Monthly Maths

Monthly
Maths
I s s u e
Maths and Art
The
Mathematical Art
Of M.C. Escher:
4 minute video
extract from a
BBC documentary
about the
relationship
between maths
and art as
explored by
Escher.
Escher's
Tessellations:
6 minute video
compiled by a
teacher who says:
“Nothing fancy.
Just needed
something to play
while my students
were working on
their tessellation
project.” You
might find it
similarly useful.
NRICH has a
section on
Maurits
Cornelius
Escher.
3 0
Tessellations
www.mei.org.uk
S e p t e m b e r
2 0 1 3
Tessellations in everyday life
The word "tessellate" is derived from the
Ionic version of the Greek word
"tesseres," which in English means "four."
Tessellations are also sometimes known
as tilings, but the word "tilings" usually
refers to patterns of polygons (i.e. shapes
with straight boundaries), which is a more
restrictive category of repeating patterns.
‘Tessellations’
illustrates examples in art, nature, food
and everyday life, and could provide a
useful starting point for the topic.
Types of tessellation:
Regular – comprised of one type of
regular polygon, for example:
Semi-regular – comprised of more than
one type of regular polygon and with the
same polygon arrangement at every
vertex, for example:
Demi-regular - comprised of more than
one type of regular polygon and with two
or three different polygon arrangements,
for example:
Beyond - comprised of more than one
type of regular polygon and with unlimited
polygon arrangements
Click here for the MEI
Maths Item of the Month
Totally Tessellated is a site produced as
a student competition entry, and is a
helpful and interesting resource providing
a basic introduction into the complex
world of tessellations and tilings. The site
authors explain the role tessellations
have had in history and in cultures
around the world. The site provides
background information about Polygons
and Angles, Symmetry and
Transformations, Tessellations and
Colour Usage. The use of regular and
semi-regular polygons in creating simple
tessellations is explored, and simple
tessellations with non-regular polygons
are investigated.
One section looks in depth at mosaics
and tilings, exploring tessellations of
polygons with small numbers of sides:
triangles, quadrilaterals, pentagons, and
hexagons.
A new MEI teaching resource is at the end of this
bulletin. Click here to download it from our website.
Disclaimer: This newsletter 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.
Classroom activities
Explore Escher
A YouTube video
tutorial, How to
Create a
Tessellation,
demonstrates how
to create a
tessellated pattern
by drawing shapes
using either
geometric or
organic lines, then
cutting out and
sliding the shapes
to form
tessellations.
This could form
the basis of some
enrichment work
for your students,
especially if you
include Paul
Giganti’s excellent
videos: Anatomy
of an Escher
Lizard and
Anatomy of an
Escher Flying
Horse as
inspiration.
On the website
Encyclopedia.com,
Dana Mackenzie’s
article Making
Tessellations
explains how to
create an Escherstyle tessellation
using translation,
reflection and
rotation. She also
describes the
“kaleidoscope
method”.
Tessellation Tutorials
Suzanne Alejandre has published a
section of tessellation tutorial pages in
The Math Forum. These tutorials and
templates for making your own
tessellations include the use of software
(which has probably been superseded by
more sophisticated software) as well as
Tessellating with only a Straightedge
and Compass.
There are examples to view of
tessellations produced by students.
The section of tessellations from Hawaii
of is particularly interesting in that it uses
ancient
Hawaiian
designs as
inspiration,
such as this
design,
Birds of
Paradise.
Another group of students, this time from
New Jersey, had fun in 2004 using
software to create a tessellation in the
shape of an animal, an object, or a
person, then writing a poem to
accompany their work. Click on each
tessellation of the Student Tessellations
page to view it in close up and to read the
poem.
Tessellations for Teachers
Jo Edkins has written a comprehensive
set of web pages called Tessellations for Teachers that include a grids
webpage showing clearly how
tessellations can be built up, and the
importance of colour in creating such
patterns. This is quite basic stuff, but full
of useful resources such as grids to
download of single and multiple regular
shapes, and of irregular shapes.
As Jo warns, ‘However, the grids provide
no discipline to stop children colouring
them in any old how… So there are
interactive pages which provide this
tessellation discipline.’ See for example:
Design a tessellation online (square
grid). Jo also provides instructions for
how you can download, copy, save and
print off the resulting tessellations.
An Australian teacher has created a Wiki
that includes resources for teaching
about tessellations; although these are
targeted at the equivalent of upper
primary level, you may find some of these
useful to adapt for your own lessons.
Using software to create tessellations
This video tutorial, How to Create a
Tessellation Out of a Regular Triangle,
demonstrates how to create a tessellation
out of a regular triangle using a computer
program such as Power Point or Paint.
The resulting shape can be printed, cut
out and used as a stencil to create a
poster.
It’s quite a slow-paced
tutorial, but could be
used by students for
independent work, as
it includes a list of
steps at the end of the
demonstration.
Another
video
demonstrates
step by step how to
create rotational
tessellations with
GeoGebra.
Further investigations
Letter
Tessellations
In 1986 Scott Kim
constructed a
Tessellating
Alphabet, where
each letter fits
together with
copies of itself to
tile the plane,
using a ‘wide
range of types of
tessellations, with
different
symmetries and
different degrees
of difficulty’. Click
each letter of the
alphabet to see
how it tessellates.
Can you think how
he might have
tessellated the
letter J?
Ask students to
explore different
ways of tiling.
letters of the
alphabet; can they
design tessellating
uppercase letters
to go with the
lower case?
Puzzling
Typography:
Mazes with Letter
Tessellations has
84 mazes
constructed only
from patterns of
tessellating letters.
Both Wolfram MathWorld
and NRICH use interactive
resources to investigate
Napoleon's Theorem. Napoleon's
theorem states that if equilateral triangles
are constructed on the sides of any
triangle, either all
outward, or all inward,
the centres of those
equilateral triangles
themselves form an
equilateral triangle.
In her Plus Magazine
article From quasicrystals
to Kleenex, Alison Boyle looks at
complex geometric designs such as those
used in Alhambra tilings.
She also looks at the creation of sets of
tiles that will tile only aperiodically (so that
the pattern never repeats itself), including
those
invented by
Oxford
professor Sir
Roger
Penrose.
You can also read
about the
disrespectful use
of his rhomb design
caused Sir Roger
to sue Kimberley
Clark back in 1997!
If you want to find out
more about Sir Roger
Penrose, in the article A
Knight on the Tiles he
talks to Helen Joyce from
the Plus team about his
ideas.
Incidentally, Sir Roger Penrose is giving
a talk about crystal symmetry in
mathematics and architecture at the
Royal Institution on Wednesday 18
September 2013 – visit the Royal
Institution website for more details.
The IMA has
developed a
Large Maths
Outreach and Careers Kit containing the
Penrose Tiles, available for loan. The kit
was created for outreach work with
schools. Matt Parker demonstrates the
Penrose Tiles in this short video.
In Secrets from a bathroom floor,
Josefina Alvarez and Cesar L. Garcia
investigate the precise art of tiling and the
challenges of producing tilings for
bathroom floors, and suggest some ‘what
ifs’ as follow up.
Congruent pentagons
were used in this
unusual floor tiling of
the headquarters of the
Mathematical
Association of America.
Craig Kaplan’s article The trouble with
five explains why you won’t find
pentagons used in regular tiling patterns.
Tiles in the shape of regular pentagons,
with all five sides and interior angles
equal, inevitably leave gaps when used to
cover a surface. In contrast, regular
hexagons do cover the plane.
The article examines how a set of shapes
with five-fold symmetry might together tile
the Euclidean plane. For example, if we
introduce another shape, will that resolve
the ‘five-fold tiling problem’ or will it just
‘buy us some time before we get stuck
again’? And if we move from a flat plane
to a sphere, such as a football, does it
become easier to tile its surface?
Structural tessellations
In the video
Origami
Tessellations,
Tung Ken Lam
showed how to
fold a basic
origami
tessellation in his
session at the
2012 Association
of Teachers of
Mathematics
Conference. It’s
interesting to see
what folded
products can be
made from a
square of paper.
This may be a
basic fold, but it’s
definitely not one
for beginners!
Some spectacular
examples by
Yoshi – Paper
Artist from
Venezuela are
shown on the
Origami
Tessellations site.
More examples of
Yoshi’s amazing
work are available
on his Flickr site.
Absolute Astronomy’s page on
Tessellation provides a useful overview
of the topic of tessellations of other
spaces.
Honeycomb morphology
The web page Structural tessellations
and morphologies tells us that
“honeycomb
structures have
become a very
popular subject and
feature in
architectural
design”.
This is because “the most efficient use of
materials in a two-dimensional space is
done by tessellating hexagonal parts and
not triangular ones, hexagonal structures/
domes have since been developed
utilizing some additional structural
support.”
Software is used
to develop linear
honeycomb
structures into
more complex
forms, both in two
and three dimensions and in planar. In
planar tessellations, each hexagon of
different size follows a plane of an
underlying form. Honeycomb morphology
can involve all the dimensions. Read
more about honeycomb morphology in
this report about a 2004 architecture
research project.
Building with tessellations
A tessellated roof is one of the most
flexible framed systems to design, the
measurements and precision are complex
and commonly part of a computer-aided
design process of production.
In this architect’s blog you can read
about challenges faced in the design of
buildings using tessellations.
The Eden Project
was designed by
Architect Nicholas
Grimshaw and
Structural
Engineer Anthony
Hunt in 2011. The
Eden Project biomes use an unusual
tetrahedral-truss, which requires little
additional material and results in a
remarkably thin dome.
Geodesic dome construction
The word “geodesic” refers to the shortest
distance between two points on a curved
surface. Wolfram MathWorld describes
a geodesic dome as “a triangulation of a
Platonic solid or other polyhedron to
produce a close approximation to a
sphere (or hemisphere).”
HowStuffWorks describes geodesic
domes more simply: “In short, geodesic
domes are structures that look like half
spheres made up of many triangle
supports” in their article: How Geodesic
Domes Work.
Stephen Luttrell looks at how to automate
the design of geodesic domes using
Mathematica, with the dome at the Eden
Project as his focus. Domerama.com
explores how the biomes at the Eden
Project were designed and constructed.
Domerama.com’s web page: Sketchup
3D Geodesic Models lists over 70
geodesic dome models created with free
3D modelling software Google
Sketchup. An example of a geodesic
tessellation of an
icosahedron is the
Equal-edge Goldberg
Polyhedron.The radii
of vertices-to-spherecentre vary slightly to
achieve planar faces.
Tangrams
Tangram Table
Massimo Morozzi,
an Italian designer,
created the
Tangram Table in
1983. It is
essentially a larger
counterpart of the
regular tangram
pieces.
The Tangram
Table is made of
up the seven
separate shapes,
with specially
designed legs that
enable the tans to
stand up.
Tangrams
A tangram is an ancient
Chinese moving piece
puzzle, consisting of 7
pieces (‘tans’) made
using 3 basic geometric
shapes: 2 large, 1
medium and 2 small
triangles, 1 square and 1 parallelogram.
The classic rules are as follows: You
must use all 7 tans, they must lay flat,
they must touch and none may overlap.
Wikipedia defines the tans in greater
detail: “... the seven pieces are:

2 large right triangles
(hypotenuse

, sides
, area
)
1 medium right triangle
(hypotenuse
, sides
, area
)

2 small right triangle
(hypotenuse
Then they can be
arranged into the
varied shapes
possible as with
the traditional
tangram - you can
see some of the
arrangements on
the deconet
website.
, sides
, area
)

1 square
(sides

, area
)
1 parallelogram
(sides of
and
, area
)
“Of these seven pieces, the parallelogram
is unique in that it has no reflection
symmetry but only rotational symmetry,
and so its mirror image can be obtained
only by flipping it over. Thus, it is the only
piece that may need to be flipped when
forming certain shapes.”
There has been some
debate over the origin
of the name ‘tangram’.
It was possibly derived
from the word trangram
meaning a puzzle or
trinket, or perhaps tá ng (the Chinese
dynasty) and gram, Greek for 'writing'.
Another possibility is that it derives from
tá ng tú , with tú meaning a picture or
diagram.
In 1903,
American
Sam Loyd ,
known by
Martin
Gardiner as
"America's
greatest
puzzler", wrote his great spoof of tangram
history, The Eighth Book of Tan. This
included an extensive, but bogus history
of the puzzle that claimed that it was
4,000 years old and had been invented
by the god Tan. Despite this fictitious
claim the book did include seven hundred
unique Tangram designs.
The tangram obtained admirers such as
Edgar Allen Poe, Lewis Carroll, Thomas
Edison, and Napoleon. Tangrams (also
known as Chinese puzzles) were also
popular pastimes during World War I, in
the trenches of both sides.
Class activities
Paradoxes
Puzzles.com’s
tangram puzzles
include a handson tangram page
with an illustration
of two monks,
developed by
Henry E. Dudeney,
demonstrating a
paradox. Both men
are assembled
with all the seven
tans, but one of
them has a foot,
while the other
hasn't. Can your
students explain
why this is? The
blog "Restless
Minds" shows the
solution.
Another paradox is
the Magic Dice
Cup tangram –
from Sam Loyd’s
book Eighth Book
of Tan. Each of
these cups was
composed using
the same seven
geometric shapes,
but the first cup is
whole, and the
others contain
vacancies of
different sizes.
Can your students
think up any more
tangram
paradoxes?
Puzzles and games
The Centre for Innovation in Mathematics
Teaching (CIMT) Mathematics
Enhancement Programme has a
section on tangrams that includes
instructions for making a set of tangram
pieces, and several puzzles to work
through. Here’s the link to the first puzzle
page; links to the next page are at the
bottom of each page. Included on puzzle
page 4 is the two monks paradox
mentioned left. The Puzzles Index page
has links to some answers, while
educational institutions can apply to CIMT
for access to others via their web page.
Tangrams.ca lists tangram games and
generators to download, including
Canvas Tangrams. You can download
your own copy; the link is on the right at
the bottom of the page. SUMS
Mathematics has a free Flash tangram
game with short (4 shapes) and long (10
shapes) games available to play online.
The Stomachion
This consists of 14
tiles forming a
12x12 unit square,
where each tile's
area is a whole
number. As with its
cousin the
tangram, the object
of the Stomachion
is to rearrange the
pieces to form interesting shapes.
It is not known whether Archimedes
developed the Stomachion, though the
puzzle was definitely known by the
ancient Greeks. Because he wrote about
the puzzle extensively, however, it is also
known as Archimedes' Puzzle or the
Loculus of Archimedes.
Puzzles.com has a useful page about
the Stomachion, as does
WolframMathWorld. There is also a
printable resource that includes the
Stomachion shapes. The National
Council for Teaching of Mathematics
(NCTM) has a lesson plan for teaching
about the Stomachion, including activity
sheets and extension work.
Crafstmanspace.com’s Stomachion of
Archimedes puzzle plan may be useful.
In 2003, a retired businessman named
Joe Marasco commissioned the puzzlemaking company Kadon Enterprises Inc.,
to manufacture Archimedes’ Puzzle for
him and his friends. Joe then challenged
programmers to identify, with proven
accuracy, exactly how many solutions this
puzzle has, offering a $100 reward for the
first correct solution.
Bill Cutler's program found there to be
536 possible distinct arrangements of
the pieces into a square, where solutions
that are equivalent by rotation and
reflection are considered identical. You
can read the whole story on the
Gamepuzzles website.
A New York Times front page article on
December 14, 2003 reported the total of
solutions to be 17,152; however this
included all rotations and reflections, etc.
Dividing 17,152 by 32 would, in fact, yield
the 536 stated by Bill Cutler.
How many solutions can your students
find? How would they go about proving
that there are this many?
A new MEI teaching
resource follows, in
PowerPoint format.
Click here to download all related files
from our website.
Tangrams: making shapes
Make a tangram by drawing an 8 x 8 square and
cutting out the 7 shapes as shown below.
Tangrams: making shapes
Use all 7 shapes each time; they can be rotated
and flipped over if needed.
Individual pieces must not overlap each other.
Try to make the following:
• A rectangle
• A large isosceles triangle
• A parallelogram
• An isosceles trapezium
• An irregular pentagon …with a line of symmetry
• An irregular hexagon …with 2 lines of symmetry
Tangrams: Square
Make a tangram by drawing an 8 x 8 square and
cutting out the 7 shapes as shown below.
Tangrams: Square
Use all 7 shapes each time; they can be rotated
and flipped over if needed.
Individual pieces must not overlap each other.
How many different ways can the large square be
made?
Tangrams: area
Make a tangram by drawing an 8 x 8 square and
cutting out the 7 shapes as shown below.
Find the area of each of the individual shapes.
Tangrams: fractions
Look at the tangram below.
What fraction of the original square is each of the
individual shapes?
Tangrams: perimeter
Make a tangram by drawing an 8 x 8 square and
cutting out the 7 shapes as shown below.
Find the perimeter of each of the individual
shapes.
Tangrams: perimeter 2
Make a tangram by drawing an 8 x 8 square and
cutting out the 7 shapes as shown below.
Tangrams: perimeter 2
Use all 7 shapes each time; they can be rotated
and flipped over if needed.
Individual pieces must not overlap each other.
Make several different shapes and find their
perimeters.
What is the largest perimeter you can make?
What is the smallest perimeter you can make?
Tessellations
A tessellation is a regular pattern made of tiles
placed so that it can continue in all directions.
Every bit of the pattern should be repeated.
Anyone looking at the pattern should be able to
see exactly how it will continue.
Some tessellations are created using just one
shape, others use two or more shapes
Which of the patterns on the following slides are
tessellations?
Quadrilateral Tessellations
Which quadrilaterals tessellate?
Try using one of the following as a tile:
• Square
• Rectangle
• Parallelogram
• Rhombus
• Trapezium
• Kite
• ‘Irregular’ quadrilateral
Regular Tessellations
A regular tessellation is one in which only one
regular polygon is used. Tiles must all be the
same size and have to be placed edge to edge.
How many regular tessellations are there?
Prove that there can be no others.
Semi-regular tessellations
A semi-regular tessellation is
one in which more than one
regular shape is used.
How many different ones
can you find?
Given that the largest
number of sides of a polygon
in a semi-regular tessellation
is 12, can you prove that no
others are possible?
Teacher notes
The idea of this month’s classroom resource is to take a simple starting
point and use it in several ways to address different aspects of the
curriculum.
It is not expected that this is a sequence of lessons as different
activities will be most suitable for different groups of pupils.
Additionally, many of these activities would make for good sources of
display work, just in case anyone’s thinking about school ‘open days’…
Tangrams: overview
This ancient puzzle can be used in a variety of ways, traditionally:
• To recreate given shapes
• To create new shapes
However, additional activities and questions are:
• Area: Given that the side length of the original tangram is 8 units,
find the area of each of the smaller shapes making up the tangram
• How many different ways can all 7 shapes be used to make the
square. Rotations and reflections not permitted.
• What fraction of the original shape are each of the 7 pieces?
• If the original tangram has side length 8 units, what are the side
lengths (and/or perimeters) of each of the 7 pieces?
• Make shapes using all 7 pieces and find their perimeters. What is
the smallest perimeter that can be made using all 7 shapes? What’s
the maximum perimeter that can be made using all 7 shapes
Tangrams
This ancient puzzle can be used in a variety of ways, traditionally:
• To recreate given shapes
• To create new shapes
The usual rule is that pieces cannot overlap, sometimes an additional
rule is given that all adjacent shapes must meet at an edge/ partial
edge.
This activity is accessible for almost all pupils
Some answers, although there are probably several possibilities for
each:
Tangrams
Area: Given that the side length of the original tangram is 8 units, find the
area of each of the smaller shapes making up the tangram.
Suitable for most KS3 pupils, can be
solved by:
• Counting squares
• Reasoning - using the smaller
square and deducing that each of
the smaller triangles is half the area
and then using combinations of
these to physically recreate the
larger shapes
• Finding what fraction of the whole
each piece represents
• Using formulae
Tangrams
How many different ways can all 7 shapes be used to make the
square.
Suitable for most KS3 pupils
Just one arrangement, but there are several rotations and reflections
that can be created.
Ask pupils to make the square in as many different ways as they can
and use their responses to initiate discussions about ‘same and
different’ and transformations as an introductory activity for reflections
and rotations.
Tangrams
What fraction of the original shape are each of the 7 pieces?
Tangrams: perimeters
If the original tangram has side length 8 units, what are the side lengths
(and/or perimeters) of each of the 7 pieces?
•
•
•
•
•
•
•
A
B
C
D
E
F
G
Tangrams: perimeters 2
Make shapes using all 7 pieces and find their perimeters. What is
the smallest perimeter that can be made using all 7 shapes?
What’s the maximum perimeter that can be made using all 7
shapes?
Mathematics required: surds
Encourage pupils to leave answers in surd form for addition purposes,
although they may have to convert to decimals to compare some.
Pupils can experiment with shapes of their own or be given the shapes
on slides 12 & 13 to begin with. They might also be given a limitation of
only using certain shapes i.e triangles and quadrilaterals.
Tangrams: perimeters 2
Long thin shapes will have a larger perimeter.
A square might be expected to have the minimum perimeter, but
shapes such as the hexagon shown actually have smaller ones.
This could lead to a discussion about the relationship between area
and perimeter. For a fixed area, the closer a shape is to being circular,
the smaller its perimeter will be.
Tangrams: perimeters 2
A selection of answers are shown
Tessellations: overview
Tessellations make for engaging activities, which are accessible to
most pupils.
In Key Stage 3 the activities might begin with creating tessellations
However, they also provide opportunities for utilising dynamic geometry
software and also for reasoning and proving.
Activities:
Is it or isn’t it a tessellation
Quadrilateral tessellations
Regular tessellations
Semi-regular tessellations
Tessellations: is it or isn’t it?
•
•
•
•
•
1 – yes
2 – yes
3 – no: not a regular pattern
4 – yes
5 – no: this is a pattern, but it’s not repeated
Tessellations: Which quadrilaterals tessellate?
All quadrilaterals tessellate
This can be explored by pupils cutting out a template and drawing
round it to create a tessellation.
Another way to demonstrate this is to use Dynamic Geometry Software,
using the Geogebra file ‘Quadrilateral Tessellation’. (free Geogebra
software required). The quadrilateral in the top left hand corner of the
page is the driver. Move the vertices of this shape and all others will
change with it, maintaining a tessellation. Hence it can be
demonstrated that all quadrilaterals tessellate.
Tessellations: Regular tessellations
Moving into reasonably simple proof, it can be shown that there are only 3
possible regular tessellations. It would be helpful for pupils to be given time to
think about how they could prove this and perhaps have a class discussion
rather than telling them how to prove it.
Tiles are fitted edge to edge and hence meet at points.
Since the angle sum must be 360°, the interior angle of the regular polygon
must be a factor of 360.
There are (at least) two ways to approach this.
• Find all the factors of 360 and work out which ones are interior angles of
regular polygons
or
• For a tessellation there must be 3 shapes meeting at a point - 2 wouldn’t be
a point. Therefore the largest angle it could be would be 120° (hexagon)
and the smallest regular polygon is a triangle 60° . This means that only 4
shapes need to be checked to determine if their interior angles are factors
of 360°
Tessellations: Semi-regular tessellations
There are 8 semi-regular tessellations, although there is a 9th if a mirror image
is permitted.
Proof by exhaustion can be used to prove that there are no others, but the
entire proof would be daunting. Providing the information that the largest
number of sides for any regular polygon in a semi-regular tessellation is 12
makes the problem more accessible.
Finding a logical and systematic way to identify combinations of interior angles
of regular polygons which have a sum of 360° allows another proof by
exhaustion.
7 piece tangram
MEI is a registered charity, number 1058911
MEI is a registered charity, number 1058911