Geometry is an important part of Medieval and Renaissance fencing and in the swords they employed. Knowledge of the rules of geometry permeated Western civilization for hundreds of years. It built cathedrals and castles, enabled siege weapons, and inspired everything from illuminations to sculpture and painting. It was even seen as expressing the divine. Geometry was one of the medieval Trivium, part of the artes liberales taught within the classical humanist curricula, which itself included as its physical education program the artes martiales –or "martial arts". It's not hard then to grasp that long proven ideas for using geometry in architecture and artwork would crossover into the realm of self-defense.
In 1482, the master Filippo Vadi linked fencing and geometry for use of all weapons from dagger to sword to spear. In 1553, the master Agrippa considered geometry fundamentally important and based his rapier method on Euclid while adding a reference to Pythagoras. The image of the geometry compass or divider (referencing the principle of measure) appears symbolically in many Renaissance martial arts sources, including the important mid-15th-century Germanic fighting manuscript, the Codex Wallerstein. For the noted Spanish master, Carranza in the 1580's, fencing was, through its concern with lines, angles, triangles, and circles, subordinate to geometry. The Spanish masters increasingly focused their theories on geometry for their court fencing well into the 1600s.
Circles and triangles in the geometric arrangement were a feature of the Italian soldier Ghilsiero's "theoremi" on the rapier of 1583. In 1606, the Italian rapier master Fabris also saw it as the principle foundation of swordsmanship. That same year the work of the master Giganti described the use arms as being a speculative science that was essentially geometrical. In the 1620s, the Flemish master Thibault famously included pages of text and illustrations dealing with geometric ideas for using the rapier.
There has long been a recognition that familiar spatial relationships exist for how weapons can cross, legs and arms bend, and feet step. Just how useful it was to relate geometry to fencing in this way is something speculative. Yet, the Pythagorean visualization of geometric forms (such as expressed in cryptic diagrams) would ideally assist a fencer with understanding the spatial relationships critical to mastery of fighting motions —especially the angled thrusts of the slender rapier's new foyning method. But geometry was not emphasized in Germanic martial art sources nor by the great master, Fiore die Liberi, in his treatise of c.1410. It wasn't a factor within the rapier teachings of Capo Ferro in 1610 nor was it directly addressed by every 17th-century fencing master. By the Baroque era, as the old martial arts had become obsolete and firearms caused swordplay to be reduced to a limited form, application of geometry in Western fighting arts was all but abandoned.
It should not surprise us, though, that given the symbiotic relationship between swordsmen and swordsmiths they may have very often shared the same working knowledge of geometry. While Medieval and Renaissance swordsmiths may not have been formally trained in geometry, as any master craftsmen or artisans might, they surely understood how measuring aspects of shape, dimension, and ratio applied to producing symmetry and harmony in good weapon design. For the most part, the overall dimensions of a sword blade –which includes its cross-sectional shape– is what is going to help determine its center of gravity. That in turn, determines its center of rotation which then determines it's center of percussion, which together lets you know something of how to best use it. The handling qualities that characterize different swords to optimally cut and thrust and ward are less a scientific matter of "balance point" than they are a holistic and intuitive one. This feeling of how to wield a blade differs with the functional intention of each sword type. That means every swordsman had a subjective choice to make in selecting a sword that was best for them. That they could do so with regard to the "natural laws" found in geometry is curiously comforting.