.

 

I have watched your Arctic Fire presentations several times (and am still waiting for the DVD's, Dave... ) but I can't recall - is there any evidence that the makers used geometry to design, or do we find geometry when we look for it? I'm speaking only of swords, of course, I recall your statement that old architecture plans had evidence of this, suggesting intentional layout and use of these kinds of rules... Is there anything at all in the old writings that suggests this, or are we simply seeing what is there to be found without understanding the rules by which it may have been used?

 

I'm asking this not to offend, but to understand - it appear to be such a graceful and unifying theory of how medieval swords were designed, it almost seems too good to be true. I figure you would know best among us, though, having handled so many originals and spending so much of your life working with these items.

Edited by Christopher Price

The Tidewater Forge

Christopher Price, Bladesmith

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Hi Peter.

 

I think this is really great, your hypothesis on geometrically is spot one.

So this could be used for just for one piece of sword too...like pommel...we could get some info about

the whole sword...also if its just tang,hilt...?

 

Do you think that this is similar to golden ratio?

 

Really amaising, thank you Peter this is really cool

 

BR

Niko

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Lovely! A beautiful and pragmatic explanation...personally I suspect many different crafts used these sorts of design rules "back in the day". In my Lutherie days I got interested in how violins were originally designed, and found that most of the important aspects of fiddles could be constructed with a compass in a similar way. (Interestingly I found a lot of odd-number ratios, which I have come to believe has something to do with de-emphasizing certain harmonics. But I digress...)

My hand-forged knives and tools at Etsy.com: http://www.etsy.com/shop/oldschooltools

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The compass and the square were the main design tools for millennia. Techniques like these can be found in all sorts of trades and were the "secrets" of the trades.

 

Amazing to see you unlock them and I'd love to see this analysis on many more blade types. I'd also love to see how the golden ratio works in there as well, as I'm sure it does. One great tool that was used was a Fibonacci gauge. Lost tools like these made designing beautiful and effective things an expression of function.

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Thanks guys!

 

Artis: yes the Ljubljana sword is a good example even if it is perhaps the most clear either. There are others down the pipeline that will be shown over the coming months. They just have to be paced to fit other things. For now, the Ljubljana sword is an OK starting point for an introduction.

 

Christopher: I absolutely do not take offense by your question! It is a very important one. If you can find a geometric basis for everything it proves nothing. The thing is that you *can* find a geometric correlation for just about anything if you keep drawing one form and line after the other, so you have to be ever critical to your own conclusions. It is important to keep looking for simple solutions.

 

Below are two good examples. The first is an analysis of the proportions of Notre Dame cathedral in Paris made by Frederic Macody Lund in his "Ad Quadratum" published in 1921. I think Macody´s solution to the geometry of the Notre Dame is an obvious result of wishful thinking and a loose and sloppy application of geometry.

MacodyLund.jpg

Second is an analysis by prof. Robert Bork published in "Geometry of Creation" Ashgate 2011. He has studied many original architectural plans drawn on parchment, looking for pin pricks left from the compass and uninked construction lines. He has found geometric structures like the one below being the principle for the design of cathedrals and abbeys through the Gothic period and into the renaissance.

In the original plans, the geometric structure is not inked in but if you look for the marks of the drawing tools, it can be reconstructed with clarity. His analysis is not guesswork but a demostration of the actual underlying design grid.

Below is the original geometric structure for the Strassbourg cathedral. Note the rhythm of reoccurring geometric forms that define both overall proportion as well as placing and dimension of details in the whole.

Strassbourg.jpg

 

 

I think there is a vast difference between the structure found by Robert Bork and the one suggested by Macody Lund. The one discovered by Bork is characterized by clarity, simplicity and coherence, while Lund only succeeded in making a cobweb of hap-hazard lines.

 

If you go about geometric analysis like Macody Lund you can absolutely prove that a phone booth in fact is an instrument for astronomical observations.

But if the geometry has to be subtle and clear like the ones found by Bork, you are restricted in your options. You cannot simply pick and choose any given form that will explain the proportions of an object. The strucure has to have a certain clarity and simplicity. This is difficult to define and describe in a short post on a forum like this. I hope to return to this later on as it is a *Very* important aspect of this kind of design.

 

Looking at medieval use of geometry as a tool for design we find that the building principles are very strict and restrained. The relation between lines and forms show a certain beauty and clarity.

 

From the creator´s or designer´s perspective, the application of geometry is absolutely free, as long as some basic rules or principles are observed. Keep it simple. Define as much as possible with the least of elements. Look for harmony and rhythm. It is a little bit like a game of chess. Infinite variations, but only according to a few simple principles.

For a geometric design to be fully valid, I think it has to cover the overall plan as well as the details with the same set of stratagems. Finding a geometric definition of a single detail is not difficult, but finding one that defines *both* all important details *as well as* the overall proportions within the same simple framework is actually rather difficult.

 

Niko: The beauty of this method is that you can use a structure of interlocked circles to define proportions that follows the Golden Section. The Circle inscribed in the Square is in fact a kind of drawing machine for the golden Section, if you know how to cut it with the compass! You *can* choose to define only parts of a design with this method of course: you are free to do whatever you want and works for you.

*But* I think that the medieval craftsmen used geometry to establish a coherent wholeness of their work. The definition f a detail was meaningless without a relation to the whole.

Geometry was also meaningful. It was not just dead classroom mathematics. It was a mystical key that unlocked the hidden treasures of Creation.

 

This is how you use the Circle and Square to define the Golden Section:

Step-y-StepGoldenSection.jpg

 

Orien: Yes odd number relations and irrational number relations can easily be established with this method. Proportions based on the Golden Section, Root 2, Root 3 and Root 5 are all pretty common in designs based on geometric structures since they follow the basic platonic geometric shapes.

The proportions of medieval swords follow both whole number ratios as well as specific irrational numbers that relate to harmonic ratios. Irrational ratios are impossible to establish with graded measuring tools. You can easily establish them with a fair degree of exactness using geometric drawing tools however.

 

There is unfortunately no guarantee that a sword designed according to geometry will be better than one that is made without any conscious thought on proportion and geometry at all.

In fact an absolutely lousy and ugly sword can be designed based on ingenious geometry!

Geometry is just a tool like any else. It can be used well or badly, and you still have to know what makes a sword function well regardless.

*But* it is a powerful method to organize and structure the design process. If used well, it can result in a certain kind of beauty that is difficult to achieve otherwise. I think medieval craftsmen were interested in this specific kind of beauty and that it is an important theme in the European medieval tradition of sword making.

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The obvious thing to keep in mind is the function of the geometry. Is it determined by the function? By esthetics? By the manufacturing process? Having a geometry that can be measured by simple measuring tools makes it easier to see if it's straight, symmetrical etc. Calipers are probably amongst the earliest measuring tools used in production, which will explain how circles fit very neatly. A very early example is a type of bronze age ceremonial dirk of the Ommerschans-Plougrescant type. I don't have the actual drawings at hand, but they also include very exactly repeating circles defining the shape. In this case it was probably used to create a very symmetric mould. But as you mention, there's also the danger that if you go far enough, you can fit regular geometric shapes in even the most randomly shaped artifacts. So the fact that you can do that doesn't say anything, unless you can support it by a good reason that it is intentional. In case of buildings, it is vital to measure everything up. Otherwise you'll get a very shoddy uneven building, which isn't structurally all that good either. In how far it was applied to swords, I can't say for myself. It is interesting to look at though, as it is information that doesn't show by just looking at it, and it may tell us a bit more about how they were designed. But it's important to be skeptical about conclusions drawn from it.

Jeroen Zuiderwijk

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Very cool stuff Peter. I'm beginning to understand, I've watched your arctic fire videos more than a few times, it's very interesting stuff. Are there any set rules for using geometry for design, or is it more just the use of geometry instead of numbers for your measuring stick?

As far as simplicity is concerned, some of the simplest things I've studied, are only simple in operation, once you dig into the why's of how they work, you begin to understand the amount of thought put into such "simple" items, I believe it was da Vinci who said "Simplicity is the ultimate sophistication"

 

Zeb

 

 

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Thank guys!

 

Jeroen: What you say is very true. Being skeptical and critical of ones results is very important for good research.

Geometric analysis as a way of backwards engineering is not of the best reputation in academia very much because it often has been used in sloppy and fanciful ways.

It is important to stress that my idea is a hypothesis: it is a suggestion of how this could have been done.

 

Here is a quote from my short article in the site of the National Museum of Slovenia where the sword in this thread is presented:

 

"The coherence of elements of design and unity of proportions was very important to medieval aesthetics.

Analysis of the proportions of medieval swords according to design techniques known from surviving architectural drawings from the period indicate that geometric drawing was also known and used by cutlers or sword smiths.

The use of geometry in design has several benefits. It is a practical way to establish specifications for work that is divided between several expert craftsmen, as was the case in the production of swords. Geometric structures are also simple to scale to any size and easy to memorize.

Geometry and number were to the medieval mind charged with symbolic meaning and this may be important for a weapon that was an emblem for worldly power and prowess but also a symbol of the spiritual fight against evil. Letters and symbols inlayed in blades served as religious invocations and magical talismans. Likewise, the use of geometry in the design of the sword transformed it into a divine instrument, possessed of a perfect wholeness derived from the unity and harmony of its parts."

 

I add this quote on this thread to point out some of the reasons why I think geometry was used in the design of swords.

 

They can also be listed like this:

-Geometry was a method of design in other qualified crafts of the period and an accepted and widely used principle for organizing concepts and ideas. For a sword maker it would not have been far fetched or strange to reach for geometry if he needed to define or specify a design. The sword maker, or sword cutler, would have been an urban craftsman who socialized with masters of other trades, like goldsmiths, masons, sculptors and painters. We know masters of these other crafts knew the use of geometry in their work. Geometric drawing and geometry based design would not have been an alien concept to a sword maker. On the contrary: it would have been rather familiar.

 

-Designs based on geometry would have been practical for manufacture of swords since it tended to be divided between several specialist craftsmen. They had to work according to a shared plan if the parts was to come together well. Geometry allows for precise definitions in a way that the varying standards of measurements of medieval times does not.

 

-Geometric constructions can easily be memorized. This is something we know was important to the medieval way to go about things: games of memory and methods for memorization was popular in a time where literacy was not as widespread as it is today. You do not have to know maths to be able to perform very subtle and advanced geometric designs. Nor do you have to be literate. Geometric drawing can be learned by observing another master perform it. It can be absorbed as a direct experience and there are only a few basic theoretical rules for how to perform it.

 

-Geometry can easily be scaled to suit specific purposes and situations. Medieval swords seems to be a result of this. It is famously difficult to judge the size of a sword from a photo alone: they can have very similar proportions regardless of their physical size. That is the hallmark for a scaled up or scaled down geometric design. This means that a master can have any number of completed geometric designs memorized, so that they are readily at hand for any given situation. Having a memorized material like this is very powerful for an artist, craftsman or engineer. He can work quicker, with greater certainty as it allows him intuitive access to a "library" of design solutions.

 

-Geometry was charged with meaning in medieval times. Not every craftsman had to be a philosopher or a mystic, it was just an accepted notion of the period. Christian faith was expressed in images and symbols. Number and geometric form was part of this language of faith. Regardless of the degree of piety of the individual craftsman, he lived in a world where geometry was an accepted language for expressing core concepts or what was real, right and righteous.

 

-A sword is not simply a weapon. It is not just a tool for killing other human beings. The sword is an emblem for power, status, privileges and duties. It is the sword given to the knight in his sworn duty to defend the Church. It is a vivid and powerful symbol for philosophical, religious and moral ideas. Symbols were a way to understand and interact with reality in the medieval period. We know from inscriptions in the fullers of swords that the sword was seen as something that could carry religious signs or even magical symbols in a way that was purposeful and crucial to its function. This is much less common on other types of weapons, even if it does happen. It clearly shows that the sword was special and regarded as a weapon that transcended its role as killing tool and also took on the purpose of religious and/or magical talisman.

Geometry was a way to "work magic" since geometric forms and number carried meaning. According to some prominent researchers in current architectural history, the use of geometry in the architecture of abbeys and cathedrals was not simply a method for secure and sturdy construction, but also a way to make sure the building was true to its holy purpose. Geometry was in fact instrumental for this purpose. This is a point of debate of course. Some still maintain that we cannot know what symbolic purpose geometry served, but this reservation seems to me overly timid. By being so restrained some very important aspects of history are ignored.

We have to be clear that we will never know for certain *exactly* what the symbols or their meaning were or what the case *for certain* is in individual situations, but it would be a mistake to dismiss the importance of symbols and their central use in medieval life and work.

 

-Beside these factors the medieval sword itself can reveal some points of argument. When analyzing swords from other time periods I have not found the same strong correlation to geometric patterns or proportions. Viking period swords for example does not seem to follow the strict geometric proportions that are commonly found in later medieval swords. They are more "organic" in their design. Much more work still remains to be done of course: this is just early preliminary results I am working hard to get access to more data. Geometric analysis cannot be performed on photos of swords. You have to have a good quality tracing that is true to the dimensions and proportions of the original. A photo is always suffering from the effects of perspective changing the proportions of the object. Archaeological drawings in publications *may* also be tracings of photographs rather than direct tracings of the objects. That means that unless you are *certain* that the drawing is true to actual proportions it cannot be used for geometric analysis. This makes work more laborious and time consuming: I have to rely on my own tracings to be sure about the quality and nature of my material.

 

All what I have said are reasons why geometry makes sense as a design tool for swords, but it does not prove for a fact it was the case. I do think it shows it is not only possible but perhaps even probable. I have a lot of material to show, but I cannot do so in a meaningful way on a forum on the internet. I have to publish somehow. It is a big subject that brings along many questions. I am aware of the limitations and restrictions for this work: I cannot say for sure that what I propose is the truth. Perhaps if an original drawing of a sword is found, the case will be clear. Perhaps I can gather enough circumstantial evidence to show a convincing case even if such a drawing is never found.

If not, I have at least developed some skills in geometric design that are very useful for me in my work :-)

 

I know already for a fact that the chance a sword does not fit a geometric structure is greater than the chance it does. Since the majority of the medieval swords so far analyzed seem to follow geometric designs, I think this is already a strong case for there being something to what I suggest. Pure chance would not give this result. Critical evaluation still needs to be applied in every case, however.

 

If you are less than strict about precision and tolerances to you can indeed squeeze and stretch a geometric analysis to fit an object. You can make the analysis complex enough so that in the end some crossing line or curves touch some parts of the object (see the alanysis of the Notre Dame by Macody above: it is a very good example of how it should *not* be done).

This is *not* what I am trying to do. I am as happy when I find a sword that does not follow a geometric pattern as I am finding one. I dismiss those structures that I think are too complex. I always strive to keep in mind the perspective of the original maker: why would he have done this? Why go about the design solution in this way? Is it meaningful? Does it make things easier? Is it beautiful? Is it simple to memorize? May it carry symbolic meaning in a way that makes sense? -These are things to consider every time a structure is found that seems to define the proportions of a sword.

 

Then it is also a question of developing my own skills in geometric design. As I have already said, there are "good" and "bad" ways to do this. Geometric design is not simply a logic puzzle to be solved. There is an art to it. This makes it all the more difficult, but also more interesting.

 

Geometry is like a language. But simply because you begin to be familiar with a language, it does not mean you will be able to speak fluently or compose poems in that language. This is something that makes geometric analysis difficult. I have a feeling that part of the reason geometric analysis is regarded with suspicion in academia is that it will always remain partly incomprehensible to some people. For some it will always seem far fetched and strangely complicated. I can identify with that: I have a problem with mathematics and statistics. My mind does not work along those principles. Still, I know statistics are important, but have t work around this and find visual graphic methods to show the chance that proportions accidentally fit to geometric structures or not. I am currently working on this and have made some progress that I think sheds a light on this important question. I will return to this later.

 

We know geometry was an important principle for thought and work in medieval times. We have to study medieval art and philosophy to get an idea of how it was understood and how it was put to use back in the day. It is very important to remain critical and skeptical to our conclusions, but I think it is a mistake to dismiss the notion that swords were designed with the help of geometry simply because there are unknowable factors involved.

I am not suggesting that one single principle or method was always used. I am not suggesting all makers were masters of geometry, nor am I suggesting that all swords were made with a great degree of precision. I have touched upon these questions in the articles I have published, but this need further discussion and clarification.

 

Not all swords would have been made according to these principles, and the skill, degree and method by which it was used would have varied. It makes for a difficult situation for backwards engineering these things today, but I think it is a worthwhile quest. I think it is possible to unlock some of the secrets of the old masters this way and I think the case is strong enough that we should not ignore the possibility that this was perhaps exactly how and why the medieval sword came to look the way it does.

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Very cool stuff Peter. I'm beginning to understand, I've watched your arctic fire videos more than a few times, it's very interesting stuff. Are there any set rules for using geometry for design, or is it more just the use of geometry instead of numbers for your measuring stick?

As far as simplicity is concerned, some of the simplest things I've studied, are only simple in operation, once you dig into the why's of how they work, you begin to understand the amount of thought put into such "simple" items, I believe it was da Vinci who said "Simplicity is the ultimate sophistication"

 

Zeb

 

Zeb: you bring up some really good questions!

Geometry can absolutely be used instead of a graded measuring tool. Also if you lack standardized stock material, it is very handy if you can establish your own standards. To be able to do this directly with the help of geometry is very powerful.

 

We must recognize the importance of this: using a micrometer to set the dimensions of your work means that your work relates to an outside more or less arbitrary source. Using a system of interrelated proportions to set the dimensions of your work means that it will be complete in itself: it is independent and "self sufficient".

-Please not that I am not suggesting that medieval craftsmen did not use a yard stick or a ruler graded for inches. I am only saying they had access both to a system of design that was independent of measurements as well as basic tools for dimensions, using each for what the situation demanded.

 

We also have to put ourselves in the mindset of medieval craftsmen: What did they think about making things? How did they make distinctions between good and bad work? How did they make sure they met up to standards and followed accepted and credible principles for respected craftsmen? Perhaps geometry was not only practical but also about doing things The Right Way?

 

Ones mindset and ideas about the creative process is also important. There is a very telling quote from a poet, Geoffrey of Vinsauf, who around the year 1200 explains how you should go about visualizing your work in detail and overall scope so that it is complete in your mind before you even begin practical work:

 

“...Let the mind´s interior compass first circle the whole extent of the material. Let a definite order chart in advance at what point the pen will take up its course...

As a prudent workman, construct the whole fabric within the mind´s citadel;

let it exist in the mind before it is on the lips”.

 

This he said about composing a poem, but it is striking how he uses the image of a master craftsman with a compass to bring home the importance of planning and visualizing your work.

 

The Goldsmith Hanns Schmuttermeyer (in the 1480´s) wrote a handbook in geometric design and explained why he did this like follows:

 

“...for the edification and instruction of our fellowmen and all masters and journeymen who use this high and liberal art of geometry, (so that) their feeling, speculation, and imagining can be better subjected, after memorization, to the true basis of measured work.

Fundamentally, this art is more freely and truly planted and developed out of the centre of the circle, together with its circumference, correct rules, points and setting out.”

 

It is clear that geometry was seen as a way to formalize and bring to awareness those intuitive thought processes that drive the creative mind, and subject them to a logic structure. Schmuttermeyer states that there are indeed rules for how geometry should be used. Reading his text you see that an important principle for this is to define all parts of a design back to a basic system of proportion. These proportions are set out and used so that the whole of the work will relate to the same principles in all its scales. In a way it is not too different from the modern concept of fractal geometry.

 

Some of the basic rules are: To construct a new form, draw a new line or define a new proportion, it has to be based on already existing features in the structure. For example: the centre and circumference of a circle has to be chosen from those lines that are already set out. Whatever you add always has to be anchored in the structure, it cannot float free and undefined. It has to relate to the already existing geometry. And you have to strive to do this with the best economy: keep it simple. Use what you already have to best effect.

 

In a way, it is like composing a poem: it has to follow rhyme and measure.

 

Also, the structure has to be complete in such a way that it defines the complete object. It has to do this in a way that alows you to also define the parts with the same system.

 

You have to be aware there is a hierarchy of forms, an order by which you define things. It is advisable to define important features like blade width and proportions of pommel and guard in a way that are independent of each other (each important definition must trace directly back to the basic structure and not depend on trivial things for its proper definition), but as much as possible also referring back to the same simple basic forms (keeping things simple will also help to make them beautiful).

 

If for example the definition of the blade width depends upon the *exact* placing of the guard, rather than a direct harmonic subdivision of the first square and circle, you have set up an awkward situation for yourself: unless the guard sits *exactly* like so, the blade no longer make sense in the structure and you have missed the whole point of using geometry as basis for your design.

 

Important aspects of the design therefore have to be independently defined, but in a way that makes their proportions related in a harmonious way.

 

I know I am guilty of mistakes to this rule of independent definition of important features. In my first article I presented an analysis of a sword where the blade width was a result of the exact placing of the guard. This is not good: it is a solution that is improbable for a medieval craftsman to make. I have since seen my mistake and am now being careful not to make the same kind of mistakes again.

 

The art of using this method for design is to establish structures that allow for richness and subtlety while at the same time being transparent and coherent with a beauty all of its own.

 

In the words of Thomas of Aquino (1225-1274):

 

“...When the parts are arranged in this way,

they all combine into the whole;

so that out of all the parts (...)

there emerges one single wholeness of things”

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A system of proportions would be especially useful as swords would have blades made in one country and very often finished in another, and measurements would not only change from country to country but very often from one province to the next.

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  • 2 weeks later...

This is really incredible and interesting work, Peter. I'm regretting only seeing this now, thank you for taking the time to show us and explain things in such an understandable way.

 

John

Not all those who wander are lost. -J.R.R. Tolkien

-Shards of the Dark Age- my blog
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I'm really enjoying comparing the two architectural drawings :)...the first one, to my eye, is over-elaborate and 'noisy', and it's clear that the point of the excercise is to generate that pentagram! The second one is much simpler and prettier, and has an almost musical repetition of a 'triple square' theme. Simple is good when it comes to these things, IMHO...you can fit a lot of decoration into a simple layout, and still get it to 'read' well.

My hand-forged knives and tools at Etsy.com: http://www.etsy.com/shop/oldschooltools

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Very cool Peter!!

 

My mind has always kinda worked the way of your theory, from childhood.
When I was very young, I was a competition sailor. I was always thinking in terms of angles, and circles, and seeing the ocean from above in my mind, to know where my competitors were at any given time.
When you added in wind speed/direction, currents/tides, and any number of other variables, it just made it easier. In other sports, like hockey, soccer, and lacrosse, and SCA fighting, all games of angles, and speeds, thinking that way helped a great deal as well.
I feel it is just something in some peoples genetics.
I suppose I do it in my metal arts as well, although I have never thought about it. I have always been big with the "see where it goes" style of art.
I know, I never have planned out a blade, but I haven't really made a blade worth planning out. Yet.

Well, I guess I have planned out some fittings.

 

Very interesting stuff, it has got me thinking a lot about my own design thoughts/flaws/examples.

 

 

Mark

Mark Green

 

I have a way? Is that better then a plan?

(cptn. Mal)

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Peter, is it possible you've understated the importance of geometry as a means of calculation? It seems necessary to remember that the medieval designer did not have our base 10 numerical system (post 13th C, they probably did), and the closest approximation of pi that I know of (Archimedes) was solved geometrically using the exhaustion method. Without numeric values for pi, the golden ratio, or an irrational hypotenuse a designer's hands are completely tied. Furthermore, they had no numeric way to calculate a lot of the stuff we take for granted in modern design today--say goodbye to trigonometry, calculus, and most of algebra. Even if they did have initial given dimensions--such as fitting a sword to a man's height and hand, they would have to solve for all the other dimensions of the sword to arrive at proportionality. This seems to be to be where an internally sufficient system of proportion would come in handy, not unlike how Gulliver was measured for clothes by the Lilliputians, something for which Gulliver praised them:





The sempstresses took my measure as I lay on the ground, one standing at my neck, and another at my mid-leg, with a strong cord extended, that each held by the end, while a third measured the length of the cord with a rule of an inch long. Then they measured my right thumb, and desired no more; for by a mathematical computation, that twice round the thumb is once round the wrist, and so on to the neck and the waist, and by the help of my old shirt, which I displayed on the ground before them for a pattern, they fitted me exactly.



It's precisely the type of proportional study we're talking about here, I think. It reminds me of high school math problems where we're asked to "solve" something "geometrically."

Geometry's original and most useful application was these sorts of calculations. Euclid's Elements becomes a much more important book when you realize that his axioms would have been tremendously useful in determining property lines and dividing land (geometry is literally "land measurement" in Greek).

What I find fascinating, Peter, is that your geometrical analysis shows care on the part of the medieval craftsman for things like the point of balance and percussion. Perhaps thats where the most useful, and concrete revelations lie in this sort of study--determining what points of design medieval craftsmen cared about when putting together their blades.

Edited by Tyler Miller
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Hey guys! Thank you for your great feedback!

 

-Orien: the medieval use of geometry is really quite close to music. Music was composed according to the same ideas of harmonious proportion. It is another example of the unity of parts that was so important to medieval aesthetics.

 

-Mark: yes, geometry rhymes well to our intuitive sense for proportion, line and space.

 

-Tyler: thank you! Those are very good remarks. I shall work to make this aspect clear.

To be able to appreciate how practical and necessary geometric construction was to the medieval craftsman, we have to remove all those things we take for granted in our modern mindset.

 

One thing that is not clear in my results so far is to what degree, or if at all, the dynamic aspects and functional properties of swords were part of this geometric design. Sometimes point of balance and pivot points correlate to the basic geometric structure, but I cannot say if this is a coincidence or not.

The material so far does not seem support that. It is something I will work more on.

 

The dynamic properties of a sword largely depends on what happens in thickness. It is absolutely possible to design distal taper and even distribution of mass with geometric methods. I play around with that when I design swords from the ground up: it is great fun. I am just not sure they actually did that back in the day. Perhaps some masters did? But that would indicate a very tight control of every step of manufacture.

 

Normally I suspect the cutler gave the blade smith some kind of template for length and width and possibly other details. I think the blade smith knew best how to shape a blade for best balance according to the dimensions he was given. This knowledge could well have been a secret kept from the other craftsmen involved in the making of swords.

 

It is fascinating to think *every* detail was part of a plan, but I am afraid that might be reaching too far. What I currently have does not suggest that dynamic properties and function was a priority for the geometric design of the proportions of the sword, but proportions and function are not completely separate matters either: there will be some carry over effect, but it might be secondary.

 

As for calculating the size of a sword, it is easy to take a complete geometric design and scale it exactly to the need of a customer. You can choose just one dimension, most probably blade length, and scale the design so that all parts follow the given blade length to *exact* proportion. There is no need for numbers or calculation. It is all done with a compass and a straight ruler. It is one of the most practical and powerful benefits of this kind of design that focus on proportion, not dimension. I will show how this is done in a post below.

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Peter, I enjoyed reading your reply. I had a thought: what if the craftsmen used an 'incorrect' (by current standards) theory to establish the dynamics of a blade? This is something that plagues scholars in understanding ancient philosophers' ideas--in order for their theories to even appear coherent, you need to uphold a bizarre world-view that often runs directly contrary to our intuition. Greek mathematics and Aristotle would form the basis of their understanding of physics, and there's a lot empirically lacking there. It could be that an obsolete theory of harmonics, that brings about a desirable result but in a way we can never reverse engineer, is behind the design of the blade.

 

My other curiosity was whether these designs weren't used to establish blade designs, but the uniformity of blade design. Consider the conservative nature of medieval architecture. There are literally an infinite number of solutions to the design challenges medieval architects faced, but only a very few of those solutions can be drafted without the benefit of modern theories. I can't think of a way to construct a parabola with a compass and rule, for example. It's obviously the case than you can't design something you can't draw, and if their only tools were compass and rule, then they can only draw things they can construct geometrically.

 

Scaling up production requires a greater attention to material expenditure and such templates could be money saving measures rather than innovative design efforts.

 

Thanks again for your work on this, it really is fascinating.

Edited by Tyler Miller
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Tyler, thanks again for your thoughts.

I like what you propose, that medieval smiths could have strived for quality features that only make sense if you believe in magic and divine intervention. Partly, I think that is a reason you want to use geometry as a basis for design: the fact it carries deep meaning and conveys ideas of religious nature and possibly changes the nature of the weapon, blessing it for a holy purpose and calling down divine favor upon the warrior might be a very important aspect.

I absolutely agree that there might have been parameters of design that seem very strange to us, but were natural, rational and important in medieval times. I would love to hear what was important to them and how they explained these things.

It is absolutely something to keep in mind: we can be pretty sure that some of the criteria we use define a good sword by would have been strange to medieval craftsmen.

 

I am not sure they ever talked about "harmonics" for example. How and where a blade vibrates might not have been something they placed much importance on.

-I would really like to know how sword smiths talked about a good sword in the gothic period!

 

From descriptions in period literature of swords we know they valued the brightness of the steel, the sharpness of the edge, the cutting power and the flexibility and the strength of the sword. So these would have been things they kept in mind when making or shopping for swords.

I have not read enough of medieval literature to be very familiar with how swords are described other than that. Perhaps other aspects of their quality are also lauded? Is the balance ever described? I don´t know.

The closest I know of is from Kormak´s saga (I think) he is (reluctantly) lent a sword for a "holmgång". The sword is an old heirloom, that is described as being a "slow and cold" weapon, while Kormak is quick and hot. It is not a happy match. Kormak puts a nick in the valuable sword but survives the fight.

 

Uniformity of design: yes! I touch on this in the two articles so far published (Park Lane Arms Fair catalogue 2012 and "The Noble Art of the Sword" catalogue). I think that the ability to set out specifications for the production of parts in a way that allowed them to come together into a useful end product was one of the most important things with geometric definition of proportions.

I think there is also a way to define very slow and gradual curves, parabolas, with the help of the compass and straight edge. You can get a series of dimensions spaced over the length of the blade that with a pretty high degree of precision will give you the kind of curve you find in the outline of medieval blades. The precision in the drawing would probably be greater than most smiths could deliver, in any case.

But would this have been a rational thing to define by geometry, or was it left up to the blade smith to decide the curvature of the edge and outline of the blade, once its length and base width was defined by the designer of the sword? There are arguments for both I think.

 

I am currently looking into different ways that distal taper and outlines of blade could have been defined. It is a lot of work to then see if there are any correlations between these methods and the actual situation with surviving swords.

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Peter, thanks for your prompt and thorough reply. I'll have to check out the articles you've had published--all this makes for very exciting reading.

 

The only reason I mentioned harmonics is that it's something that was pretty well understood, and because the image of dividing a line with circles looks remarkably like the nodes of a natural harmonic. Beyond that, it's just me wishing for a strong classical connection. This conversation has got me wondering about how the virtues of a classical sword were described. One more thing to look into, I suppose.

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This geometric method is obviously a heritage from the classical world translated through Christian Platonist ideas. There are *very* few roman swords surviving intact: either hilts (or hilt parts) or blades but rarely both. Otherwise roman swords would be a good topic for research on earlier applications of these ideas in sword design.

 

I have a strong feeling you would find good geometric proportions in the ancient greek Xiphos. There are several surviving in decent state of preservation...

 

I have yet to document a greek Xiphos. it is on the to-do list ;-)

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Peter,

it is really nice to see that you are sharing your thoughts and findings with us all,

 

since you gave me a glimpse on your notes in Solingen I thought that there is more in these than just for one book

time seems to be ready for sharing and more discussions,

good that you can publish you findings in this way

 

I think the real transition starts when one uses the hints you gave

making any new sword in the old way

 

gives me lots to think about the design of my own knives and puukkos

 

best regards

Jokke

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