Rose Windows

Contextualised task 38 Rose Windows

Teaching notes This task explores the mathematics behind the design of rose windows: large, intricate, and often extravagant, circular windows in churches and cathedrals. Students will apply straight edge and compass techniques. A large supply of plain A3 paper and good compasses will be required.

Task A: The Basilica of San Zeno Outline Students explore how to construct a basic rose window design using straight edge and compass techniques.

You will need:  Teachers’ script  PowerPoint  Question sheet  Mark scheme  Plain A3 paper

Task B: Chartres Cathedral Outline Students develop the ideas in Task A to construct a more complex design. In reality, this is only the central section of a much larger rose window at Chartres Cathedral.

You will need:  PowerPoint  Question sheet  Mark scheme  Plain A3 paper

Task C: Church of St Etienne, Beauvais Outline Students consider an alternative way to create a twelve-pointed star. With less guidance than the first two tasks, they use it to construct the most complex design here.

You will need:  Teachers’ script  PowerPoint  Question sheet  Mark scheme  Plain A3 paper

Task A: Teachers’ script for PowerPoint presentation

The text in the right-hand boxes provides a possible script to be read to students. However, it is probably preferable to use your own words and elaboration. When questions are asked, time for discussion in pairs / groups should be provided. Ensure that students are given opportunity to explain their reasoning in response to these questions. All students need to understand the concepts in order to make progress with the task.

Slide 1 Keep this slide on the screen until you are ready to start the presentation Rose windows

Slide 2 This is the Basilica of San Zeno, in Verona, Italy. It is considered to be the place of the marriage of Romeo and Juliet in Shakespeare’s play.

It has a Rose window above the door.

Many European cathedrals built between the 12th and 16th centuries have rose windows. It was part of the Gothic architectural style.

Slide 3 From inside the building, a rose window with stained glass is spectacular. This example is from a church in Australia.

Both the windows you have seen here have a shape with rotational symmetry of order 12. The design is based on a regular dodecagon.

In this task you will use mathematical techniques to construct two basic designs of rose window.

Slide 4 There are several ways to construct the regular dodecagon. One way uses a perpendicular bisector construction (see here: https://en.wikipedia.org/wiki/Dodecagon#Dodecagon_construction) Does anyone know how to construct a regular hexagon? (Construct a circle. Using the same radius, span six arcs around the circumference. The points of intersection between each arc and the circle is a vertex of the hexagon) Advance six clicks to demonstrate arcs Three intersections can be joined to make an equilateral triangle. Advance one click And so can the other three. Advance one click This makes a ‘Star of David’. Now we need to find the point on the circumference of the circle that are exactly half way between the vertices of this star. How could we do this? Students might suggest an angle bisector construction. Advance three clicks to show one way to do this. We now have 12 equally spaced points around the circle. Advance one click These can be joined to make a regular dodecagon These 12 points are also needed for many rose window constructions. The constructions lines to find them should be very faint as they are, mostly, not part of the final design. Slide 5 The 12 points can be joined in different ways. Advance 12 clicks to show initial construction star This arrangement can be used to create the design of the Basilica of San Zeno. Advance one click to reveal centre circle This centre circle has a radius such that it intersects the vertices of an inner hexagon. Advance two clicks to reveal lines Each ‘petal’ of the rose window needs two other construction lines as shown here. Again they are based on existing points of intersection – the ‘internal vertices’ of the star. Advance one click to reveal next circle This circle is centred on a vertex of the star. The two (blue) lines are tangents to the circle. Note that the blue lines are only thicker for emphasis. They are still construction lines and should be faint. These lines can now be used to trace each individual window. Advance four clicks to reveal arcs and line segments.

Slide 6 This window would now be repeated twelve times. The rest of the design is built on the same original construction lines.

Slide 7 This slide can be used to support task B

Slide 8 This slide can be used to support task C

Slide 9 This slide can be used to support task C

Slide This slide can be used to support task C 10

Rose Windows – Basilica San Zeno

Task A: Question

Create a copy of this diagram.

Repeat the process to create all twelve windows of the complete rose window.

Remember

This diagram will help you find the equally spaced points around the circle Task A: Mark scheme

The information below is intended as a guide only

Partial credit Creates at least one of the twelve sections accurately

Limited credit Demonstrates at least three of the following  Creates the twelve pointed star accurately  Identifies how to construct an angle bisector  Uses compasses with consistent accuracy  Identifies the centres of circles that need to be drawn  Appreciates the use of tangents to set the radius of circles  Can explain some of the features of the construction using accurate mathematical language

No credit Any other response.

Rose Windows – Chartres Cathedral

Task B: Question

Chartres Cathedral in France is a World Heritage Site. The building contains three rose windows.

The inner section of the west window is a similar design to the example at the Basilica of San Zeno. It uses some additional construction lines.

The diagram below shows how one of the twelve separate windows can be constructed.

Circle P has a radius 3/5 of the radius of the large circle.

Circle Q is centred on the intersection of circle P and the line AB. Two lines of the twelve-pointed star are tangents to circle Q.

Create a copy of this diagram. Repeat the process to create all twelve windows of the complete rose window.

B Task B: Mark scheme

The information below is intended as a guide only

Full credit Creates an accurate diagram of the rose window that demonstrates the following:  Consistent accurate use of compasses  Faint, but sharp and accurate, construction lines  Twelve equally spaced points on the circumference of the circle  Small circle in the centre  Circle P has a radius 3/5 of the radius of the large circle  Twelve-pointed star created by joining the twelve points correctly  Joins opposite points on the circumference of the circle with straight lines (e.g. AB)  Creates the angle bisector of these lines (e.g. XY)  Circle Q centred on point of intersection of AB with lines of star as tangents  Circle R centred on point on circumference with angle bisectors as tangents

X Q

Partial credit Creates at least one of the twelve sections accurately

No credit Any other response. Rose Windows – St Etienne, Beauvais

Task C: Question

The rose window in the church of St Etienne, Beauvais is more complex. The twelve points on the initial circle are joined differently

The diagram below shows how one of the twelve separate windows can be constructed.

Create a copy of this diagram.

Repeat the process to create all twelve windows of the complete rose window.

Task C: Marks cheme

The information below is intended as a guide only

Full credit Creates an accurate diagram of the rose window that demonstrates the following:  Consistent accurate use of compasses  Faint, but sharp and accurate, construction lines  Twelve equally spaced points on the circumference of the circle  Circle A in the centre  Four equilateral triangles created by joining the twelve points  Joins opposite points on the circumference of the circle with straight lines (e.g. PQ and RS)  Creates the angle bisector of these lines (e.g. XY)  Circle B centred on point of intersection of two equilateral triangles with lines PQ and RS as tangents  Circle C centred on point of intersection of circle B and angle bisector XY with line of equilateral triangles as tangents  Circles D and E centred on points of intersection of circles B and C with angle bisector XY as tangent X P

R D E B A

Q Partial credit Creates at least one of the twelve sections accurately

No credit Any other response. Progression in reasoning Identify processes and connections  prioritise and organise Identify a next step towards Identify some steps towards Identify a sequence of steps the relevant steps achieving a solution. achieving a solution, not towards achieving a solution. needed to complete the e.g. follows the instructions in necessarily in sequence Identify priorities – the tasks task or reach a solution task 1 and creates a rose e.g. can establish the that are essential, and those window design techniques that have been tasks which may embellish used to create one of the the solution if there is time. designs e.g. creates concentric arcs and parallel lines to demonstrate how the mathematical construction is used to create the actual stonework boundary

Represent and communicate  explain results and Explanations are clear – both A wider range of appropriate Orally and in writing: use procedures precisely orally and in writing, using mathematical vocabulary is mathematical vocabulary using appropriate some mathematical used in explanations. precisely, e.g. following the mathematical language vocabulary. If written as Arguments are supported conventions for labelling in instructions, they will lead to with evidence. geometrical proofs, the intended correct result. e.g. uses correct vocabulary appropriate notation in e.g. can explain the processes consistently when explaining algebra. that they are applying the processes that they are e.g. can produce accurate applying written instructions for creating the third rose window

Review  select and apply e.g. ensures compasses are not e.g. take action to remedy e.g. checks every centre and appropriate checking loose before drawing circles situation if any circles set to radius combination at every strategies meet with one potential point of contact before tangent overlap another starting to draw a circle

GCSE Content GCSE Mathematics – Numeracy and GCSE Mathematics GCSE Mathematics only Understanding and using properties of shape  The geometrical terms: point, line, parallel, right angle,  Use of ruler and pair of compasses to do perpendicular, horizontal, vertical, acute, obtuse and reflex angles. constructions.  Vocabulary of triangles, quadrilaterals and circles: isosceles,  Construction of triangles, quadrilaterals and equilateral, scalene, exterior/interior angle, polygon, hexagon, circles. radius, diameter, tangent, circumference, and arc.  Constructing angles of 60, 30, 90 and 45.  Accurate use of ruler, pair of compasses and protractor. (Lengths  Order of rotational symmetry. accurate to 2mm and angles accurate to 2.)  Bisecting a given line, bisecting a given angle. Constructing the perpendicular from a point to a line.  Constructing 2-D shapes from given information. Understanding and using properties of position, movement and transformation  Constructing the locus of a point which moves such that it satisfies certain conditions, for example, (i) a given distance from a fixed point or line, (ii) equidistant from two fixed points or lines.  Solving problems involving intersecting loci in two dimensions

Key Foundation tier content is in standard text. Intermediate tier content which is in addition to foundation tier content is in underlined text. Higher tier content which is in addition to intermediate tier content is in bold text.