Tracery

In a small town in the Bergisches Land stands one of the most important Gothic church buildings in Germany.

Figure 1: Altenberg Cathedral

The construction of the Altenberg Cathedral began in 1259 and was completed around 1400. The famous west window of the church is considered the largest Gothic tracery window “north of the Alps”. Designed by an anonymous artist, called the “Master of the Berwordt Retable”, the stained glass of the window reflects the medieval idea of the “Holy Jerusalem”, as a place of Christian end-time expectations. Altenberg Cathedral, built as a monastery church, served as a burial place for the dukes and counts of Berg and Jülich-Berg until 1511, and today is the common parish church for both the Protestant and Catholic communities of the town.

A tracery or tracery window — as shown by the exhibit (as a spatial puzzle) in MATHEMATICS ADVENTURE LAND (cf. Figures 2 and 3 below) — is understood to be a geometrically constructed ornamental form of the Gothic period.

Figure 2: Unsolved puzzle
Figure 3: Solved puzzle

The filigree architecture of the arches, points and circles made and joined by stonemasons in the past initially served to design large windows, especially on sacred buildings. Later, so-called blind tracery was also used to structure wall surfaces and gables, and openwork tracery was used for parapets. In the late Gothic period, tracery became increasingly complex and varied. The term “tracery” itself is derived from the term “measured work”. It is a form that can be geometrically constructed exclusively from circles, as shown in our exhibit. The tracery serves to subdivide the arch field (“couronnement”, see Figure 4) located above the “impost line” (the connecting line of the “imposts”, i.e. the load-bearing stones of an arch).


And now … the mathematics of it:

The geometry of the tracery is limited to the use of compasses and rulers. The individual constructions use the Pythagorean theorem (see also the exhibits [“Pythagoras”] and [“Proof without words: Pythagoras for laying”] and) and the 2nd ray theorem, as well as the Vieta root theorem for solving quadratic equations. Three of the basic constructions are shown below.

(I) Pointed arc with an incircle

In this construction, the incircle of radius r touches both arcs from the inside (each as parts of the arc of circles of radius R\gt r) and the base side of length R (cf. Figure 4).

Figure 4: Pointed arch with one incircle

Following the Pythagorean theorem we get (cf. Figure 4):

    \[r^2+\left(\frac{R}{2}\right)^2=(R-r)^2,\]

i.e. r^2+R^2/4=R^2-2Rr+r^2. Thus 2Rr=\frac{3}{4}R^2 or more precisely r=\frac{3}{8}R. So the radius of the incircle relates to the radius of the arcs as 3:8.

(II) Pointed arc with a small circle over a semicircle

In this construction the small circle with radius r touches from the inside both arcs, which are parts of circles with radius R\gt r, and from the outside a semicircle with radius R/2 (cf. Figure 5).

Figure 5: Pointed arch with a small circle over a semicircle

Following the Pythagorean theorem we get:

    \[\left(\frac{R}{2}\right)^2+\left(r+\frac{R}{2}\right)^2=(R-r)^2,\]

d.h. R^2/4+r^2+Rr+R^2/4=R^2-2Rr+r^2. From this follows: 3Rr=R^2/2 and thus r=R/6. The radii of the small circle and the semicircle below it are 1:6.

(III) Pointed arc with circle over two semicircles

In this construction, a circle of radius r from the inside touches both arcs, which are parts of circles of radius R\gt r, and — lying below them — two semicircles of radius R/2 from the outside (cf. Figure 6).

Figure 6: Pointed arch with circle over two semicircles

Following the Pythagorean theorem (in triangle CDE), R=\lvert\overline{AB}\rvert, r'=\lvert\overline{CD}\rvert=R/4 and h=\lvert\overline{CE}\rvert yields on the one hand the equation

    \[\left(r+\frac{R}{4}\right)^2-\left(\frac{R}{4}\right)^2=h^2\]

and on the other hand (in triangle ACE)

    \[(R-r)^2-\left(\frac{R}{2}\right)^2=h^2.\]

That is

    \[r^2+Rr/2+R^2/16-R^2/16=R^2-2Rr+r^2-R^2/4.\]

So 5Rr/2= 3R^2/4 and thus r=\frac{3}{10} R, i.e. the radii of the “resting” circle and the circles whose parts form the arcs behave as 3:10.


Literature

[1] Binding, G.: Maßwerk, Darmstadt, 1989.

[2] Helten, L.: Mittelalterliches Maßwerk. Entstehung — Syntax — Topologie, Berlin, 2006.

[3] Schnellbächer, I.: Das Altenberger Westfenster, seine Botschaft im Licht der Bibel, Mariawald, 2009.

[4] http://www.janschuster.net/kirchenfenster/