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Peter Johnsson
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PostPosted: Fri 25 Nov, 2016 1:15 am    Post subject:         Reply with quote

J. Nicolaysen wrote:
If you guys aren't inscribing your doodles on vellum with compass, calipers and square, I don't think you can trust your accuracy.

http://thomasguild.blogspot.com/2013/01/the-m...liper.html

All in good fun. It's above my head anyhow. Sure appreciate the level of intelligence and precision tho.


So you are saying.... What?
... That using a compass on vellum is *more* exact than a vector program?

But you bring up an important point: it *is* important to understand that compass and straight ruler were the tools of the trade. I have stressed the importance and consequence of this in just about every article I have published on this hypothesis, and it is worth stressing again.
Any attempt at reconstructing a system of design using geometry needs to take into account the degree of precision (or lack thereof) that is attainable with drawing tools. This precision is trainable but limited to the thickness of the lines and the quality and character of the medium. The quality and precision of a construction can also vary from person to person.
When we use a digital tool to construct geometry, we are certain about absolute precision. This is helpful in that we negate any drawing error on our own part.
However, it is crucial that the geometry that is constructed with the computer is of a kind that typically and realistically could be made with compass and ruler. This has been a guiding principle for me in my work on the geometry of the medieval sword.

It is also very important that we use just such "strategies" of construction that we know were used in period work. Geometry can be performed in many ways and the way it is constructed will have an effect on the look and character of the object that is designed. If we do not use the strategies of construction the medieval masters used in their work, our work is likely going to look more or less odd.
Nigel Hiscock has studied what symbolic significance numbers and geometric forms carried through out the medieval period. He has also compared geometric correlations in both medieval architecture and modern neo-gothic architecture. Interstingly, he can show clear fundamental differences between the authentic gothic architecture and the neo-gothic style. While both share a superficial similarity in aesthetic style, the neo-gothic architecture does not show the commensurable harmonic proportions that are typical for medieval gothic architecture.
-I think we can see an interesting correspondence for this is replica swords made today. We all recognise the typical brazil nut swords of the high medieval period. They are simple in form, but they are actually pretty hard to make to look right. There is a subtlety in their proportions that is elusive. I think this is a sword form that has its roots in the use of geometry for construction. Without awareness of this and without applying the same principles, it is a challenge to nail the form and proportions of these weapons.

We have proof of the "strategies" that were used in geometric construction of gothic ecclesiastical architecture. Lime washed floors and large scale parchments survive to our day. In these mediums we see traces of the tools used in creating the geometry. We also see the traces of *how* the geometry was constructed, what steps were used and what elements of geometry was involved in the construction. The circle, the square, the equilateral triangle together with the octagon, the hexagon together with the other regular geometric forms were like the letters of the alphabet in this creative language.
Professor Robert Bork of Iowa university has done important research on the construction methods and use of geometry in gothic architecture. "Geometry of Creation" is a very impotent publication anyone interested in this should study. Over the years I have discussed the use of geometry by medieval engineers, architects and artisans and showed him my results in the study of swords.

If we are serious in this study of the sword: trying to understand principles for design, we must also study other medieval art forms and constructions.

I take your comment made in jest to reach for something like this?
If so, I fully agree. I have made an effort to make these aspects central in this project.
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PostPosted: Fri 25 Nov, 2016 2:10 am    Post subject:         Reply with quote

Thank you Peter for giving my jest more thought than I intended. Your response shows how much time and effort you have put into this. I do value the conversation. Perhaps with some training and better review of your previous work, the diagrams would look less like a mystic mandala beyond my comprehension, and something I can appreciate more. And I would have more useful things to say...or just sense to stay quiet.

First of all it was indeed a simple joke, maybe poorly made, about the idea that you all are using computers and drawing programs where the medievals could use hand tools with far less precision and create such intricate patterns you have rediscovered. To me, that is humbling that there is a level of geometric design and understanding from those days that we have lost...and you all are finding again.

But it's also truly an appreciation of your work, seriously, that I am not blessed with much of a geometric mind to be able to grasp all that you are sharing through this work. And Mark and others can, so it's very intriguing.

Quote:
Any attempt at reconstructing a system of design using geometry needs to take into account the degree of precision (or lack thereof) that is attainable with drawing tools. This precision is trainable but limited to the thickness of the lines and the quality and character of the medium. The quality and precision of a construction can also vary from person to person.
When we use a digital tool to construct geometry, we are certain about absolute precision. This is helpful in that we negate any drawing error on our own part.
However, it is crucial that the geometry that is constructed with the computer is of a kind that typically and realistically could be made with compass and ruler. This has been a guiding principle for me in my work on the geometry of the medieval sword.

It's reassuring you not only take this into consideration, you've planned everything around it. If others who follow your work don't do as well, it's quite open to reductive thinking, and error.

Quote:
Nigel Hiscock has studied what symbolic significance numbers and geometric forms carried through out the medieval period. He has also compared geometric correlations in both medieval architecture and modern neo-gothic architecture. Interstingly, he can show clear fundamental differences between the authentic gothic architecture and the neo-gothic style. While both share a superficial similarity in aesthetic style, the neo-gothic architecture does not show the commensurable harmonic proportions that are typical for medieval gothic architecture.
-I think we can see an interesting correspondence for this is replica swords made today. We all recognise the typical brazil nut swords of the high medieval period. They are simple in form, but they are actually pretty hard to make to look right. There is a subtlety in their proportions that is elusive.


The understanding of geometry and proportion the medievals had, that you are revealing and others today are studying, seems to me like a key to the mind and world-view that we have lost since then. It shows also in this sense the past is a foreign country, and we cannot ever fully live there, vellum and compass or vector program. Perhaps this is too obvious now to bring up, we all appreciate it already I suppose.

So that I am not just a useless jester, here's another question you may have answered elsewhere that I have not seen: At what point in the process in sword making would these things have been done, by who? It's my understanding that most swords would have been a collaborative project between smith, cutler, etc. Who's working the geometry out along the way, or designing it first? Because if this is widespread, albeit gnostic, planning and design, it makes a difference where the swords are forged, fitted, finished. If it is not widespread, are we accurate to apply it to swords of all time periods, or only those of a certain time and place of making? Because there is a long ways between Solingen and Soborg, etc. Perhaps the knowledge was transmitted through the guild system...but this was not developed by the time all the swords we are studying were made, right? Only high medieval again, so perhaps earlier swords cannot reveal as fine geometry, or it is still being developed.

Of course I do have more basic questions and philosophical queries about all this, but I'm sorry to distract from Mark's thread. The fact that he and others can take up this tool and use it in this way is actually quite a testament to your work---and perhaps another testament to the reality of what you've re-discovered as well.

Thanks again, all the best to you, Mark Lewis, Vincent and others who are pursuing this.
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Peter Johnsson
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PostPosted: Fri 25 Nov, 2016 2:56 am    Post subject:         Reply with quote

J. Nicolausen,

Thank you for kind words and great questions!

The Why and Who in this are of course very important questions. I have touched on this in published articles (Park Lane Arms Fair catalogue of 2012, "The Noble Art of the Sword" (catalogue for an exhibition at the Wallace Collection in 2013) and most recently the catalogue for the exhibition "The Sword - Form & Thought" that I was part in curating).

We must remember that we are looking at something that spanned several centuries and a whole continent. There must be variations and exceptions. I have also as of yet only studied some 200+ swords from this perspective. It is only a beginning.

"There is a long way between Søborg and Solingen" Yes! A good observation. But swords were often made in a few centres of manufacture. The Søborg sword was found in the moat of the castle of a Danish bishop, but was probably made in the Solingen region, Passau or some other of the few places where quality swords were made.
We must not forget that swords were never really made by local blacksmiths. They were made by specialists who tended to work close to economic and politic centres of power with good supply of raw material located at hubs of communication and long distance trade.

Over the years since 2010 when I stumbled upon this thing, I have formed some kind of general idea of the Whos and Whys. This idea will likely change as I keep working.
I think there is a beginning sometime in the period of Charlemagne. The emperor brought scholars to his court and as a result we saw the carolingian humanist revival, with the seven liberal arts at the core of this (geometry and arithmetic being two of these arts). It is widely accepted in academia that this had far reaching effects in most all expressions of Eurpean culture: architecture, art, engineering, philosophy and the politics of the Church and the state were all revolutionised as a result. We have the romanesque and gothic styles as a result of this. It is really strange than no one has thought to put the development of the sword into this perspective. Especially since the sword changes from what we recognise as the "Viking sword" to the cruciform knightly sword exactly during this period. From what I have seen so far in my studies, swords of the old traditional form (the "viking" swords) do not readily follow geometric definitions of their proportions (with the exception of Anglosaxon weapons of this period that seem to possibly build on geometric principles, but more study is needed!)
The tea cosy and brazil nut pommel swords of the 10th century commonly follow simple geometric structures that define their proportions. Moreover, they tend to share some basic similar lay outs and often make use of some common similar strategies in the geometry.
Charlemegne stated in law that the Church was responsible in procuring/producing weapons for his warriors. The Church was well capable of this, since it was the one institution that had scholars and literate men with a mind for organisation. These men influenced and brought on the romanesque style of art and architecture, they started to make books that were written with letters we recognise and use today and they were involved in creating an identity for the warrior as a sworn protector of the Church in the form of knighthood. These new warriors needed weapons worthy of their task. We know many big and famous monasteries and abbeys produced weapons. I think that the swords that were produced were designed to harmonise with the Creation and Will of God. That is why they were designed according to geometric principles, just like other important things (abbeys cathedrals, books and religious art) were made to express Truth, Glory and Beauty through harmonic proportions that governed heaven and earth. Geometry was like the finger print of God in creation and the Holy scriptures could be better understood by someone who knew the meaning of numbers and geometric forms. Swords were inscribed with letters and religious (and magical) symbols. We know the sword was a religious talisman with deep symbolic significance both for the individual warrior, but also in culture as an attribute to power and righteousness.

If you are tasked to make such a weapon, you surely want to do it *right*. If you are a medieval scholar who is tasked to set artisans to produce these, you are perhaps well served to set out the ideas with geometry? Perhaps geometry is the only language both the scholar and the artisan share? Writing will not do, since the smith is likely illiterate. Measurements are not really all that reliable. Geometry is exact and scaleable. Most skilled artisans know the use of the compass and straight ruler. Now is a time to put this into work to do Gods will for King and Country.


That is sort of how I envision the start of this.

Later on I think that perhaps the use of geometry in the sword smiths craft become more a tradition with less obvious religious significance. In the 12th century we see a economic revolutions with guilds in towns. The church and the abbeys are no longer the largest or most important economical centres. It is accepted in academical studies that the liberal arts that started as an influence from the church gradually became a matter for not only univerisites but also to some extent the trades and experts in the crafts.
The masons are famous examples of this, but we have proof that also other trades had geometry as an important part of the intellectual and theoretical toolbox.

Again, the sword has not been studied from this perspective. When I have done so, I sometime get the feeling of opening a door that has been closed a long time. I may of course be mistaken in my observations.
It is just striking how well and readily swords follow the same geometry that has been proven to hav been used in the construction of architecture of the period. The idea also goes well with what academic study has shown us about the way medieval artisans in other trades went about constructing work and educating new generations. It is important to have a theoretical base also in manufacturing. There are ideas that needs to be communicated.

When sword manufactureing started to spread from workshops organised around cathedrals and abbeys, we see craftsman's guilds stepping up to take ownership of power of production, organisation of labour and also training of coming generations of craftsmen. The theoretical aspects of the craft became secrets of the trade that were passed on from master to journeymen. Sometimes upon oaths of secrecy. I therefore think that it was the cutler who was the most likely person to use geometry as a design tool for swords during the later part of this period. In different places and in different times this person had different titles. By cutler, I mean the person who oversaw production of swords. He made orders for blades from blade smiths, and hilts from hilt smiths. If the parts were to fit together both aesthetically and functionally they had to be made according to some kind of specifications.
Geometry is uniquely suited for this role in the medieval period.


Last edited by Peter Johnsson on Fri 25 Nov, 2016 3:45 am; edited 2 times in total
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PostPosted: Fri 25 Nov, 2016 3:25 am    Post subject:         Reply with quote

To further give example of the significance of numbers and geometric forms are some examples (and I take this from the work of Nigel Hiscock and Robert Bork).

The Circle was seen as an image for the Universe, but also a symbol for God.

The square and number four signified stability, justice and the four evangelists, as well as the framework of the physical world in the four corners of the world and the four humours of man.

The triangle is obviously a symbol for Holy Trinity.

Six and the hexagon are Perfection like the numbers of days of labour during creation. The six pointed star has connotations to the notion of microcosm and macrocosm since it is made up of two triangles that mirror each other, one pointing upwards and the other downwards: "As above, so below. As below so above."

Seven is the full cycle and the first completed week. Seven also signified virginity and wisdom.

Eight is the New Begining, since the 8th day is the first day of the new week. It is therefore the number for baptism and resurrection. It is also the number for the Martyrs. Chapels dedicated to martyrs are often octagonal in plan. Baptismal fonts are likewise often octagonal in shape.

It is therefore thought provoking that the equilateral triangle is often involved in defining the hilt of many swords. Likewise is the square a common element in the definition of hilt proportions and blade widths.

The octagon and eight pointed star with a circle are very common strategies in the design of gothic architecture. It is also a common method for defining hilt proportions and blade widths.

(In a way we may perhaps see Geometry and its meanings as the Zen of Medieval Europe: it is at the core of the mystic vision of the world that was predominant at the time.)

The use of geometry in design may be an intentional method to infer meaning, imbue power or call down blessing to the weapon. Or it might simply be a modus operandi of ancient craftsmen who follows a tradition. Looking at a groups of swords, there certainly are patterns within the patterns to be observed. Perhaps this is because certain geometrical structures inherently involved a set of symbolical meanings?
Short of finding a manual for sword smiths written in the 12th century, we will never know for certain.


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J. Nicolaysen




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PostPosted: Fri 25 Nov, 2016 7:08 am    Post subject:         Reply with quote

Thank you Peter for your generous explanation and context. That was the most thorough and reasoned explanation of it I have seen. From my small background in liberal arts and religion, I think you are very accurate to see the medieval mind exploring and reasoning out the totality of God's will in this time and way. I am a little familiar with the developments at the same time in music and theology and what I know of religious studies, it makes perfect sense to follow this with something as special as the sword. I'll try to find your earlier works from the catalogs to follow up.

Just some small questions if you don't mind. How do you and others using this system choose the swords to deal with? Are you able to narrow it down to those you believe made and fitted in places like Passau and Solingen, or must you only work with the ones you can handle, or those swords you like the best? Because if this gnostic sword language was shared in those places, but not available in others, you may find problems in the analysis, I think.

What is the end time for your swords? At what time period does the geometry no longer factor in the creation of swords at all? I have a guess, but I would like to hear your thoughts. Thank you for giving me such a great idea of the beginnings of the process.
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PostPosted: Fri 25 Nov, 2016 8:45 am    Post subject:         Reply with quote

This work on analysing the proportions is a new thing. From videos of lectures, demos and interviews on YouTube, the few articles I have published and now the catalogue from the exhibit in Solingen (with contains the most material on the topic so far published) interest has now begun to grow and people start to try analysing sword using some of the methods I have outlined.

I have analysed every sword I have gotten access to and in some cases made preliminary analysis of swords that others have documented and shared dimensions and data with me. Since the late 1980`s I have been travelling to museums to study and document swords. Slowly and very gradually at first. During the last 16 years I have made more than 60 research trips to museums in Europe and the States. This is the material I base my study of proportions on.

After tracing a sword, measurements and pencil lines are used to construct a faithful vector image of the sword.
This vector image can be scaled and compared to various geometric construction models I have prepared over the years.
I do not choose swords that I think will confirm my hypothesis. I am as happy seeing a sword that does not fit any modular or geometric definitions as I am when I find swords with proportions that are clearly harmonic and commensurable.
It is important to keep an open mind as to the results, since this is a new line of research. I may be mistaken in my conclusions, but I also want to show the results I have found. I think this study offers an important aspect in our understanding of the medieval sword. If the hypothesis is deemed credible and my results seen as trustworthy, we may get an insight into the thought patterns of the makers of the medieval swords and form a deeper understand of this iconic weapon.

From what I have seen so far, swords from the earliest time of the knightly cruciform swords up to and including the renaissance correspond to geometric definition of their proportions. There are exceptions of course. Not all swords have harmonic commensurable proportions and there may be local traditions influencing this (as well as the obvious situations when swords have been remounted, repaired after being broken and so on).
Even many rapiers conform to geometric definition of proportions to a surprising degree. The geometry of rapier hilts naturally become complex because of the complexity of the forms. Because of this, the results are less convincing but non the less still a possibility.

There is one rare literary text that mentions geometry or at least mathematics and swords and it dates from the 1630´s.
It is a play "The Weeding of Covent Garden" by Richard Brome.
A father makes a surprise visit to his son who studies law in London. The son has squandered the money he was given from his rather on women, wine and song and therefore puts on a farce to try and convince the father he is a diligent student in the hope to squeeze more money. Unbeknownst to the son, the father believes his son wastes too much time pouring over books, while he instead should learn important lessons in the facts of life from the taverns and the streets. The following dialogue unfolds:

Crosswill (the father): Mad. Are they not? And so will you be shortly, if you follow these courses. Mooting, do they call it? You shall moot nor mute here no longer. Therefore, on with your cloak and sword, follow me to the tavern, and leave me such long-tailed company as these are, for I do not like them.

Mihil (the son): No more do I, sir, if I knew how to be rid of ’hem.
 
Crosswill: I think thou hast ne’er a sword, hast thou, ha?
 
Mihil: Yes, sir.
 
Crosswill: Where is it, sir? Let me see ’t, sir.
 
Mihil: ’Tis here, under my bed, sir. –[Note: CROSSWILL reaches under the bed and brings out MIHIL's sword.]
 
Crosswil: Why there’s a lawyer’s trick right, make his weapon companion with his piss-pot. Fie, fie, here’s a tool indeed. [CROSSWILL offers MIHIL money There’s money, sir, buy you a good one, one with the mathematical hilt, as they term it.
 
Ta Daaaa!
Mathematical hilt, indeed.
:-)
The editor of this text wrote to various museums of arms and armour without getting a satisfactory explanation as to what a mathematical hilt might be. Toby Capwell of the Wallace Collection showed it to me: -"I think you might be interested to see this..."
I naturally have my theory what is implied... ;-)


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PostPosted: Fri 25 Nov, 2016 9:42 am    Post subject:         Reply with quote

This thread is by the way very interesting.

Have you got an example of a "real" historical sword that is not made with harmonic commensurable proportions? I'm curious to know if I could see a difference just with the eye.
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PostPosted: Fri 25 Nov, 2016 10:58 am    Post subject:         Reply with quote

Below is an example from the museum of Chalons that fail to meet any of the basic lay outs.

The thick lines mark the discrepancies:
In the first position, the guard does not meet the vesica.
In the second example with the guard at the vesica, the point fails to meet a circumference.
In the third example, the guard is placed in the middle of the circle and the point fails to meet a circumference
In the last example, the guard is placed at the circumference of the first circle and the point still does not fall on a circumference.
(the two last examples are basically the same, but in two different scales).

This may be a sword that was never designed based on geometry, or it may have been refurbished, remounted or repaired after its point got broken.

However, since the guard seems to fit the structure so well in the second example, perhaps we may venture a guess that the blade was broken so that a part of the point was lost. We cannot know this, but if I were to make a reconstruction of this sword, I would be tempted to see what it looked with a longer blade that met the geometry of the second example.



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PostPosted: Sat 26 Nov, 2016 6:14 pm    Post subject:         Reply with quote

The "mathematical hilt"... what a great reference!

Peter Johnsson wrote:
If you are a medieval scholar who is tasked to set artisans to produce these, you are perhaps well served to set out the ideas with geometry? Perhaps geometry is the only language both the scholar and the artisan share? Writing will not do, since the smith is likely illiterate. Measurements are not really all that reliable. Geometry is exact and scaleable.

I think this is a compelling theory, and fits well with the suggestion that clergy may have been instructing smiths as to the text of some inscribed sword blades, and the fact that they had an active role in defining the content and iconography of so many other arts and crafts.

I've begun revising and correcting the diagrams from my first post, and comparing with published measurements. The hilts of the swords from Utrecht and Veste Coburg do not have a 1:1 ratio as I first suggested, instead both are close to 13:12. The guard of the Coburg sword seems like it may be defined with the two octagons and by the diameter of the basic circle. The Utrecht sword seems different: the thickness of the guard seems to follow the inscribed hexagon, but it's width matches the inscribed octagon.





Thinking about this discussion about the use of geometry in medieval arts and crafts brought to mind an illumination from the Gospel Book of Otto III (which happens to include a brazil-nut pommel... tweaked my memory). Was this illustration made according to an underlying geometric design? I had in mind that I could identify the equilateral triangle and inscribed square which align with the emperor and his throne, but then found much more! The progression of square root dimensions was inspired by one of Peter's diagrams in Form and Thought.



https://en.wikipedia.org/wiki/Gospels_of_Otto_III
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PostPosted: Sun 27 Nov, 2016 2:02 am    Post subject:         Reply with quote

Mark,

I enjoy seeing your continued experiments!
There is a number of simple geometric cuts of the square that may also define hilt details. Several of these are demonstrated in the Solingen catalogue. You may want to they some of these to define pommel and guard widths & heights.

Inscribing the regular geometric forms in the circle is a common strategy. Direct cuts of the square is the other approach. By combining these two principles you make use of both the square and the circle.

The square express the unit 1 in its side and the irrational root2 in its diagonal. The circle express the rational 1 in it diameter and the irrational in its circumference through the number Pi.
I find it fascinating that the use of geometry as a principle for design will naturally shift the form between the rational and the irrational.

See below a preliminary analysis of an icon of Saint Clement of Alexandria, who said: "Righteousness is quadrangular". Geometry was so closely incorporated into the understanding of the nature of Creation that geometric forms could be used to express ideas on moral philosophy.



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PostPosted: Sun 27 Nov, 2016 2:53 am    Post subject:         Reply with quote

A type XIIIa sword from the National Museum of Slovenia (of German origin with a nice "AGLA" inscription in the blade) conforms to a beautiful, clear and simple (but subtle) geometry. I published this illustration in the German knife periodical Messer Magasin.

Hilt to blade ratio is 1:4, a very common proportion for war swords (fig I).
The hilt is defined by simple geometric cuts of the basic square.
The blade width is defined by arcs that divide the side of the square in proportion of the golden section (fig III).
The guard width and the pommel diameter are defined by the same double pair of tangent arcs (fig IV).



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PostPosted: Sun 27 Nov, 2016 5:35 am    Post subject:         Reply with quote

Mark Lewis wrote:

Thinking about this discussion about the use of geometry in medieval arts and crafts brought to mind an illumination from the Gospel Book of Otto III (which happens to include a brazil-nut pommel... tweaked my memory). Was this illustration made according to an underlying geometric design? I had in mind that I could identify the equilateral triangle and inscribed square which align with the emperor and his throne, but then found much more! The progression of square root dimensions was inspired by one of Peter's diagrams in Form and Thought.



https://en.wikipedia.org/wiki/Gospels_of_Otto_III


Hi Mark.

What a great example you have uncovered - it becomes very clear in your explanation.
It really highlights that idea that as God had created the Earth using sacred geometry, so humans could imitate in God's honour these sacred geometric designs and thus "christen" the surroundings.

The idea of sacred geometry is off course much older than both Judaism and Christianity (Neolithic age if not older), but in the classical pagan world these ideas was very much tied to astrology. Especially the hellenistic Mithras religion where each planetary sphere had a guardian, you had to pass to reach heaven (Mithraism being a syncretistic mix of primarily Platonism and Iranian religion).
It explains the great distaste Christians had for astrology; where they instead explain numbers and secret being exclusively from God; but the basic idea of sacred geometry being divine secrets (mysteries) which the faithful can unlock are the same.

So we should expect these religious experts to use this geometry for christening their surroundings and convey other religious points to the viewer:

Peter Johnsson wrote:

a preliminary analysis of an icon of Saint Clement of Alexandria, who said: "Righteousness is quadrangular". Geometry was so closely incorporated into the understanding of the nature of Creation that geometric forms could be used to express ideas on moral philosophy.


Thanks Peter for this example, which was exactly what I had been looking for Big Grin
So if "Righteousness is quadrangular", then what about the triangle that Otto III sits in [In fact his head actually is the focal points of five triangles of different sizes].
3 being only the number of the trinity is probably too simple an explanation -> a moral point could be included as well.

So swords inscribed with circles, triangles, hexagons or octagons would likely have moral points as well !

and with Otto III, Mark have included a hexagon around his head. The specific choice of that geometric form around his head almost shouts for a "moral explanation".
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PostPosted: Sun 27 Nov, 2016 10:29 pm    Post subject:         Reply with quote

Hi Niels,

If you look at my post a few steps up this thread you will see a list of some accepted symbolic meanings of numbers and their corresponding geometric forms as they were understood in the period.

In the individual case it is naturally difficult to know if geometry was intentionally developed to convey a symbolic meaning. I think it would be a mistake to discount the possibility that it was a factor, however. "The search for inner meaning of things was a medieval habit" as is said by Nigel Hiscock.
I wonder if you cannot describe the use of symbols in objects from anything like obvious decorative devices to the subtle structures of design in geometric grids as a kind of "software" that elevated the commonplace into the powerful.

Perhaps we can understand the combined geometric forms and numbers as a kind of prayer or a way to call down a blessing upon the weapon and its user? The square for righteousness and stability, the triangle as a blessing, the octagon as a hope for resurrection or that death may be an act of martyrdom?
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Niels Just Rasmussen




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PostPosted: Mon 28 Nov, 2016 7:04 am    Post subject:         Reply with quote

Peter Johnsson wrote:
Hi Niels,

If you look at my post a few steps up this thread you will see a list of some accepted symbolic meanings of numbers and their corresponding geometric forms as they were understood in the period.

In the individual case it is naturally difficult to know if geometry was intentionally developed to convey a symbolic meaning. I think it would be a mistake to discount the possibility that it was a factor, however. "The search for inner meaning of things was a medieval habit" as is said by Nigel Hiscock.
I wonder if you cannot describe the use of symbols in objects from anything like obvious decorative devices to the subtle structures of design in geometric grids as a kind of "software" that elevated the commonplace into the powerful.

Perhaps we can understand the combined geometric forms and numbers as a kind of prayer or a way to call down a blessing upon the weapon and its user? The square for righteousness and stability, the triangle as a blessing, the octagon as a hope for resurrection or that death may be an act of martyrdom?


Hi Peter.
I saw these (for some numbers more general) symbolic meanings for the different numbers, but I think there is way more to it, as your example of with the square shows. The triangle is likely tied to more meanings, than just the Trinity especially when it is in relation to other numbers.

Your quote by Nigel Hiscock is something I fully agree with from what I have read so far. As your searched the Bible for hidden secrets you also looked at the world (created by God) in the same way. I think it very likely that a least an educated medieval person would automatically look at the geometry of swords to find the hidden multifaceted symbolic meanings (as they would also expect artists to put them there in the first place). A meaning that goes way beyond the simple understanding of 3= trinity etc.

I found some quote from Isidore of Seville's (~560-636) Etymologies that are interesting for the way the educated Christian mind would search for numbers.

iv. What numbers do for us (Quid praestent numeri)
1. The reckoning of numbers ought not to be despised,
for in many passages of sacred writings it elucidates how
great a mystery they hold
. Not for nothing it is said in
praise of God (Wisdom 11:21), “Thou hast ordered all
things in measure, and number (numerus), and weight
.”
2. The [number] that contains six units (senarius), which
is perfect in its own parts, declares the completion of the
world by a certain signification of its number
.2 Likewise
for the forty days during which Moses and Elijah and
the Lord himself fasted: without an understanding of
numbers, the span of days is unintelligible.
3. So also there are other numbers in the Sacred
Scriptures whose figurative meaning cannot be resolved
except by those skilled in the knowledge of the mathematical
art
. It is even our lot to depend on the discipline
of numbers to some extent when through it we name the
hours, when we dispute about the course of the months,
and when we recognize the duration of the turning year.
4. Indeed, through numbers, we are provided with the
means to avoid confusion. Remove numbers from all
things, and everything perishes. Take away the computation
of time, and blind ignorance embraces all things;
those who are ignorant of the method of calculation
cannot be differentiated from the other animals
.

Source: http://sfponline.org/Uploads/2002/st%20isidore%20in%20english.pdf

Or St. Augustine of Hippo's (354-430) "On Christian Doctrine".

25. "Ignorance of numbers, too, prevents us from understanding things that are set down in Scripture in a figurative and mystical way. A candid mind, if I may so speak, cannot but be anxious, for example, to ascertain what is meant by the fact that Moses and Elijah, and our Lord Himself, all fasted for forty days.(7)
And except by knowledge of and reflection upon the number, the difficulty of explaining the figure involved in this action cannot be got over. For the number contains ten four times, indicating the knowledge of all things, and that knowledge interwoven with time. For both the diurnal and the annual revolutions are accomplished in periods numbering four each; the diurnal in the hours of the morning, the noontide, the evening, and the night; the annual in the spring, summer, autumn, and winter months.
Now while we live in time, we must abstain and fast from all joy in time, for the sake of that eternity in which we wish to live; although by the passage of time we are taught this very lesson of despising time and seeking eternity. Further, the number ten signifies the knowledge of the Creator and the creature, for there is a trinity in the Creator; and the number seven indicates the creature, because of the life and the body.

For the life consists of three parts, whence also God is to be loved with the whole heart, the whole soul, and the whole mind; and it is very clear that in the body there are four elements of which it is made up.

In this number ten, therefore, when it is placed before us in connection with time, that is, when it is taken four times we are admonished to live unstained by, and not partaking of, any delight in time, that is, to fast for forty days. Of this we are admonished by the law personified in Moses by prophecy personified in Elijah, and by our Lord Himself, who, as if receiving the witness both of the law and the prophets, appeared on the mount between the other two, while His three disciples looked on in amazement.

Next, we have to inquire in the same way, how out of the number forty springs the number fifty, which in our religion has no ordinary sacredness attached to it on account of the Pentecost, and how this number taken thrice on account of the three divisions of time, before the law, under the law, and under grace, or perhaps on account of the name of the Father, Son, and Holy Spirit, and the Trinity itself being added over and above, has reference to the mystery of the most Holy Church, and reaches to the number of the one hundred and fifty-three fishes which were taken after the resurrection of our Lord, when the nets were cast out on the right-hand side of the boat.(1) And in the same way, many other numbers and combinations of numbers are used in the sacred writings, to convey instruction under a figurative guise, and ignorance of numbers often shuts out the reader from this instruction."
Source: http://faculty.georgetown.edu/jod/augustine/ddc2.html

So these two major Christian figures and huge authorities in the Middle Ages certainly takes the numbers game seriously.
To put it frankly you are either Mathematician OR an ignorant Animal.

My point is really that the medieval mind seemed to expect the "computation of scripture" to be complex and not simple!
I would expect their art to be also complex and not simple -> Otto III is surrounded by geometry and what does that "canvass of geometrical numbers leads us to conclude? Augustine finds a sum of 153 fishes after some mental number gymnastics, so what is the sum of Otto III?

Peter, you actually said it very well in a former post, what I wanted to highlight with these quotes:
"That is why they were designed according to geometric principles, just like other important things (abbeys cathedrals, books and religious art) were made to express Truth, Glory and Beauty through harmonic proportions that governed heaven and earth. Geometry was like the finger print of God in creation and the Holy scriptures could be better understood by someone who knew the meaning of numbers and geometric forms".
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Peter Johnsson
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PostPosted: Mon 28 Nov, 2016 9:58 am    Post subject:         Reply with quote

Niels,

I fully agree with you that there is much more to be done in this area.
So far, I have avoided digging too deep into medieval numerology and magic even though it has played a role in the background. Since this is a completely new hypothesis it seemed best to proceed with some caution. ;-)
Claiming that geometry might have played a role in the design of swords is bad enough.
In some of the articles I have published I touch lightly on the symbolic aspect of geometry and numbers and in some cases I have suggested a possible meaning involved in the proportions of some swords. So far, I have hardly even skimmed the surface of this.

Looking deeper into the additional layers of possible meaning could well reveal possible explanations for some combinations of forms and modules. That could be important in finding new angles. In a way, I think that the symbolic meaning of numbers and forms is a necessary aspect if we are to understand the reason and importance of this method of design.

-Thank you for the fine quotes, by the way! I really enjoyed those.
:-)
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Mark Lewis





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PostPosted: Tue 29 Nov, 2016 10:00 am    Post subject:         Reply with quote

I want to follow up on several trains of thought, so this post is going to ramble a bit...

Peter Johnsson wrote:
I enjoy seeing your continued experiments!

Thanks Peter, I'm glad you are still following along! I definitely view all of my posted diagrams are experimental, and have put my learning curve on display for the sake of gaining some feedback and stimulating discussion... so that's been successful at least Happy. However, on the other hand I do not want to be responsible for spreading any misleading information, so as time permits, I'm trying to review my first diagrams and correct any elements or ratios that are clearly contradicted by new information.

Peter Johnsson wrote:
There is a number of simple geometric cuts of the square that may also define hilt details.

A type XIIIa sword from the National Museum of Slovenia (of German origin with a nice "AGLA" inscription in the blade) conforms to a beautiful, clear and simple (but subtle) geometry.


I made a few basic attempts along those lines, but just haven't gotten a "hit" yet. I also haven't yet tried analyzing any late medieval swords... with the longer swords like XIIIa's, foreshortening becomes hard to avoid in photos, so I don't have as many good candidates to even try to work on.

Thanks for posting that sword in particular - I knew about it's very interesting inscription (Niels: did we talk about it a few months ago?) but had not seen any good, clear photos of the entire sword.

Is there any noticeable progression in time as to what type of geometric designs are used? Are "simple" inscribed shapes superseded by more subtle geometric cuts, or do designs remain consistently varied throughout the whole period?

Peter Johnsson wrote:
See below a preliminary analysis of an icon of Saint Clement of Alexandria, who said: "Righteousness is quadrangular".

Looks great! Is this your analysis also, or is there someone else working more specifically on icons and artwork?

Something I notice in the icon and the illumination of Otto III is that the placement of hands is also very deliberate. With the exception of the the soldier at far right, every hand is holding/pointing/revealing part of the geometric design.

If I had to guess, I would suggest that the progression of square root dimensions is more practical than symbolic - a tool for guiding the use of space in the illustration. The progression is easily constructed by tracing a sequence of arcs which increase in size, and the final size of the illustration can be changed by using more or fewer in the sequence. The area of the illustration is then divided up into several "registers" used for different elements of the picture... the lower register is waist height for the human figures, and then their heads fall in the third register. The upper register is used for the sky and distant background.

Niels Just Rasmussen wrote:
"The [number] that contains six units (senarius), which is perfect in its own parts, declares the completion of the world by a certain signification of its number."

Isidore is using the word "perfect" here in a very specific mathematical sense: six is the smallest perfect number, a number that is equal to the sum of it's own proper divisors:

6 = 1 x 2 x 3 = 1 + 2 + 3

The next perfect number is 28 = 1 + 2 + 4 + 7 + 14, the number of days in a lunar cycle. Perfect numbers are very rare, there are only two more below 10,000. Perfect numbers were known to the ancient Greeks and to medieval Christian and Muslim scholars. The number six is not important merely because it numbers the days of creation... rather Augustine reverses the priority and says that the world was created in six days because six is the first perfect number!

https://en.wikipedia.org/wiki/Perfect_number

Peter Johnsson wrote:
Anglosaxon weapons of this period that seem to possibly build on geometric principles, but more study is needed!

Charlemegne stated in law that the Church was responsible in procuring/producing weapons for his warriors.

I think that the swords that were produced were designed to harmonise with the Creation and Will of God. That is why they were designed according to geometric principles

This is very interesting if Anglo-Saxon swords are distinct in having geometric designs... "Anglo-Saxon" meaning Petersen's type L specifically? If this is really the case, there is a possibly a common thread here, very speculative... Many continental monasteries were rejuvenated by the arrival of Anglo-Saxon and Irish monks. For a time the master of Charlemagne's own Palace School was Alcuin of York, thought to be the author of the collection of mathematical and logical puzzles Propositiones ad Acuendos Juvenes. Several of the puzzles are essentially problems of ratios and divisions of a whole into different parts.

So if geometric design first appeared among Anglo-Saxons, it may also have been one of their contributions to the Carolingian Renaissance... Very interested to know if you have considered this possibility!

https://en.wikipedia.org/wiki/Alcuin
https://en.wikipedia.org/wiki/Propositiones_ad_Acuendos_Juvenes
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PostPosted: Tue 29 Nov, 2016 3:46 pm    Post subject:         Reply with quote

The companion illumination on the page opposite from Otto III uses the same division of space.

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PostPosted: Wed 30 Nov, 2016 8:26 am    Post subject:         Reply with quote

Mark Lewis wrote:

Something I notice in the icon and the illumination of Otto III is that the placement of hands is also very deliberate. With the exception of the the soldier at far right, every hand is holding/pointing/revealing part of the geometric design.

If I had to guess, I would suggest that the progression of square root dimensions is more practical than symbolic - a tool for guiding the use of space in the illustration. The progression is easily constructed by tracing a sequence of arcs which increase in size, and the final size of the illustration can be changed by using more or fewer in the sequence. The area of the illustration is then divided up into several "registers" used for different elements of the picture... the lower register is waist height for the human figures, and then their heads fall in the third register. The upper register is used for the sky and distant background.

Niels Just Rasmussen wrote:
"The [number] that contains six units (senarius), which is perfect in its own parts, declares the completion of the world by a certain signification of its number."

Isidore is using the word "perfect" here in a very specific mathematical sense: six is the smallest perfect number, a number that is equal to the sum of it's own proper divisors:

6 = 1 x 2 x 3 = 1 + 2 + 3

The next perfect number is 28 = 1 + 2 + 4 + 7 + 14, the number of days in a lunar cycle. Perfect numbers are very rare, there are only two more below 10,000. Perfect numbers were known to the ancient Greeks and to medieval Christian and Muslim scholars. The number six is not important merely because it numbers the days of creation... rather Augustine reverses the priority and says that the world was created in six days because six is the first perfect number!


Mark, thanks for the perfect number explanation!

Furthermore I don't think you can make a clear dividing line between practical and symbolic and having the progression of square root dimensions being "only" a tool. I think BECAUSE it is mathematical, then it would also be symbolic, at least for the viewer of the art piece. If a practical tool was used without being mathematical, I would think it likely to have no symbolic meaning.

You mentioned the idea by Augustine of Hippo in this City of God [De Civitate Dei], that 6 is not holy "in itself" because God created the world in 6 days, but because 6 is a perfect number. God created the Earth in that number of days, because God is the omniscient Mathematician. Likely that's also - for Augustine and Isidore - why God chose to make the lunar cycle 28 days, because that is also a perfect number.

So only by knowing math can you observe God's beauty and God's plan in the scriptures and in the created world, and by applying math in earthly crafts (and that is also a very practical tool) you create the "earthly City of God" as a mirror image of the "heavenly City of God".

For the "math-churchfathers" an "only focus" on the literal meaning in the Bible is how the ignorant read the Bible. "Those who knows" will use an allegorical reading to find God's mysteries (secrets of numbers). That is clearly still the idea in the high middle ages as well [I will return with Hugh de Saint Victor later as he has a lot of allegorical interpretation of mathematical material - but that will be a enormous post].
Perhaps protestant people being very used to things being holy because it "says so in the Bible" will have a tough time understanding this allegorical tradition in the Roman church, but it can explain why it is prudent for Peter to be careful with his hypothesis as this kind of thinking is quite alien for the modern mind and also doesn't really conform to the general idea some modern people have about knowledge in the middle ages (~being primitive, which is clearly not the case for the elites).

Origines in the "De Principiis":
Preface 8.
"Then, finally, that the Scriptures were written by the Spirit of God, and have a meaning, not such only as is apparent at first sight, but also another, which escapes the notice of most. For those (words) which are written are the forms of certain mysteries, and the images of divine things. Respecting which there is one opinion throughout the whole Church, that the whole law is indeed spiritual; but that the spiritual meaning which the law conveys is not known to all, but to those only on whom the grace of the Holy Spirit is bestowed in the word of wisdom and knowledge".
Source: http://www.newadvent.org/fathers/04120.htm

A literal and superficial reading of the Bible are for the ignorant. Those with wisdom and knowledge (of math) can unearth the spiritual meaning of the "forms of certain mysteries and the images of divine things"!
In that regard Christianity is really a Hellenistic Mystery-religion in its early period when Clement and Origines lived in Alexandria. You have a superficial meaning in the scriptures, but after Baptism the newborn christian (child) was ready for "adult" things - the mysteries.

So whether you craft-technique is practical or not is really not so important - it's whether your technique are in the form of certain mysteries (very likely mathematical) so you get a view into divine things (God's idea/plan for the cosmos).

If you make your layout according to mathematics, then you are in effect doing a mirror image of God's creation, as he also applied math to the creation.
So to also take up a notion that Peter had in a former post in this thread:
By applying math to the layout of the art-work being designed (whether an image, a sword, or a chuch-building) the craftsman is engaged in contemplation about the mysteries of God (perhaps even prayer as Peter stated). When actually creating the thing you then mirror the heavenly City of God down to the earthly City of God (which for Augustine is no longer Rome itself but more broadly the Christian Community = The Roman Church). Is it practical? "Yes off course, since math is from God".
The created thing makes it then possible for the "viewer in the know" to also contemplate the mysteries of God and perhaps even getting a mirror-view into the heavenly City of God (icons as windows into paradise is at least a thing in the Orthodox church).
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Mark Lewis





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PostPosted: Tue 06 Dec, 2016 7:40 pm    Post subject:         Reply with quote

I've made another attempt at analyzing the broad-bladed sword in Paris... A type II structure may work better than my first construction, with the first vesica aligning better with the mid-line of the cross. The width of the blade seems to coincide with a square inscribed inside the vesica, which in turn defines ad quadratum another square that is close to inscribing the circular pommel.



Here is a new diagram that I think developed in an interesting way...the sword is formerly from the Christensen collection, and was auctioned a few years ago. The angular profile of the type E pommel seems to match the inscribed hexagon, while the upper and lower arcs of the curved cross (which is quite asymmetrical) follow circles with lower edges aligned with vesicas of the type II structure.



A third illumination from the Gospels of Otto III has Christ standing before a literal Golden Rectangle. Many details seem to align with the diagonals of the boundary rectangle, or their perpendiculars... The asymmetry of the image makes it harder to find an underlying design, but here is my sketch of a few possible connections.

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Niels Just Rasmussen




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PostPosted: Mon 12 Dec, 2016 6:41 am    Post subject:         Reply with quote

Mark Lewis wrote:

Here is a new diagram that I think developed in an interesting way...the sword is formerly from the Christensen collection, and was auctioned a few years ago. The angular profile of the type E pommel seems to match the inscribed hexagon, while the upper and lower arcs of the curved cross (which is quite asymmetrical) follow circles with lower edges aligned with vesicas of the type II structure.

A third illumination from the Gospels of Otto III has Christ standing before a literal Golden Rectangle. Many details seem to align with the diagonals of the boundary rectangle, or their perpendiculars... The asymmetry of the image makes it harder to find an underlying design, but here is my sketch of a few possible connections.


Hi Mark.
Interesting that the sword from the Christensen collection also shows an asymmetrical crossguard that doesn't really fit the geometry - just like on the Esrum sword! So do we have a later repair/fix on the crossguard here as well since they also here, as the Esrum sword - are "flaring" out.
But to see the inscribed hexagon (a perfect number 6) outlined by the exact pommel shape makes you think -> is it that shape because of function and/or because the need to make a hexagon?!

With Christ you have an octagon and square around his eye connected with his right hand (and the divine halo circle within the square is already painted from the start).
Still a good result, even if no Pythagoras Wink
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