The Geometry of the Thirteenth Century Façade

(Figs. 61-75)

The most obvious geometrical forms presented by the façade are the three circular windows. The central window is octagonal in design, while the north and south windows are hexagonal in design. This arrangement brought to mind the geometry discovered by this author in the great Hagia Sophia building in Istanbul (Fig. 61). There the three circles would seem to symbolise the Holy Trinity, This may be the case here, or it may be that the octagon here represents the Risen Christ, whereas the hexagons may represent the human beings created on the sixth day, male and female, or possibly the twelve apostles.

The second geometrical element of importance is the arrangement of the three triangles of the gables, which, enclosing the three circles of the windows, spread themselves downwards, suggesting the enlargement of the three circles. Following this idea we arrive at three lower and larger circles overlapping each other by half their diameter (Fig. 63), exactly as in the Hagia Sophia plan.  Eventually we will arrive at a clear confirmation of this arrangement in certain hitherto unexplained elements in the fabric, but this was not appreciated until the analysis of the geometry had been completed. 

In accordance with the rule that any sacred geometrical construction must begin with a circle, we create a circle of the width of the complete façade, 156 English feet (47.54m.) in diameter. Various centres for this circle can be tried, but the only centre that leads forward is the niche at the top of the Perpendicular porch c.1380, which at first seemed surprising, but, as will be seen in the consideration of the porch geometry, is exactly as it should be, for it relates to the fact that the geometry of the c.1380 porch will be found to derive from the geometry of the whole of the c.1230 façade (See the slide show 14: The Geometry of the Perpendicular Porch Elevation).

 When this large circle is developed through inscribed squares (Fig. 62), the second square provides the line of the pavement and the underside of the string-course 78 feet above. It also provides the central vertical to each of the three portals. The third inscribed square provides the centres for three overlapping circles, each of 78 feet (23.77m.) diameter, and all tangent to the string-course line, the pavement level, and the primary circle.

It then appears that the diameters of these three circles are almost exactly three times the width of the north and south portal openings, i.e. 26 feet.  But the division each of the diameters of these circles into three equal parts must not be done arithmetically, for the medieval geometers had no measuring rules.  The division must be done geometrically using only unmarked rule and compasses, and the means by which this is achieved are shown in Fig. 64.  Centred on point A a circle is drawn passing through the intersections of the three overlapping circles. Where it crosses the centre vertical at D is the centre of a smaller circle that if drawn tangent to the second square within the primary circle has a diameter one third that of the larger circles. From here it is a simple matter to develop the two small circles below by swinging the diameter down the centre line with the compass/dividers. 

The same procedure applied from points B and C divides the north and south circles into three small circles, and all of these are 26 feet in diameter.

The rows of 26-ft circles are then extended  above the ‘string-course’ line, to form three circles above that line, and here one encounters one of the incalculable fortuities that occur in sacramental geometry, for the north and south circles of this upper row pass exactly tangent to the primary 156 foot circle.

In Fig. 65 the hexagrams within the large 78-ft circles north and south are completed, and also the octagram within the central large circle, plus another octagram and a square developed within.

Although the 26-ft circles fit the width of the north and south portal openings, they do not immediately provide the 18.4 foot width of the central portal opening. But 26 is 18.4 times the square root of two, 1.414.  If we therefore inscribe the square within the circle we arrive at the width of the central opening (Fig. 66).    

As prescribed in the chapter concerning the techniques and rules of sacramental geometry every square should be turned to create an octagram.

Also in Fig. 66 the square circumscribing the primary 156ft. circle is constructed.

The octagram constructed within the first octagram in the central gable circle produces a circle centred on the circular window of the gable and of similar though not identical diameter Fig. 67.  A simple calculation shows that the circle around the window in the geometry must be 13 feet in diameter, but the diameter of the actual window is 12 feet. However the geometry is correct and the development of the 12-foot window from the 13-foot geometry is fully presented in the following chapter.

The circles of the north and south portals and gables are similarly inscribed with their hexagrams in accordance with the design of their windows, and here again they create 13-foot diameter circles centred on their windows. The development of the design of the windows from these circles is also presented in the next chapter.

The six verticals tangent to the nine small circles need to be drawn, as also the two horizontals tangent to those same circles (Fig. 66).  At this time the horizontal diameters of all the 26-foot diameter circles are drawn. It will be seen (Fig. 73) that three of these horizontals provide the lines of the junctions of the shafts of the thirty-six detached colonettes enhancing the three portals, and marked by the ring-shafts holding them in place.


The geometry of the arches

The geometry of the central arch is displayed in Fig.68. A circle is drawn circumscribing the 26ft. square immediately below the string-course. From the square inscribed within the smaller circle the base line is extended across the whole design. This gives the level of the terminations of the detached shafts with their capitals, and also the springing of the arches. The vertical sides of the next smaller square are extended down to meet the base-line of the larger square at the two points C and D.  These two points become the centres for the drawing of the arch, the inner arcs springing from the lower corners X and Y of the 18.4ft square inscribed within the 26ft circle. They thus create the 18.4ft opening of the central arch.  The outer arcs spring from the intersections of the larger (36.7ft) circle with the springing level, Z and E, and create the outer width of the central arch of 32 feet.

The geometry of the north and south arches is shown in Fig. 69. It is developed from an intriguing combination of hexagon and square both inscribed within the 26ft circle. The baseline of the square relates to that of the central arch that defines the springing level of all three arches.

The intersections of the apex of the inverted triangle with the base of the square, F and G provide the radial centres for the arcs.

The outer arcs swing from the apex P of the upright triangle down to the springing line, N and O, creating a width of 40 feet.

The inner arcs run from the intersections of the smaller square with the verticals of the 26ft square up to the centre vertical. This gives an inner width of 26 feet.

The strictly geometrical design found here in these arches negates the theory of numerical proportions in arch design that has been advanced hitherto.


The geometry of the façade towers (Figs. 70, 71, 72).

     The full width of the north tower between the centres of the attached shafts is 16 feet 3inches, whereas that of the south tower is 8 inches less at 15 feet 7 inches. The narrowing of the southern tower is inexplicable, but could have resulted from a need to widen the passage between the tower and buildings of the Great Court, X in Fig. 42. A document of 1245 concerning the Abbey of Winchcombe quoted by Salzman (1952, p. 383) may have relevance:

   15 December. Inquiry whether it would to the injury of the town or of the Abbey of Winchcombe, or of anyone else, if the King should allow Master Henry, rector of the church of St.Peter of the same town, to lengthen the church 12 ft. eastwards,and to enlarge an aisle begun on the south side of the church to the length of 30 ft. and the breadth of 12ft. [The jury] say that if the chancel were lengthened 12 ft eastwards it would be to the damage of the Abbot and Abbey, because the Abbot could not have a free way in and out for his carts and for carrying his timber.


      In view of the narrowness of the passage at Peterborough between the south tower and the Bishop’s wall, it may be judged more likely that the south tower was built narrower for reasons similar to those at Winchcombe, than that the north tower was built wider for no possible reason.  Therefore the north tower was considered to have the more geometrically correct dimensions, and was selected for geometrical analysis.   One could also surmise that the towers were not very holy places and that exact geometrical correctness would not be demanded of them.  The full width of the north tower, excluding the attached shaft and plinth mouldings projecting on its north side, is 18 feet 9 inches. The width of the wall between the attached shafts on the west face is 13 feet 3 inches. The ratio of 18.75 to 13.25 is 1.415, almost exactly the square root of two. The design of this face of the tower must therefore be based on the square within the square.

    It will be further observed that the arrangement of blind arcading and colonnettes decorating the tower from top to bottom divides the width of the wall into four equal parts. Both the hexagram and the octagram can divide a diameter vertically into four, but only the octagram additionally provide the necessary horizontals. Thus we arrive at an octagram within the 13.25 foot square within an 18.75 foot circle.

     The vertical space between the verticals of this square and the limit of the north portal, 18 feet 9 inches, is then divided in half, using as starting levels the three horizontals tangent to the twelve small circles:

     Three circles are drawn tangent to the verticals, and within each circle two

octagrams and a square are progressively constructed.

    It will then be observed that linking these constructions squares have been created that are exactly similar to those within the circles.  When these squares are developed in the same way (Fig. 71), they are found to be centred on the horizontals through the diameters of the four rows of circles in the main façade.

   A final similar construction is made centred on the horizontal tangent to the tops of the gable circles.

     The geometry of the north tower in relation to the fabric is shown in Fig.72.

     Now the centre verticals of the large north and south circles are extended to meet the primary circle at X and Y (Fig. 71).  A horizontal is drawn between X and Y crossing the central vertical at Z.  Straight lines are then drawn from X, Y, Z through the intersections of the 78 foot circles and extended to their horizontal diameters.   These lines provide the outlines of the gables.

      Finally a circle, dotted in Fig. 73, is drawn centred on the string-course, radius to the pavement, and a square drawn around it.  This gives the height of the tower spires.

     The complete  geometry in relation to the fabric of the façade is shown in Fig. 73.


The geometry of the gable windows

The geometries of the three gable windows as they are developed from the general geometry of the whole façade is provided in Figs. 76-112.

    The diameter of the circle in the general geometry surrounding each window is 13 feet, being that of the second square within a 26 foot circle.  But the diameter of the actual window is 12 feet. The geometrical means by which the difference between the two dimensions is resolved will be demonstrated in the later analysis.

The geometry of the spires

      Protractors for the measurement of angles in degrees did not exist in the Middle Ages.  Angles other than right-angles were defined numerically as gradients, as they are today on road and rail, and an example is demonstrated in Villard de Honnecourt’s notebooks.   The gradient of the large spires upon the staircase towers at Peterborough is one in eight.   The gradient of the small spites on the turrets is one in seven. The fabric of the spires corresponds exactly with these gradients, measuring base width to height.

    At the same time all the spires are geometrically developed from the larger geometry of the facade, and their geometry originates from nodal points on the line of the string-course.

      It will be observed that the angles of the gable profiles do not correspond exactly with the angles of the geometry.   The geometry gives angles of 60 degrees, while the fabric has angles of 58 degrees.  The explanation of this anomaly is that to produce an angle of 60 degrees one requires, in whole numbers, a gradient of 64 in 111, or 2 to 3.368, which was beyond the limits of medieval expertise.  But a simple gradient of 2 to 3, well within the grasp of any mason, produces an angle of 58 degrees, and that is what they used.  One needs to appreciate that neither the stone masons nor the master mason had any means of measuring or checking the angle.


The verification of the analysis

    It will be observed that the north and south geometries arrive at the height of the stringcourse. But the geometry of the central arch achieves a height  a foot below the string-course.  In Fig. 44, it will be seen that this unexpected difference in height is in fact presented in the fabric, where there is a gap of one foot between the string-course and the upper limit of the central arch. This circumstance proves the veracity of the geometrical analysis.

A second powerful proof of the truth of the analysis is provided by the four false gargoyles. The function of these apparently unique objects has never been fully explained although Stallard comes very close.  They are larger than the usual true water-spout gargoyles, and they are not waterspouts. Each head is surmounted by a form of moulded abacus that provides a flat surface about 21 inches below the string-course and seemingly unrelated to any other feature architectural feature. They appear in fact to be sited independently of the geometry. And each gargoyle has a lengthy appendage pointing downwards to nothing in particular (Fig. 74).

But when the geometrical analysis is presented superimposed upon the fabric (Fig. 75), it is seen that each of the gargoyles stands over an apex of a hexagon inscribed within the north or south 78ft circles. In the case of the two central gargoyles these apexes are also the intersections of the three 78ft circles. The appendages are now seen to be indicating the exact points of intersection.

The flat tops of the gargoyles are then seen to provide a meeting place for the linear extensions of the gable profiles.

Thus we realise that in the false gargoyles the architect of the c.1220 façade provided indications in the fabric that would enable future architects to deduce his geometry and thus be able to ensure that any alterations or additions that they might make would conform to his existing geometry.

We shall find other deliberate indications of the underlying geometries in other parts of the cathedral. These clues are designed to be recognisable and interpretable only by those practised in the art of sacramental geometry.