m e i . o r g . u k
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.
I s s u e
N o v / D e c
2 0 1 5
There have been many recent
collaborations between artists and
mathematicians, and for several
“Mathematics, rightly viewed,
reasons. Sophia Chen’s article: Get lost
possesses not only truth, but supreme
in the internet’s mind-bending mathbeauty.” (Bertrand Russell, Mysticism
inspired art (June 2015, Wired)
and Logic, 1919)
considers whether it’s the order in
mathematics that appeals to artists, or
In this issue we explore the maths in art
“simply because math describes nature,
and the art in maths.
and nature is beautiful”. Chen looks at
Joseph Malkevitch of York College (City five mathematically-inspired artists and
organisations that use both new
University of New York) wrote for the
technologies such as 3-D printing and
American Mathematical Society in
traditional media such as textiles.
April 2015, entitled ‘Mathematics and
Art’. Melkevitch writes: “There are, in
The mathematical artwork of Robert
fact, many arts (music, dance, painting,
Bosch is interesting, in particular TSP
architecture, sculpture, etc.) and there
Art and Simple Closed Curves,
is a surprisingly rich association
illustrating Jordan's Jordan Curve
between mathematics and each of the
Theorem. “ Bosch can draw the Mona
arts.”
Lisa with a single line. First he lays
down some dots on a grayscale version
It appears that rather than being two
separate disciplines, art and maths are of the image, and then he uses an
algorithm to connect the dots in a way
very much bound up together. As this
looks like the original.”
New York Times article Putting Art in
The mathematics in art...or the art
in mathematics?
Steam explains, “being able to quickly
sketch to
communicate an idea
is an enormously
Included as an appendix to
useful tool,” says
the consultation document
James Michael
is the DfE’s proposed
Leake, director
appendices to the subject
content which contains of a
of engineering
list of notation for A level
The first ever printed
graphics at the
Mathematics and Further
version of the
Mathematics and a list of
icosidodecahedron, University of Illinois.
formulae which must not
by Leonardo da Vinci “To do engineering
be given in examinations.
as appeared in the
you’ve got to be able
''Divina Proportione''
Responses should be
to visualize.”
th
submitted by 11 January
5 0
by Luca Pacioli 1509
Click here for the MEI
Maths Item of the Month
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
M4 is edited by Sue Owen, MEI’s Marketing Manager.
We’d love your feedback & suggestions!
Disclaimer: This magazine provides links to other Internet sites for the convenience of users. MEI is not responsible for the availability or content of these
external sites, nor does MEI endorse or guarantee the products, services, or information described or offered at these other Internet sites.
Maths and art
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.
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
To access the ‘Beauty
practitioners and artists
is in the Eye of the
Beholder’ PowerPoint
who are crossing
teaching resource
mathematics-arts
that accompanied this
boundaries, there is an
issue, visit
the M4 Magazine web array to be seen on the Mathematical
Art Galleries website, the online
page, scroll down to
home of the mathematical art exhibits
Archived Monthly
Maths at the bottom
from the annual Bridges Conference
of the page and look
and Joint Mathematics Meetings. The
for February 2013 in
exhibition comprises 2D and 3D
the menu.
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.
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.
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.
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.
Beauty in Mathematics
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.
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
according to their comprehension of the
equation, from 0 (no comprehension
whatsoever) to 3 (profound
understanding). The distribution of postscan 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:
1+eiπ=0
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.”
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
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”.
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 Maths and Art package includes:
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:

Maths and the visual arts

Circles

Maths and design

Islamic Art

Maths and music

Tessellation

Maths and film

Fibonacci

Maths and theatre

Pythagoras

Maths and writing

Celtic Knot

Try it yourself with NRICH

3-D Models

Random Art

Fractals
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”.
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.
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.
Hugh’s Views
Arches and architecture
Dr Hugh Hunt is a
Senior Lecturer in the
Department of
Engineering at
Cambridge University
and a Fellow of Trinity
College.
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
In his previous
London. Designed
column Hugh talked
by the unbeatable
about the different
engineering team
types of arch:
of Robert Hooke
parabola, catenary
and semicircle. In this and Christopher
column he goes on to Wren. From the
inside and the
look at how arches
outside you see
have been used in
architecture and how hemispherical
domes and if you
they have evolved.
didn’t know it then
you’d never guess
the cleverness of
the construction
within.
Click to view the
Sept/Oct 2015 M4
Magazine
View Hugh’s
videos on
his YouTube channel:
spinfun
Follow Hugh on
Twitter:
@hughhunt
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.
The M4 Editor
found plenty of
arches on a
recent trip to the USA.
See the MEI
Facebook page.
Hooke thought
that the shape of
the perfect dome was the cubicoparabolical conoid, i.e. the cubic y = ax3
rotated about the y axis. He was very
close!
By Lwphillips. Shape of hanging chain versus
arch. [CC BY-SA 3.0], via Wikimedia
The Wren Library in Trinity College
Cambridge has arches, but where are
they?
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 Andrew Dunn. The Wren Library,
Cambridge. [CC BY 2.0 ], via Wikimedia
Commons.
By Bernard Gagnon.
Dome of Saint Paul's
Cathedral seen from
Tate Modern, London.
[CC BY 2.0 ], via
Wikimedia Commons.
By Dp76764.
Dome of St Paul’s, via
Wikimedia Commons.
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
McGraw-Hill Dictionary of
he created a Architecture and Construction.
raft made up S.v. "inverted arch."
Retrieved October 1 2015
of inverted
from http://
arches.
encyclopedia2.thefreedictionary.
com/inverted+arch
Hugh’s Views
Buildings and bridges
By Jacques Heyman.
Hooke's CubicoParabolical 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
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).
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!
Via Wikimedia Commons.
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
From the outside the King’s Chapel the Thames. I imagine
looks kind-of square and maybe a Wren and Hooke are
By Alexandre Buisse.
looking down from their St Pauls Cathedral and
bit dull. But those big exterior
Millennium Bridge. [CC
pillars are part of the engine room lofty dome. They see
BY-SA 3.0], via
the
cables
of
the
bridge.
of the fan vaulting inside. The thin
Wikimedia Commons.
spidery stone filaments inside (see “I told you so” Hooke
photograph, left) are like lines on a mutters. “Ut pendet continuum flexile, sic
stabit contiguum rigidum inversum, as hangs
graph, illustrating the lines of
the flexible line, so but inverted will stand the
thrust. The forces are channelled
rigid arch”.
into the pillars in an orderly
By Dmitry Tonkonog.
King's College Chapel. [CC BY-SA
3.0], via Wikimedia Commons.
By Lofty. Ceiling of
King's college,
Cambridge.
[CC BY-SA 3.0], via
Wikimedia Commons.
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.
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.
The MEI Maths
Item of the
Month is a
monthly problem
aimed at teachers
and students of
GCSE/A level
Mathematics.
Each month a
He has a strong
mathematical problem is added to the
interest in the use of
technology in maths, home page of the MEI website. The
especially at A level,
MEI staff are all mathematics
and has delivered
enthusiasts and putting an interesting
many professional
problem on the front of the site is a
development courses
technological way of wearing our
on this. He is the
mathematical hearts on our sleeves.
chair of MEI’s
GeoGebra Institute
and runs the MEI/
The first Maths Item of the Month
Casio Teacher
appeared in September 2006 and there
Network.
have been over 100 items since then.
Tom has also recently A full archive of the problems is
developed MEI’s new available on the site and they can be
technology-based A
used for enrichment, problem solving or
level unit: Further
as a way to encourage mathematical
Pure with
thinking/proof.
Technology.
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
Follow Tom’s blog:
Digital technologies fairly lengthy set of miscellaneous
for learning
problems at the end!
mathematics
One of my favourites is one of the
You can also earliest ones from December 2006:
follow Tom
“19 not out” – Some positive numbers
on Twitter at
add up to 19. What is the maximum
@tombutton
product?
There is also usually a Christmasthemed 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 elibrary.
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 Art
It 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 Art
Using 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 Art
You 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 angle
These skills are outlined on the next slides.
Divide a Circle Equally into Six
• Draw a circle and keep the compasses at the
same radius throughout
• 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
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
B
and draw an arc
• Draw a line from the vertex of
A
the angle through the
intersection of the arcs
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 Art
This 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 Art
One 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 Art
Slides 2 – 8
When 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 Art
Slides 9-13
These 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 Art
Slides 14-21
These 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 Art
Challenge 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 Art
Challenge 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 Art
Challenge 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 Art
Challenge 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
Acknowledgements
Rangoli 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/st
ained-glass