Architecture – Page 3 – Geometries of Creation

June 9, 2020April 12, 2024

Ground Plan Geometries in Suger’s St-Denis: A Prototype for Altenberg

Author: Robert Bork

Ground Plan Geometries in Suger’s St-Denis

Bellver Castle, Palma

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.

Petra Deir- “Monastery”

Begin with a square, aligned with the axes of the main temple front, and call its side length 1.000. Set its equator even with the baseline of the main lower entablature, i.e. with the top edges of the lower capitals at height .500. Its top edge will align closely with the top edge of the natural wall behind, seen at left. Draw a circle around the square, noting that its top point at height 1.207 locates the baseline of the second story entablature.
Circumscribe an octagon around the circle, and note that its facet length of .586 establishes the notional span between the axes of the main tetrastyle subtemple; the left hand column is too far to the left, for reasons to be considered below. Within the .586 facet space, step down by quadrature to find the width of the door opening (.148) and the width of the door frame (.207), though this time the alignments are better at right. More precisely, note the vertical alignments at the following heights:

.086---baseline of the central door
.207---baseline of the lateral “windows”
.354---molding on the lateral “windows”
.396---molding on the central door frame
.574---the base of the first story entablature
.646---bottom edge of the column bases in second story
.707---molding on upper statue pedestals
Following up on the octagon theme, step inward twice via octature the circle circumscribing the original unit square, and twice more from the circle of diameter .586 framed by the facets of the first octagon introduced in step Bork2. From the smaller, one finds the following heights:

.388---baseline of the main portal lintel
.750—main molding in the central tholos niche
From the larger, one finds:
.038---baseline of the central portal area
.962---moldings on the upper lateral “windows”
1.104---baseline of the second main entablature.
Note also that the vertical dropped from the diagonal at height .962 conforms closely with the edge of the “window” and the column below on the right-hand side, and that a similar relationship can be made with the next column inward, but that the relationships differ slightly on the left side, suggesting possible confusion on the builders’ part between axes and edges, or between successive steps in the octature process, or both.
To describe the upper part of the structure, step outward with octature, and then build diagonals off of that toward the apex of the composition. More specifically, if one steps out by a single octature step from the original circle seen in step Bork1, the midpoint of its upper facet will be at height 1.041, locating the tops of the column shafts in the second story. This lateral facet of this octagon corresponds closely with the right margin of the structure, and this relationship is perfect in its extension into the Doric frieze of the second entablature. Again, this is less precise on the left side, but the frieze steps inward with respect to the column below, suggesting that the builders had realized their error and were trying to fix it. The diagonal extensions of this octagon converge at height 1.500, a key point on the tholos elevation, and the raking edges of the broken pediment are defined by lines dropping from this convergence point to the points on the outer frame at height 1.153, which is where the original circle cuts its own upper octagon ray. Another horizontal level just above on the outer frame is at height 1.265, one ocature step up from the previous. The tips of the broken pediment are at height 1.367, where their slope is cut the by verticals rising from the intersections at height 1.104. There are key points on the obelisk at height 1.500 and 1.582, where diagonals from the previously described octagons converge, and its tip was probably originally at height 1.672, where an analogous structure once octature step larger would converge.

Et voila…
This is the so-called Monastery from Petra, Jordan. The source image comes from Judith McKenzie, The Architecture of Petra. British Academy Monographs in Archaeology 1. Oxford: Oxford University Press, 1990, plate 139.

Petra- “Treasury”

I started with the axes of the outer columns. I will call the distance from those axes to the centerline 1 unit, so that the width between those axes is 2 units. The height of the column shafts in the first story is 1 unit, so that this area forms a double square (slightly smaller than the one that you found around the exterior of the composition; more on that below…)

I stepped in by quadrature, and found the width of the door opening, which is (root2)/4, or .354, in addition to noticing that the right-hand column flanking the door sits further outboard than its pendant to the left.
I create a shaded orange square whose height matched that of the column shafts.

I circumscribe a circle around it, and I notice that the intermediate column axes are halfway between the sides of the square and the vertical tangents to the circle. The span between those orange vertical axes is therefore 1.207 = (1 + root2)/2.
The same circle reaches up to height 1.207, at the base of the figural frieze in the entablature.
I also create an arc whose center is the centerpoint of the façade baseline, so that it sweeps through the upper corners of the shaded square. The top of this arc locates the top edge of the capitals. This geometry relates to the Golden Section (Phi), such that the top edge of the capitals lies Phi-1 = 1.618 -1 = 1.118 units above the baseline. In other words, the height of capitals is 11.8% as great as the height of the shafts. The big double square that you discovered in your analysis, similarly, is 11.8% bigger than the one I got from the pier axes.
I also infer from this structure that the inner column axes are supposed to be .500 units apart, and that the greater displacement of the right-hand one is a mistake. I carry those orange axes upward to the top of the composition.
To climb upward, I first draw 45-degree lines from the corner of the shaded square, so that they converge at height 1.500. The slope of the main pediment seems to have been set by connecting the vertex of the resulting triangle with the points at height 1.207 on the outer column axes.
The principal horizontal crease just above the pediment is at height 1.618, i.e. exactly the Goldne section of the first story’s column height. This height can be found by unfolding the half-diagonal of the shaded square, and the heavy orange arc indicates. This will be a big deal for later.
Meanwhile, I also note that the orange diagonals launched from the column axes at height 1.000 converge at height 1.802, locating the top of the main blocks in the statue bases. These lines continue upwards to meet the column axes at height 2.618, or 1+Phi, which locates the base of the second, broken, pediment.
The top of the second story capitals falls at height 2.427, i.e. 3/2 Phi, as the three green semicircles along the left margin of the graphic indicate.

The base of the figured frieze seems to fall at height 2.496, which is Phi (1.118), found by unfolding the big green arc from the corner of the big double-square in first story, whose corner is the outboard corner of the capital.
The middle of the urn above the tholos appears to be at height 3.236, which is 2 Phi, so that the whole composition up to that level is a big Golden Rectangle, of which two subquadrants are shown gold-shaded in the graphic.
Although the photogram source image doesn’t clearly show the horizontal lip cut into the cliff above the main façade structure, it is tempting to imagine that it is 1.118 times larger, so that the whole composition would be an even larger Golden Rectangle precisely framing the outer corners of the façade structure.
Cool, eh?
This is the so-called Treasury from Petra, Jordan. The source image comes from Judith McKenzie, The Architecture of Petra. British Academy Monographs in Archaeology 1. Oxford: Oxford University Press, 1990, plate 80.