Medieval – Page 3 – Geometries of Creation

Here is Charles Bouleau’s analysis of a miniature from the Tres Riches Heures of Jean de Berry, made by the Limbourg Brothers early in the fifteenth century.
Here Adam and Eve are shown in the Garden of Eden, whose circular perimeter was surely drawn with a compass.
More interestingly, Bouleau showed that the height of the Gothic fountain of life in the image could be found by unfolding the diagonal of the square framing the garden’s circular border.
Bouleau also demonstrated that the width of the fountain was set by the intersection of two pentagons inscribed within that circle. In the lower-left of the image, moreover, Adam’s foot locates the corner of the pentagon with the horizontal base.
Bouleau extends that baseline to the right, as shown in green, and then connects it back to the tip of the fountain.
The intersection of this green diagonal with the circle’s equator, he claims, sets the width of the Gothic portal through which the first couple exit, while the blue diagonal to its top sets the sloping pose of the expelled Eve.
This page is folio 25v of the Très Riches Heures du Duc de Berry, which is MS 65 from the Musée Condé in Chantilly, France. This particular image is from: https://commons.wikimedia.org/wiki/Tr%C3%A8s_Riches_Heures_du_Duc_de_Berry#/media/File:Folio_25v_-_The_Garden_of_Eden.jpg

A slideshow of Bellver Castle in Palma visualizing a series of steps that display the geometry behind the medieval architecture

Here is the basic plan of Bellver Castle, in Palma. Its form can be explained in a series of geometrical steps of the kind familiar to the medieval builders and craftsmen.
Start with the radius between the center of the castle and the center of its donjon. A circle of this radius will be tangent to the lobes of the moat at left and right. Construct a hexagon within the circle, and a slightly smaller circle inside the hexagon; its radius equals that of the main moat structure. The difference between the two large circles gives the radius of the donjon.
Next, construct a Star of David within the hexagon, and a dodecagon framed by the interior faces of the star; the corners of this dodecagon lie on the outside surface of the inner wall ring.
Circumscribe a circle around the dodecagon to describe the outer surface of the inner wall ring. The radius to the inner column screen is smaller by a factor of √2, and the inner surface of the outer wall ring is greater by a factor of √2, as one can see by the sequence of nested circles and squares that describe them.
The inner surface of the inner wall can be found by inscribing two dodecagons and a circle within the middle yellow circle. The outer surface of the outer wall plan, correspondingly, can be found by circumscribing two dodecagons and a circle outside the outer yellow circle.
In the area around the donjon, the outer face of the walkway around the moat is described by a circle centered on the donjon and passing through the corners of the green dodecagon previously used to locate the outer wall surface of the castle itself. The inner face of the walkway corresponds to a slightly smaller circle tangent to the inner surface of the castle’s outer wall. To find the splayed base of the donjon, begin by constructing a Star of David within that middle circle, and then a final blue circle within the star...
…the actual donjon base is then described by a slightly larger circle, circumscribed around a hexagon circumscribing the inner blue circle. The wall thickness of the donjon, similarly, can be found by circumscribing a circle around the core of a smaller Star of David inscribed within the red circle describing the outer wall surfaces of the donjon. The splayed base of the castle itself meets the ground at the large violet circle, which is halfway between the outer castle wall surface and the inside edge of the walkway around the moat, as the small violet circles within the moat zone indicate.
The geometrical centers of the left and right turrets appear to stand one wall thickness out from the outer wall surface of the castle. From these centers, the edge of the walkway around the moat can be found by striking circles whose inner points are tangent to the outer surface of the inner castle wall. From the same centers, the sloping bases of the turrets can be found by creating middle-sized circles whose radii pass through the points where the inner surface of the outer castle wall intersects the large orange Star of David. The left turret is slightly smaller than the right one. The left appears to coincide closely with a circle inscribed within a hexagon reaching to the inner surface of the castle’s outer wall, while the right one coincides closely with the circle circumscribed around such a hexagon. Both, however, are slightly narrower than this ideal construction suggests.
The bottom turret is even more irregular, even in is placement, with a center notably to the left of the graphic’s middle axis. The inner surface of the walkway around the moat in this zone, though, coincides closely with a circle centered along the edge of the castle’s splayed base, and tangent to the corner of the large orange Star of David. The donjon and the three other main turrets thus offer a variety of permutations on the same basic theme of hexagons and circles that governs the geometry of the castle as a whole.