Geometry of castles
The "Geometry of Castles" refers to the architectural and defensive design principles applied in the construction of medieval castles, which served as residences and fortifications for European nobility. Initially built from wood, castles evolved into stone structures as the need for stronger defenses grew due to advancements in warfare, particularly with the use of artillery. Defensive geometry played a crucial role in their layout, exemplified by concentric designs that included multiple walls to trap attackers between them, as well as circular keeps and towers that minimized vulnerable corners and maximized shooting coverage. Notable features such as moats and drawbridges were engineered to enhance security, while the strategic placement of castles on elevated terrain provided additional defensive advantages.
The construction process was complex and costly, often requiring significant financial resources and meticulous planning to ensure adequate supplies during potential sieges. Additionally, experimental archaeology has emerged as a method for better understanding these ancient structures by recreating them using historical techniques and materials. Castles not only signify military ingenuity but also reflect the cultural and artistic expressions of their time, influencing modern perceptions of medieval life and architecture.
Geometry of castles
Summary: Mathematics has been used to both construct and study castles.
Castle are fortified structures, used as residences by European nobles in the Middle Ages. Early castles were often made of wood, but with the development of better attack methods, castle builders switched to stone as the main building material. With the extensive use of artillery, residential castles became indefensible. They were replaced by purely military forts (not used for administrative and residential purposes) and decorative residences resembling castles (not used during wars). The geometry of a castle was often dictated by defense considerations. Architect Benjamin Bramer fortified castles and published a work on the calculation of sines.
The Alhambra, a fourteenth-century palace and fortress, is well known for its mathematical tiles. In the early twenty-first century, the American Institute of Mathematics proposed a headquarters in California that would be modeled after the Alhambra, popularly referred to as a “math castle.”
Castles are frequently found in fantasy and horror literature. One common image is that of Dracula’s castle. Dracula author Bram Stoker earned a degree in mathematics. Some mathematics teachers use castles like Cinderella’s castle or sand castles to explore concepts such as ratios, fractions, volume, statistics, and geometric shapes. Scientists, including physicist Mario Scheel, explore the physical properties of sand-like material, and researchers in experimental archaeology model and design castles.
Geometry of Castle Defense
Both the layouts of castles and the shapes of their parts were dictated by defense needs. For example, concentric castles consisted of several concentric walls. The barbican (the outer wall) had relatively many entrances, while the inner wall had few, making the attacking army crowd between walls and, thus, become vulnerable to defenders.
Keeps and towers were mostly round to allow for a larger arc of shooting coverage from each arrowslit. In addition, the isoperimetric theorem states that for a given area, the circle has the least perimeter among all shapes, thus minimizing the amount vulnerable walls (not to mention reducing the costs of building materials). Each corner introduced blind spots where enemies could avoid arrows, and circles have no corners. Also, corners are more vulnerable for mining.
Cylindrical towers led to the invention of spiral staircases. Most castle staircases were built so attackers would ascend clockwise, making the central shaft of the staircase interfere with their right hands—often the hand that held the sword.
Stonemasons building castles used simple tools, such as compasses, dividers, and straightedges. Their manuals included descriptions for creating a variety of shapes with these tools. For example, pointed and rounded arches, including Tudor, lancet, and horseshoe arches, could be traced with compasses and straightedges.
Shooting from high towers allowed for better view, and also used gravity to add acceleration to arrows and other projectiles. When glass windows were installed in circular towers, they were made by blowing glass inside a cylinder, cutting it, and then connecting multiple pieces with lead to match the curvature of the castle wall.
Castle builders used terrain geometry to support defense. In addition to the height advantage of the castle walls and towers, castles were frequently situated on hills (either natural or artificial) or on earthen mounds called “mottes.” Defensive ditches around castles, called “moats,” prevented siege towers from coming close. When moats were filled with water, they could also make digging tunnels for mining the walls more difficult.
The construction of moats led to the invention of drawbridges and the mechanisms of raising and lowering them. The drawbridge mechanisms involved levers and pulleys.
Logistics and Finance
Building a large castle was a major financial undertaking spanning many years, and occasionally bankrupted the ruler attempting it, such as King Edward I. Supplying the castle, especially with enough supplies to withstand lengthy sieges, presented another organizational problem. A siege was a common method of castle attack in which the attackers surround the castle grounds and waited for the defendants to starve. The siege process could sometimes last for months or even years.
Experimental archaeology is a new field of study that combines archaeological research, computer modeling, and actual building. Observations in building experiments allow for conclusive results of how models can be made to work. For example, Project Gueledon is a real-size castle built recently to help give people a deeper understanding of how castles were constructed in medieval times. The researchers used building methods and materials similar to those used by thirteenth-century castle builders, with a team of 50 workers from various professions.
Bibliography
Holden, Constance. “A Castle Fit for a Mathematician.” Science 314, no. 6 (October 2006).
Whitney, Elspeth. Medieval Science and Technology. Westport, CT: Greenwood Press, 2004.