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Curriculum Update

Development of new Mathematics and Further Mathematics AS and A levels for first teaching in 2017 continues. The report from the A level mathematics working group has just been published, focusing on areas of mathematical problem solving, modelling and the use of large data sets in statistics. It contains expert advice from group members on how these key aspects of the new content could be assessed and provides examples of questions. Ofqual’s open consultation on subject-specific rules and guidance has also just been launched, seeking views on:

the revised version of

the assessment objectives

the proposed approach

to regulating new AS and A level qualifications in mathematics and further mathematics;

the subject-specific

Conditions requirements and guidance Ofqual propose to introduce to implement that approach. Included as an appendix to the consultation document is the DfE’s proposed appendices to the subject content which contains of a list of notation for A level Mathematics and Further Mathematics and a list of formulae which must not be given in examinations. Responses should be submitted by 11th January

M4 is edited by Sue Owen, MEI’s Marketing Manager.

We’d love your feedback & suggestions!

The mathematics in art...or the art

in mathematics?

“Mathematics, rightly viewed,

possesses not only truth, but supreme

beauty.” (Bertrand Russell, Mysticism

and Logic, 1919)

In this issue we explore the maths in art

and the art in maths.

Joseph Malkevitch of York College (City

University of New York) wrote for the

American Mathematical Society in

April 2015, entitled ‘Mathematics and

Art’. Melkevitch writes: “There are, in

fact, many arts (music, dance, painting,

architecture, sculpture, etc.) and there

is a surprisingly rich association

between mathematics and each of the

arts.”

It appears that rather than being two

separate disciplines, art and maths are

very much bound up together. As this

New York Times article Putting Art in

Steam explains, “being able to quickly

sketch to

communicate an idea

is an enormously

useful tool,” says

James Michael

Leake, director

of engineering

graphics at the

University of Illinois.

“To do engineering

you’ve got to be able

to visualize.”

There have been many recent

collaborations between artists and

mathematicians, and for several

reasons. Sophia Chen’s article: Get lost

in the internet’s mind-bending math-

inspired art (June 2015, Wired)

considers whether it’s the order in

mathematics that appeals to artists, or

“simply because math describes nature,

and nature is beautiful”. Chen looks at

five mathematically-inspired artists and

organisations that use both new

technologies such as 3-D printing and

traditional media such as textiles.

The mathematical artwork of Robert

Bosch is interesting, in particular TSP

Art and Simple Closed Curves,

illustrating Jordan's Jordan Curve

Theorem. “ Bosch can draw the Mona

Lisa with a single line. First he lays

down some dots on a grayscale version

of the image, and then he uses an

algorithm to connect the dots in a way

looks like the original.”

In this issue

Curriculum Update

This half term’s focus:

Mathematics and Art

Hugh’s Views: Guest writer Hugh

looks at Arches and Architecture

Site-seeing with... Tom Button

Teaching Resource: Maths and

Art

The first ever printed

version of the

icosidodecahedron,

by Leonardo da Vinci

as appeared in the

''Divina Proportione''

by Luca Pacioli 1509

Maths and art

Maths-Art Seminars at London

Knowledge Lab was a monthly series of

maths-art seminars held in central

London. The idea for these seminars

grew out of the London Knowledge

Lab’s work in hosting the annual

international Bridges

Conference in London in August

2006. Regretfully, the seminars are no

longer organised, but the site remains

online as an archive.

The University of Oxford’s Art and

Oxford Mathematics web page

outlines the connection between

mathematics and art at its Mathematical

Institute. Included is a podcast of

Marcus du Sautoy's talk on the

connection between mathematics and

art. The Mathematical Institute has a

large collection of

historical mathematical models,

designed and built over a hundred

years ago. “The aesthetic beauty of the

models should be enjoyable for anyone

with an interest in mathematics, art or

history, regardless of your level of

mathematical training.”

Dr Ron Knott, Visiting Fellow in the

Department of Mathematics at the

University of Surrey, has created a web

page Fibonacci Numbers and The

Golden Section in Art, Architecture

and Music, which has a wealth of

information, examples and links about

the Golden Section, including

Miscellaneous, Amusing and Odd

places to find Phi and the Fibonacci

Numbers.

The Virtual Maths Museum presents

artists who use mathematical ideas as

subject matter and/or inspiration: “In

recent years, the advent of computers

has made possible the development of

various forms of digital art that allow

artists and mathematicians to cooperate

in a highly synergistic fashion.”

For examples of

innovative and creative

work by mathematics

practitioners and artists

who are crossing

mathematics-arts

boundaries, there is an

array to be seen on the Mathematical

Art Galleries website, the online

home of the mathematical art exhibits

from the annual Bridges Conference

and Joint Mathematics Meetings. The

exhibition comprises 2D and 3D

mathematical art, ranging from

computer graphics to quilts to

geometrical sculptures. The

Mathematical Art Exhibits page links

to photos of artwork from upcoming and

past conferences.

MoSAIC is a collaborative effort

sponsored and funded by MSRI

(Mathematical Sciences Research

Institute) and administered by

the Bridges organisation. Together,

they are creating a series of

interdisciplinary mini conferences

and festivals on mathematical

connections in science, art, industry,

and culture, to be held in colleges and

universities around the United States

and abroad.

The Golden Ratio The February 2013 edition of Monthly Maths looked at the Golden Ratio in Nature, Art and Architecture, and activities include finding Phi by experimentation and calculation. To access the ‘Beauty is in the Eye of the Beholder’ PowerPoint teaching resource that accompanied this issue, visit the M4 Magazine web page, scroll down to Archived Monthly Maths at the bottom of the page and look for February 2013 in the menu.

See also this 4 minute BBC video presented by Carol Vorderman (originally on the One Show), hosted on TES Resources: The beauty of the golden ratio.

Beauty in Mathematics

according to their comprehension of the

equation, from 0 (no comprehension

whatsoever) to 3 (profound

understanding). The distribution of post-

scan ratings showed that there was a

highly significant positive correlation

between understanding and scan-time

beauty ratings.

The mathematical subjects were asked

four questions about emotional

responses to equations. All answered

affirmatively to the question: “Do you

derive pleasure, happiness or

satisfaction from a beautiful equation?”

By contrast, out of 12 non-mathematical

subjects (with experience of

mathematics up to GCSE level), the

majority (9) gave a negative response

to the same question. The researchers

supposed that non-mathematical

subjects who had rated any equations

as ‘Beautiful’ “did so on the basis of the

formal qualities of the equations—the

forms displayed, their symmetrical

distribution, etc”.

The formula most consistently rated as

beautiful (avg. rating of 0.8667), both

before and during the scans, was

Leonhard Euler's identity:

Most consistently rated as ugly (avg.

rating of −0.7333) was Srinivasa

Ramanujan's infinite series for 1/π:

Study author Semir Zeki writes,

“Relegating beauty to the study of art

and leaving it out of science is no

longer tenable.”

The experience of mathematical

beauty and its neural correlates is a

2014 original research article by Semir

Zeki, John Paul Romaya, Dionigi M. T.

Benincasa and Michael F. Atiyah, who

suggest that: “Art and mathematics are,

to most, at polar opposites; the former

has a more “sensible” source and is

accessible to many while the latter has

a high cognitive, intellectual, source and

is accessible to few. Yet both can

provoke the aesthetic emotion and

arouse an experience of beauty,

although neither all great art nor all

great mathematical formulations do so.”

Fifteen mathematicians were asked to

view a series of 60 mathematical

equations and rate each one on a scale

of −5 (ugliest) to +5 (most beautiful).

Then they scanned the subjects' brains

with functional MRI as they looked at

the equations again. The pre-scan

beauty ratings were used to assemble

the equations into three groups, one

containing 20 low-rated, another 20

medium-rated, and a third 20 high-rated

equations, individually for each subject.

These three allocations were used to

organise the sequence of equations

viewed during each of the four scanning

sessions so that each session

contained 5 low-rated, 5 medium-rated,

and 5 high-rated equations.

Each subject then re-rated the

equations during the scan as Ugly,

Neutral, or Beautiful. The frequency

distribution of pre-scan beauty ratings

for all 15 subjects was positively

skewed, indicating that more equations

were rated beautiful than ugly. After

scanning, subjects rated each equation

Mathematical art

The Virtual Math

Museum has a

gallery of samples

mathematical art,

including that by

Robert Bosch (see

page 1):

“Mathematics and the

graphic arts have had

important

relationships and

interactions from the

earliest of times, for

example through a

common interest in

concepts such as

symmetry and

perspective that play

an important role in

both areas. In recent

years, the advent of

computers has made

possible the

development of

various forms of

digital art that allow

artists and

mathematicians to

cooperate in a highly

synergistic fashion.

Our goal in this

gallery is to show how

beautiful

mathematical objects

can be, and also to

present artists who

use mathematical

ideas as subject

matter, inspiration, or

both.

1+eiπ=0

Classroom resources

Maths In

Art is a

new online

resource

providing a

programme of

activities that suggest

ways that you can

explore areas of

mathematics through

an arts-themed

project. The scheme

may be adapted for

Key Stage 3 students.

It’s free to use the

resources, by signing

up on the Maths In

Art website. You can

view a video about

the scheme on

YouTube.

In 2016

the

Science

Museum

will open

a new permanent

mathematics gallery

suitable for students

aged 12-16: You can

view a YouTube

video design

animation created by

Zaha Hadid

Architects of

designs for the

gallery.

This 2010

nrich resource

looks at connections

between maths and

National curriculum links: the activities

based on geometric Islamic patterns in

this booklet support learning about

shapes, space and measures. Students

at Key Stage 3 can study

transformational and symmetrical

patterns to produce tessellations.

The maths2art website promotes the

teaching of mathematics in ways that

are visually stimulating, for pupils of a

wide range of abilities in Key Stages 3,

4 and 5. “Using a project to teach

maths can be more meaningful for

pupils than teaching the curriculum

areas of shape and space, number and

algebra in distinct units of work. It can

allow them to explore their own ideas at

a pace which suits them whilst making

meaningful connections between

different areas of maths.” Projects

include:

Circles

Islamic Art

Tessellation

Fibonacci

Pythagoras

Celtic Knot

3-D Models

Random Art

Fractals

Included on the site are links to some excellent resources from TROL (Teaching Resources On Line)at Exeter University; these can be printed for use in the classroom.

Maths in the City provides a look at maths in context,

with a maths trail around London.

Plus Magazine’s Maths and art: the

whistlestop tour provides a brief look

at “some of the types of art with a

strong mathematical component, or

conversely where a mathematical

visualisation has an astonishing

beauty”.

Plus’s Teacher package: Maths and

art is one of a series of teacher

packages designed to give teachers

(and students) easy access to Plus

content on a particular subject area and

“provide an ideal resource for students

working on projects and teachers

wanting to offer their students a deeper

insight into the world of maths”.

The Maths and Art package includes:

Maths and the visual arts

Maths and design

Maths and music

Maths and film

Maths and theatre

Maths and writing

Try it yourself with NRICH

Maths and Islamic art & design

This teachers’ resource provides a

variety of information and activities that

teachers may like to use with their

students to explore the Islamic Middle

East collections at the V&A. It can be

used to support learning in Maths and

Art.

Hugh’s Views

Dr Hugh Hunt is a

Senior Lecturer in the

Department of

Engineering at

Cambridge University

and a Fellow of Trinity

College.

In his previous

column Hugh talked

about the different

types of arch:

parabola, catenary

and semicircle. In this

column he goes on to

look at how arches

have been used in

architecture and how

they have evolved.

Click to view the

Sept/Oct 2015 M4

Magazine

View Hugh’s

videos on

his YouTube channel:

spinfun

Follow Hugh on

Twitter:

@hughhunt

The M4 Editor found plenty of arches on a

recent trip to the USA. See the MEI Facebook page.

There’s something very pleasing about the shape of an arch. Classically an arch ought to be a semicircle, but over the centuries arches have evolved, especially in churches and cathedrals, into an amazing array of shapes.

Take St Paul’s

Cathedral in

London. Designed

by the unbeatable

engineering team

of Robert Hooke

and Christopher

Wren. From the

inside and the

outside you see

hemispherical

domes and if you

didn’t know it then

you’d never guess

the cleverness of

the construction

within.

Hooke had worked

out that a hanging

chain and a

perfect arch have

the same shape:

“as hangs the

flexible line, so but

inverted will stand

the rigid arch”. It

was this key

understanding that

powered the

design of Wren’s

amazing buildings.

Hooke thought

that the shape of

the perfect dome was the cubico-

parabolical conoid, i.e. the cubic y = ax3

rotated about the y axis. He was very

close!

The Wren Library in Trinity College

Cambridge has arches, but where are

they?

Amazingly, Wren hid them away in the

foundations – but upside down! He

recognized that the muddy reclaimed

land by the river Cam was too weak to

take the

weight of his

proposed

building so

he created a

raft made up

of inverted

arches.

Arches and architecture

By Bernard Gagnon.

Dome of Saint Paul's

Cathedral seen from

Tate Modern, London.

[CC BY 2.0 ], via

Wikimedia Commons.

By Samuel Wale and

John Gwynn. Engraving

of a cross section of the

dome of St. Paul's

Cathedral in London via

Wikimedia Commons.

By Dp76764.

Dome of St Paul’s, via

Wikimedia Commons.

By Andrew Dunn. The Wren Library,

Cambridge. [CC BY 2.0 ], via Wikimedia

Commons.

By Lwphillips. Shape of hanging chain versus

arch. [CC BY-SA 3.0], via Wikimedia

McGraw-Hill Dictionary of

Architecture and Construction.

S.v. "inverted arch."

Retrieved October 1 2015

from http://

encyclopedia2.thefreedictionary.

com/inverted+arch

mathematical progression. And if you

wondered what the pointy pinnacles outside

on the top of the roof are for – no, not just for

decoration – they are the weights necessary

to hold the uppermost stones in place against

the huge sideways forces generated by the

vaulting. It’s like putting your foot against a

ladder to stop it from sliding. They’re

decorated to make them look pretty.

Am I allowed another Cambridge arch?

It is the so-called

Mathematical Bridge

in Queens’ College.

So many stories are

told about the bridge

– they’re all wrong!

What is true is that the main lower arch is

made up of seven straight lines. Yet it looks

like a smooth curve. I think that’s what

Newton had in mind when he invented the

calculus (or was it

Leibnitz?!) that if you

break a curve down into

little bits of straight lines

then it becomes very

simple. This bridge is what mathematicians

call an ‘envelope’ – straight lines making a

curve. So simple, so beautiful.

My jaw just dropped

when I first saw the

Millennium Bridge across

the Thames. I imagine

Wren and Hooke are

looking down from their

lofty dome. They see

the cables of the bridge.

“I told you so” Hooke

mutters. “Ut pendet continuum flexile, sic

stabit contiguum rigidum inversum, as hangs

the flexible line, so but inverted will stand the

rigid arch”.

The bookcases inside the building

bear down directly on the pillars of

the arches as you can see in the

diagram opposite. It’s funny that

all this engineering beauty is

hidden away. I suppose nothing

has changed. The amazing

engineering under the bonnets of

our cars or in our mobile phones is

totally hidden from view. Just

imagine how much more

interesting airports would be if

there were windows into the

baggage handling area, or if there

were working models of turbofan

engines!

Perhaps the highest evolved form

of the arch is in the fan vaulting at

King’s College Chapel, again in

Cambridge (sorry for all these

Cambridge references – but I ride

my bike past these amazing

buildings twice a day and never

cease to marvel at them).

From the outside the King’s Chapel

looks kind-of square and maybe a

bit dull. But those big exterior

pillars are part of the engine room

of the fan vaulting inside. The thin

spidery stone filaments inside (see

photograph, left) are like lines on a

graph, illustrating the lines of

thrust. The forces are channelled

into the pillars in an orderly

Hugh’s Views

By Jacques Heyman.

Hooke's Cubico-

Parabolical Conoid.

Notes and Records of

the Royal Society of

London. Vol. 52, No. 1

(Jan., 1998), pp. 39-50

Published by: The Royal

Society. Stable URL:

http://www.jstor.org/

stable/532075

By Lofty. Ceiling of

King's college,

Cambridge.

[CC BY-SA 3.0], via

Wikimedia Commons.

By Dmitry Tonkonog.

King's College Chapel. [CC BY-SA

3.0], via Wikimedia Commons.

By Alexandre Buisse.

St Pauls Cathedral and

Millennium Bridge. [CC

BY-SA 3.0], via

Wikimedia Commons.

Buildings and bridges

Via Wikimedia Commons.

Site seeing with… Tom Button

Tom Button is the FMSP Student Support Leader and MEI’s Learning Technology Specialist. Prior to this he taught mathematics in a number of different sixth form colleges.

He has a strong interest in the use of technology in maths, especially at A level, and has delivered many professional development courses on this. He is the chair of MEI’s GeoGebra Institute and runs the MEI/Casio Teacher

Network.

Tom has also recently developed MEI’s new technology-based A level unit: Further Pure with

Technology.

Follow Tom’s blog: Digital technologies for learning

mathematics

You can also follow Tom on Twitter at

@tombutton

The MEI Maths

Item of the

Month is a

monthly problem

aimed at teachers

and students of

GCSE/A level

Mathematics.

Each month a

mathematical problem is added to the

home page of the MEI website. The

MEI staff are all mathematics

enthusiasts and putting an interesting

problem on the front of the site is a

technological way of wearing our

mathematical hearts on our sleeves.

The first Maths Item of the Month

appeared in September 2006 and there

have been over 100 items since then.

A full archive of the problems is

available on the site and they can be

used for enrichment, problem solving or

as a way to encourage mathematical

thinking/proof.

A curriculum mapping for the problems

has recently been completed and this

can be seen at: mei.org.uk/miotm. This

is mapping is not intended to be

comprehensive – for example many of

the algebra or geometry problems can

be used with GCSE or A level students.

There are also a number of problems

that were hard to categorise and form a

fairly lengthy set of miscellaneous

problems at the end!

One of my favourites is one of the

earliest ones from December 2006:

“19 not out” – Some positive numbers

add up to 19. What is the maximum

product?

There is also usually a Christmas-

themed problem for December.

December 2014’s was:

“A Christmas Star” – An

eight pointed Christmas

star is made with a gold

layer and a silver layer.

What fraction of the gold layer is

covered by the silver layer?

Another resource

that has had a

significant impact on

me is Improving

Learning in

Mathematics (often

known as the

Standards Unit box).

The full set of materials in available, for

free, in the National STEM Centre e-

library.

The materials form a definitive guide for

using active learning approaches with A

level or GCSE students. There is a

range of activities including open

questioning, card sorts, group work,

learners creating their own questions

and many others. All of these are

presented in the context of lesson plans

so that teachers can see how to use

these strategies effectively to improve

students’ understanding.

There is also a set of professional

development materials that can be used

by a Mathematics department to

develop teachers’ skills across a

number of areas: learning from

mistakes and misconceptions, looking

at learning activities, managing

discussion, developing questioning and

using formative assessment.

Maths, Religion and ArtIt sometimes surprises people that there exist strong links between Mathematics, Religion and Art.Look at the images on the next few slides and describe the Mathematics you see.

Maths and ArtUsing some of these designs as inspiration, during this activity you will construct some of your own.

The ones here are nowhere near as complex or beautiful as the ones shown, but can be used as a basis for something more intricate.

Maths and ArtYou will need a pair of compasses, a ruler and a pencil (or a Dynamic Geometry Package) and will need to be able to:• Draw a circle and divide it equally into six• Bisect a line (perpendicular bisector)• Bisect an angleThese skills are outlined on the next slides.

Divide a Circle Equally into Six

• Mark a point on the circumference

• Place the point of the compass on the mark and make a mark on the circumference

• Repeat until you have 6 marks

• Draw a circle and keep the compasses at the same radius throughout

Perpendicular Bisector• Open a pair of compasses to approximately

¾ of the length of the line • Place the point at one end of the line and

draw arcs above and below the line• Keep the compasses at the

same radius, place the point at the other end of the line and draw arcs above and below the line to cut the previous arcs.

• Join the 2 intersection points

Bisect an Angle• Open the compasses • Place the point at the vertex of the angle and

draw an arc to create points A and B• Put the point of the compasses

on A and draw an arc• Keep the same radius, put the

point of the compasses on B and draw an arc

• Draw a line from the vertex of the angle through the intersection of the arcs

B

A

Challenge 1

Challenge 1 Construction lines

Challenge 2

Challenge 2 Construction lines

Challenge 3

Challenge 3 Construction lines

Challenge 4

Challenge 4 Construction lines

Teacher notes: Maths and ArtThis edition looks at Maths and Art and encourages students to use precise geometric constructions to copy the given designs and/ or create their own.

Students can use pencil, straight edge (measuring is traditionally discouraged) and compasses to construct the designs or could use a Dynamic Geometry Software package.

With each of the designs, working out how it has been constructed and the geometric properties involved is the first step and may require quite a lot of discussion. Working individually with pencil and paper methods, but seated in small groups will encourage this. If using a DGS package, working with a partner should be encouraged.

For KS3 and KS4 students, use of pencil, straight edge and compass will reinforce some of the geometric constructions they should be familiar with, but A level students might enjoy these activities too.

Teacher notes: Maths and ArtOne way of extending the activity to make it more challenging, particularly for A level students, would be to ask them to use a graph plotting package to create some of the designs, which would require a good working knowledge of: equation of a circle, equations of straight lines and trigonometry.

Students may find it helpful to firstly create the design using pencil and paper methods or a DGS package. This will ensure they understand how it has been constructed before trying to create it using a graphing package.

Teacher notes: symbols

An opportunity for students to discuss something in pairs and then feed back to the class.

Teacher notes: Maths and ArtSlides 2 – 8When looking at the images, initially simply ask what students see, then probe their thinking by asking them to describe specific shapes, symmetry, and underlying structure.

Some designs are based on dividing a circle into 6 (and then 12 and then sometimes 24), whereas others are based on dividing a circle into 4 (and then 8 and then 16).

Often there is rotational symmetry.Sometimes there is reflectional symmetry in the structure, but one needs to look carefully at the colouring.

Ask students how they think the basic designs have been constructed. What mathematical or geometrical skills did the artist need?

Teacher notes: Maths and ArtSlides 9-13These slides ensure that students have the geometric skills needed to construct the designs.

Traditionally, a Geometer is only permitted a straight edge, pencil and pair of compasses to construct designs.

Teachers may wish to allow students to measure distances or angles, particularly if the students need practice at using a protractor or find using compasses difficult. Students will need to calculate the angle required in each case.

Teacher notes: Maths and ArtSlides 14-21These slides show 4 different designs for students to re-create.

Show students a design and ask how it has been created. They should then try to construct it for themselves.

If they cannot work out how to construct it there is a second slide for each which shows the construction lines. The teacher notes below describe the constructions in more detail.

Once students have constructed the design, they might like to make a more intricate version of it.

The 4 designs do not have to be completed in order, nor do students need to complete all of them. Slides 14, 16, 18 and 20 could be reproduced full size and groups permitted to choose which one(s) they work on.

Teacher notes: Maths and ArtChallenge 1: A Rangoli style pattern

• Draw a circle and divide into 6 equal sections

• Bisect one of the angles• Use this distance around the

circumference to divide the circle into 12

• Draw other circles using the centre point, these can be equally spaced, or not

• Use intersection points and lines to create a design in one section

• Reflect and repeat the design around the circle

Teacher notes: Maths and ArtChallenge 2: An Islamic style floor pattern• Draw a circle and divide into 16 equal sections. • Draw a diameter and then construct a

perpendicular bisector to obtain 4• Bisect one of the angles to obtain 8 and then

bisect again to obtain 16• Use this distance around the circumference to

divide the circle into 16

• Draw other circles using the centre point

• Use intersection points and lines to create a design in one section

• Reflect and repeat the design around the circle

Teacher notes: Maths and ArtChallenge 3: A Trefoil

• Draw a circle and divide into 6• Select 3 alternate points• Join each to the centre • Bisect the radius to obtain a

required centre point

• Draw the red circle to obtain the other centres• Draw the 3 circles

Teacher notes: Maths and ArtChallenge 4: Seven Circles (one solution)• The challenge with this construction is that

a radius needs to be divided into 3 equal sections.• The easiest way to achieve this – without measuring with a ruler – is

to draw a line, use compasses to mark off a short length, keep the radius the same, move to the new point , mark again and repeat.

• Using the red dots as the centre and radius, draw a circle

• Divide the circle into 6 sections• Draw concentric circles at the other marks• Use the intersection points shown as centres

for the other circles with the radius the same as the central circle

AcknowledgementsRangoli designs: https://www.flickr.com/photos/rejik/9758154621 and http://homemakeover.in/rangoli-designs-for-holi/

Islamic floor design: https://en.wikipedia.org/wiki/Islamic_architecture

Stained glass window: https://www.durhamworldheritagesite.com/architecture/cathedral/intro/stained-glass