Too much has been written about the magical cathedral of Chartres for me to introduce it here. It is a geometer's dream, and we will work with it more than once. Today we are focusing on one of its famous rose windows, referred to as the north transept rose window but also known by it subject matter: The Glorification of the Virgin.
Rose windows, which reached their apogee in Gothic cathedrals in France, can be found in various sizes in these structures, but the main ones were meant to evoke a vision of heaven, and bathe the worshipper in divine light. In the otherwise drab surroundings of the medieval city, there is no doubt that their beauty and luminous colours created an experience for the viewer that is unimaginable in our current, visually saturated culture.
The construction process below results in a slightly simplified version of this rose window, leaving out the smallest details. It requires working on a large sheet of paper (nothing less than A3, but larger is recommended) as the scale gets quite intricate even then!
Step 1
Begin with a circle divided into 12.
While we usually do this by dividing it into 6 first (see Working With 6 and 12), we have seen that constructing a square around a circle also results in 12 divisions of the circle (see Working With 4 and 8). This is the method chosen here, because it has the advantage of requiring no construction lines outside the circle, meaning we can draw the circle as large as possible on our sheet of paper. It will be clear later on why this is desirable.
Below, note how I also reduced the crossing arcs used to find the bisector, so that they'd fall inside the circle.
Step 2
Join every fifth point on the circle to form a dodecagram.
Step 3
Connect opposite points of the dodecagram so that each of the outer kites is bisected.
Step 4
We now need to nest a circle inside each of these kites. First I'll zoom in on a single one of them, and name some points for clarity.
Step 5
This is the method we learned in Working With Circles. Start by bisecting the angle ACB. The bisector cuts the segment AB at O, which will be the centre of the circle.
Step 6
With the point of the compass on A, opening AO, draw a partial arc outside the kite. Repeat with the point on D, opening DO. The two arcs intersect at E.
Step 7
Connect EO to find point F on the side of the kite.
Step 8
Draw the circle centered on O, opening OF.
We now have the first of our 12 circles.
Step 9
There's no need to repeat these steps for the 11 remaining circles! We have the two measurements we need for this set of outer circles (remember this denomination because we will refer back to them): the position of the centre O on the kite's bisector, and the diameter.
Transferring the position of O to the rest of the set is done by placing the compass point on the centre of the original circle (we'll refer to it as the original centre), and drawing the circle that passes through O. This cuts the bisectors of all the kites in their respective O's.
Then, returning the compass to the radius of the outer circle, draw all the circles. Having more than one compass can be handy in such constructions, because some measurements are used repeatedly, and it's always more accurate not to have to change the compass opening back and forth.
Step 10
Connect the outer point of the outer circles (it is on the bisector) to draw another dodecagram. It is exactly the same as the first one, only smaller.
Step 11
Now, here is why we worked with a circle as large as possible: only the central part actually makes up the rose window. To define the window proper, draw the triangle shown here...
... then the circle inscribed in it—in ink, as it is a final line. Everything outside this circle is extraneous to the final design!
From now on, for clarity, the diagrams will show the close-up version below, up to the outer circles.
Step 12
The combined lines of the two dodecagrams create squares. It is good to ink them now (or at least darken them) as it helps make the array of lines less confusing. They may not appear to be perfect squares, but that's an optical illusion due to all the criss-crossing lines. They are in fact each created by two pairs of perfect parallels, and cannot be anything other than perfect!
Step 13
Also ink the parts of the outer circles that are inside the defining circle.
Step 14
For what comes next, which is nesting a smaller circle into these partial circles, we'll work with a close-up again:
Step 15
Bisect the median line in this shape:
Step 16
Draw the inscribed circle.
Step 17
As we have done before, use an overall circle to mark all the centres, and draw the full set of small circles.
Step 18
Let us now zoom in on the central area of the window.
Step 19
With the compass still set to draw small circles (step 17), draw a circle in theoriginal centre.
Step 20
Move the compass to its "west" point and draw an arc to cut the first circle in two places.
Step 21
Join these two points to define the centre of a new circle, and draw that.
Step 22
Now move the compass to its "east" point and draw another circle.
Step 23
Finally, draw the circle encompassing this pair. All this was to create a circle with a diameter of 1.5 times the diameter of the small circle.
Step 24
From now on, for clarity, I am showing new lines in blue. Draw the following lines, which join a series of points so they pass between the squares.
Step 25
Keeping the same compass measurement used to draw the central circle in step 23, place the point at its north pole and make a mark on the vertical line.
Step 26
Return the point to the original centre, and use this mark to draw the circle below.
Step 27
Set the compass to the same measurement as the small circles, and draw 12 circles centered on the intersections shown here. Ink the circles.
Step 28
From the original centre, pull two lines touching a circle on either side.
Step 29
Using the circle drawn in step 26, place the compass point and draw a circle tangent to the two lines. (If things are getting very small by now, a circular template may be useful, but with a good compass it's possible to draw very tiny circles.)
Step 30
Ink the shape formed by the two lines and part of the circle.
Step 31
Repeat all around.
Step 32
Draw the circle that connects the outer point of the squares.
Step 33
Draw a circle as shown. Its centre is the intersection of this circle with a diameter of the window circle, and its radius is up to the point where two lines of the original dodecagram cross.
Step 34
Move the compass point to each of its cardinal points in turn to draw four more circles.
Step 35
Ink the outline of this quatrefoil shape.
Step 36
Repeat all around.
Step 37
The finished window, here inked with different line thicknesses to create a visually pleasing hierarchy of shapes.
Colour to your liking... Each of these shapes can also be filled in with arabesques or figures to echo the feel of the original window.
Rose windows range from fairly simple to incredibly complex, and an entire course could be dedicated to them. Next month we are traveling east again, to a land of most intricate knotwork...
