After handling literally hundreds of antique specimens of real historical swords and hundreds of replicas of all quality and accuracy and training in the authentic source teachings for almost four decades now, I take a particular view towards appreciating the subtle geometry of fighting blades. The qualities that make them handle and perform, inflict impacts with edge and penetrate with point, as well as ward off or deflect forceful blows is what it's all about for me.
In particular, the swords of Western Europe from ancient times through the Medieval and Renaissance eras reflect a certain awareness of Euclidean geometry. Just how much of the proportion and dimensions of their design is a deliberate matter of a craftsman’s intention and how much may be a matter of subjective pattern recognition on our part today is the question.
It's possible to look at a sword and make judgements about its proportions and infer relationships between them that may or may not really be there. It's possible to take near infinite measurements of a sword's shape and cross-section to then imagine we can deduce the conscious intentions of its maker. But whatever or not was known about geometry by a historical swordsmith and how it might then have been applied to any single specimen or model, the end goal was to make a durable and effective fighting weapon. A blade was only deemed of value if it could reliably serve its user in combat.
It's easy enough to make a replica copy of a historical sword by looking at a side profile of its blade and then matching its hilt components. But to do it right the hilt of the original should be detached to look at the tang and the blade itself should be turned in every dimension, especially edge on, so that its three-dimensional cross-sectional differential can be closely replicated. This full profile —intended to meet a specific function— is what a good bladesmith achieved with his knowledge and skill. Along with overall shape and length, the variety of fullers, shallows, spines, and risers that were historically used in blade profiles is absolutely enormous. Any such profile will differ from blade to blade over the centuries and even within the same historical period. Many achieve the very same results through distinctively different compositions. But whatever a blade’s profile, it ultimately had to be fitted with a handle and grip as well as some kind of guard configuration that together optimized its manner of use in combat. An awareness of this was surely factored into the blade’s design itself. It's not difficult to see how this choice would have reflected some notion of a harmonious geometric relationship to the finished piece. It's impossible to say if doing so was a matter of aesthetics, practicality, or a little of both.
We may notice geometric elements in some Medieval and Renaissance swords and wonder to what degree they may have been by conscious design according to some philosophical assumption or else merely serendipitous of the kinesthetic elements of tool use. Perhaps the most important thing about the geometry of swords is the most obvious yet most easily overlooked: design is a direct factor of their ability to inflict wounds and defend against them. Having tested, experimented on, trained with, broken, and explored the use of all manner of ethnographic sword forms for many years —cutting, thrusting, slicing, and warding with them— I can attest with certainty to this fundamental truth. …But what of it? All I can say is that, when it comes to swords there is more than one way to achieve an effective fighting blade.