The Mathematics
of Tessellation
© 2000 Andrew Harris
The Mathematics of Tessellation
1
© 2000 Andrew Harris
Contents
Defining Tessellation.......................................................................................................... 3
Tessellation within the Primary School Curriculum........................................................... 3
Useful Vocabulary for Tessellation ..................................................................................... 3
Prior Knowledge Required for Understanding the Mathematics of Tessellation................ 4
Useful Resources for Teaching Tessellation........................................................................5
Progression in Learning about Tessellation......................................................................... 5
1. Tessellation by Repeated Use of One Regular Shape........................................6
2. Tessellation by Repeated Use of Two or More Regular Shapes....................... 9
3. Tessellation of Triangles and Quadrilaterals..................................................... 11
Preparatory Knowledge for Understanding the
Tessellation of Triangles and Quadrilaterals.............................11
(a) Tessellating with Triangles................................................................ 12
(b) Tessellating with Quadrilaterals.........................................................14
4. Tessellation of Irregular Shapes obtained by Mutation of
Tessellating Shapes.........................18
The designs of M.C. Escher.................................................................... 19
5. Tessellations involving other irregular shapes.................................................. 21
Use of ICT in Teaching Tessellation................................................................................... 23
Resources
Hexagonal Tiling Mats.......................................................................................... 25
Square Tiling Mats - 1........................................................................................... 26
Square Tiling Mats - 2........................................................................................... 27
Square Tiling Mats - 3........................................................................................... 28
Useful References: Tessellation...........................................................................................29
The Mathematics of Tessellation
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© 2000 Andrew Harris
Defining Tessellation
A tessellation can be defined as the covering of a surface with a repeating unit consisting of one
or more shapes in such a way that:
• there are no spaces between, and no overlapping of, the shapes thus employed,
and
• the covering process has the potential to continue indefinitely (for a surface of infinite
dimensions).
Tessellation within the Primary School Curriculum
Primary school tessellation activities fall into two categories:
•
•
The mathematics of tessellation
Application of knowledge about tessellation
The ‘application of tessellation’ aspect is often seen in schools and features in most
published mathematics schemes. However, the ‘mathematics of tessellation’ aspect is often
overlooked. This results in poor progression with children often repeating the pattern-colouring
activities they have undertaken in previous years of their schooling. Moreover, such children
have little idea of why some combinations of shapes tessellate while others do not.
The ‘mathematics of tessellation’ aspect should focus on providing children with the
knowledge and skills to explain why tessellation is or is not possible for particular units of
shape. It is this which is the basis of good practice for teaching tessellation.
That said, it is very worthwhile to show children real-life examples of the application of
tessellation. Brick walls, paving, wall and floor tilings, woodblock floors, carpets, wallpapers,
wrapping papers, textiles and works of art are often useful resources for whole-class or group
discussions about tessellation as a concept and about its application in everyday life. Typical
activities when using these might be to identify the repeating unit of shape used to create the
tessellation, to explain why the tessellation works, to consider other ways of tessellating the
given surface or to develop mental visualisation skills (e.g. as part of a mental/oral starter or
plenary within the daily mathematics lesson). Using resources of this kind demonstrate to
children the value and purpose of understanding the mathematics of tessellation and thus
provide an incentive and reason to learn about it.
Useful Vocabulary for Tessellation
In common with other aspects of Shape and Space work, children will need to become familiar
with the following mathematical vocabulary
Plane:
Regular shape:
two-dimensional or, colloquially, ‘flat’.
a shape in which all the sides are the same length AND all the angles are
the same size.
Irregular shape: a shape in which not all the sides are the same length AND/OR not all the
angles are the same size.
Polygon:
a two-dimensional, closed shape in which all the sides are straight lines.
Interior angles: the angles inside the boundary of a shape (see illustration below).
Exterior angles: the angles through which one would turn at the corners of a shape if
walking around the boundary of the shape (see illustration below).
The sum of the exterior angles of any polygon is 360° or one whole turn.
Re-entrant angle: an interior angle of more than 180°.
Congruent shapes:shapes which are identical in terms of type of shape, lengths of sides
and sizes of angles. Their position in space and their orientation may be
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© 2000 Andrew Harris
different.
Children will also need to know the names and properties of common two-dimensional shapes.
Exterior
angles
Interior
angles
interior angle + corresponding exterior angle = 180°
Interior and Exterior Angles of a Regular Pentagon
Prior Knowledge Required for Understanding the Mathematics of Tessellation
In order to be able to understand why some combinations of shapes will tessellate and others
will not, children will need to know:
•
•
•
•
a whole turn around any point on a surface is 360°;
the sum of the angles of any triangle = 180°
the sum of the angles of any quadrilateral = 360°
how to calculate or measure the interior angles of polygons
The interior angles of regular polygons (i.e. not other polygons) can be calculated in one of
two ways:
1 Divide a whole turn (360°) by the number of exterior angles (= the number of sides)
to find the size of one exterior angle. Then use the fact that
the exterior angle + the corresponding interior angle = 180°
(because angles on a straight line add up to 180°) to find the interior angle.
e.g. for a regular pentagon (5 sides, so has 5 exterior angles)
the exterior angle = 360° ÷ 5 = 72°
so the interior angle = 180° - 72° = 108°.
2 The sum of the interior angles of a n-sided regular polygon = (n - 2) × 180°.
Once the total has been calculated in this way, the size of one of the interior angles
can be found by dividing by the number of interior angles (= n).
e.g. for a regular pentagon (5 sides, so has 5 interior angles)
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© 2000 Andrew Harris
the sum of the interior angles = (5 - 2) × 180° = 3 × 180° = 540°
so one interior angle = 540° ÷ 5 = 108°.
Note that these methods only work for regular polygons.
Measuring the angles with a protractor is possible in all cases, regular or irregular.
In summation, children will need to know or learn about the angle properties of all regular
polygons and of common irregular polygons in order to understand the mathematics of
tessellation.
Useful Resources for Teaching Tessellation
The following may be found useful when teaching about tessellation:
• a large number of various cardboard or plastic regular plane (i.e. flat/2D) shapes each
of which have sides of the same length;
• gummed paper shapes which have sides of the same length;
• tiling mats (obtainable from the Association of Teachers of Mathematics or see
Resources section at the end of this booklet)
• computer software packages which allow children to investigate tiling and tessellation
activities;
• real-life examples of tessellation patterns (such as works of art (e.g. by Escher),
fabrics, wrapping papers, wallpapers, floor and wall tilings, brickwork patterns);
• sets of different kinds of triangles
• sets of different kinds of quadrilaterals
• sets of irregular shapes (some of which tessellate and some of which do not)
• protractors
Progression in Learning about Tessellation
Several distinct stages can be identified in learning about the mathematics of tessellation. As
progress is made through these stages the degree of regularity of the shapes under consideration
reduces. The suggested stages in the progression are:
1 Tessellations involving repeated use of ONE regular polygon;
2 Tessellations involving repeated use of a unit of shape made up of TWO OR MORE
different regular polygons;
3 Tessellations involving triangles or quadrilaterals;
4 Tessellations of irregular shapes obtained by transformation of other ‘more regular’,
tessellating shapes ;
5 Tessellations involving other irregular shapes.
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© 2000 Andrew Harris
1. Tessellation by Repeated Use of One Regular Polygon
Tessellations in which one regular polygon is used repeatedly are called regular tessellations.
Initially, children should investigate using sets of 2D shapes to find out which common regular
polygons will tessellate and which will not. They should arrive at the following conclusions:
• Equilateral triangles will tessellate:
• Squares will tessellate:
• Regular hexagons will tessellate:
• No other regular polygons will tessellate in this way:
for example,
regular pentagons
The Mathematics of Tessellation
regular heptagons
6
regular octagons
© 2000 Andrew Harris
All of the above conclusions can be reached by simply trying each regular shape in turn. It is
much harder, however, for children to be able to explain why it is that, of all the regular shapes,
only the equilateral triangle, the square and the regular hexagon will tessellate.
At this point the prior knowledge outlined on page 4 must be utilised. The key piece of
knowledge here is that a whole turn around any point on the surface is 360°.
The vertices of six equilateral triangles meet at Point A. Each of the interior angles of the
equilateral triangles is 60°.
60°
60° 60°
60° 60°
60°
Point A at which the vertices
of the six triangles meet.
Angle sum around Point A is
60° + 60° + 60° + 60° + 60° + 60°
= 6 × 60°
= 360°
The sum (total) of the angles around Point A is 6 × 60° = 360°. This fact is true of all such
points where the vertices of six equilateral triangles meet and thus the equilateral triangles will
tessellate.
An alternative way to look at this idea is to note that 6 complete equilateral triangles can meet
at a common vertex at any point on the surface to be covered (because 360° ÷ 60° = 6, a whole
number) and without any gaps being left or any overlapping occurring (because 360° is exactly
divisible by 60°).
[Note that it does not matter what size the equilateral triangles are (as long as they are all
congruent) since the angles will still be 60° whatever the length of the sides.]
This tessellation may be represented by the abbreviated notation 36 (signifying that six threesided regular polygons meet at a common vertex). Note this does NOT mean ‘three to the
power six’ in this context. You may wish to avoid this notation when working with children
because of the potential for confusion with the more usual interpretation of this as ‘three to the
power six’. It is included here because of its usefulness for the speedy notation of tessellation
patterns involving regular shapes.
The same idea can be applied to the tessellation of squares and of regular hexagons:
Point B at which the vertices
of the four squares meet.
90° 90°
90° 90°
Angle sum around Point B is
90° + 90° + 90° + 90°
= 4 × 90°
= 360°
The total of the angles around Point B is 4 × 90° = 360°. Since this is true of all such points
where the vertices of four squares meet this explains why squares tessellate.
Alternatively, the four complete squares can meet at a common vertex at any point on the
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© 2000 Andrew Harris
surface (because 360° ÷ 90° = 4) and without any gaps being left or any overlapping occurring
(because 360° is exactly divisible by 90°).
This tessellation may be represented by the notation 44 (four four-sided regular polygons meet
at a common vertex).
Point C at which the vertices
of the three hexagons meet.
120°
Angle sum around Point C is
120° + 120° + 120°
= 3 × 120°
= 360°
120°
120°
The total of the angles around Point C is 3 × 120° = 360°. Since this is true of all such points
where the vertices of three regular hexagons meet this explains why regular hexagons tessellate.
Alternatively, three complete regular hexagons can meet at a common vertex at any point on the
surface (because 360° ÷ 120° = 3) and without any gaps being left or any overlapping occurring
(because 360° is exactly divisible by 120°).
This tessellation may be represented by the notation 63 (three six-sided regular polygons meet at
a common vertex).
However, if we use a similar line of argument for the other regular polygons we find that:
Regular Shape
Interior Angle Size
360° ÷ Interior Angle
Tessellates?
Regular Pentagon
Regular Heptagon
Regular Octagon
Regular Nonagon
Regular Decagon
... ... etc. ... ...
108°
128.57°
135°
140°
144°
...
360° ÷ 108° = 3.333
360° ÷ 128.57° = 2.800
360° ÷ 135° = 2.667
360° ÷ 140° = 2.571
360° ÷ 144° = 2.5
... ... ... ...
No
No
No
No
No
...
Note that, for each shape in the table above, the result of dividing 360° by the interior angle is
not a whole number. Consequently, for any of these shapes it is impossible for an exact (whole)
number of them to meet at any point on the surface to be covered. Thus, either gaps will be left
between them or overlapping of the shapes will occur and therefore none of these shapes can be
used to create a regular tessellation.
Of all the regular polygons, only the equilateral triangle, the square and the regular hexagon
have interior angles such that the result of dividing 360° (a whole turn) by the interior angle is a
whole number. Consequently, only these three regular polygons can be used to create regular
tessellations.
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© 2000 Andrew Harris
2. Tessellation by Repeated Use of Two or More Regular Polygons
Having explored the tessellating possibilities of single regular polygons children should be
asked to investigate which combinations of two or more regular polygons will tessellate and
then consider why some combinations are successful and others are not.
Tessellations in which:
• there are two or more regular polygons around each common vertex,
• the tiling around each common vertex is identical
are known as semi-regular tessellations.
and
Children should be taught to identify the repeating unit (composed of 2 or more shapes) in such
tessellations. This builds upon the skill of being able to identify the repeating unit in linear
patterns such as those made with beads on a string or with multilink which children should have
already experienced in previous work.
There are 8 semi-regular tessellations to be found. Each is shown below with the abbreviated
notation signifying how many of which type of regular polygon are located around each
common vertex.
33.42 (3 equilateral triangles & 2 squares)
Angle sum around common vertex
= 90°+90°+60°+60°+60° = 360°
34.6 (4 equilateral triangles & 1 regular
hexagon)
Angle sum around common vertex
= 60°+60°+60°+60°+120° = 360°
The Mathematics of Tessellation
32.4.3.4 (3 equilateral triangles & 2 squares)
Angle sum around common vertex
=90°+60°+60°+90°+60° = 360°
3.6.3.6 (2 equilateral triangles & 2 regular
hexagons)
Angle sum around common vertex
= 60°+120°+60°+120° = 360°
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© 2000 Andrew Harris
4.82 (1 square & 2 regular octagons)
3.122 (1 equilateral triangle & 2
dodecagons)
Angle sum around common vertex
= 150°+60°+150° = 360°
Angle sum around common vertex
= 90°+135°+135° = 360°
4.6.12 (1 square, 1 regular hexagon, 1 dodecagon)
Angle sum around common vertex
= 150°+90°+120° = 360°
3.4.6.4 (1 equilateral triangle, 2 squares &
1 regular hexagon
Angle sum around common vertex
= 120°+90°+60°+90° = 360°
For each of these semi-regular tessellations, the sum of the angles around each of the common
vertices is 360° and this is the reason why each of these combinations of regular polygons
produces a viable tessellation. Other combinations of regular polygons do not produce semiregular tessellations because it is not possible to achieve an angle sum of 360° around each
common vertex while maintaining identical tiling around each common vertex.
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© 2000 Andrew Harris
3. Tessellation of Triangles and Quadrilaterals
Up to this point all work undertaken has been with regular polygons. Once the idea has been
established that the viability of a potential tessellation is determined by the angle sum around
the common vertices of the shapes involved, the child can begin to explore the tessellating
propensities of other polygons which are less regular. Two important sets of irregular polygons
that should be investigated are triangles and quadrilaterals.
Preparatory Knowledge for Understanding the Tessellation of Triangles
and Quadrilaterals
Children will need to learn that:
• the sum of the angles in a triangle is 180°;
• the sum of the angles in a quadrilateral is 360°.
The Sum of the Angles in Any Triangle
The fact that ‘sum of the angles in a triangle = 180°’ is usually ‘proved’ to children by
asking them to draw on paper any old triangle with angles A, B and C. The triangle is cut
out with scissors and the corners of the triangle are then torn off and arranged as shown
below:
tear off corners
& arrange on
a straight line
B
C
B
A
A
C
The fact that the angles A, B and C can be arranged to lie on a straight line (check with a
ruler) indicates that for this triangle the sum of the angles is equal to a half-turn or 180°.
If several children attempt this each with different triangles it can be shown to work for
several triangles. This is then usually accepted as adequate evidence that the sum of the
angles of any triangle = 180°.
Note that this procedure does not constitute a rigorous mathematical proof of this
mathematical statement since there are an infinite number of possible triangles and therefore
not all triangles have been tested by the procedure outlined above. However, it is usually
considered an adequate basis upon which to proceed for children at this level of
mathematics. The visual nature of the ‘proof’ helps to convince children of the truth of the
hypothesis.
The Sum of the Angles in Any Quadrilateral
The fact that ‘the sum of the angles in any quadrilateral = 360°’ can be ‘proved’ in a similar
way to the procedure outlined above for triangles.
Draw any quadrilateral, tear off the corners and arrange around a common point:
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© 2000 Andrew Harris
B
C
tear off corners
& arrange around
a common point
A
D
A
C B
D
Then use the fact that a whole turn = 360° to deduce that A + B + C + D = 360° and so
‘prove’ that the sum of the angles of any quadrilateral = 360°.
Note that the same reservations expressed above (regarding the procedure for ‘proving’ the
angle-sum of triangles) about lack of mathematical rigor apply in this case also.
Triangles and quadrilaterals can be explored in relation to their tessellating propensities by
either presenting the task as a pair of investigations (i.e. Which triangles tessellate? Which
quadrilaterals tessellate?) or as a pair of hypotheses (‘All triangles tessellate’, ‘All
quadrilaterals tessellate’) which children are asked to test and so either prove them or disprove
them.
(a) Tessellating with Triangles
An initial exploration of a few different types of triangles leads one to suppose that most
triangles will tessellate:
Different types of isosceles triangles
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Right-angled triangles
Scalene triangles
In fact, any triangle can be used as a repeating unit with which to tessellate.
In order to prove this, children can make use of what they already know, namely, a whole turn
about any point on the surface is 360°. In addition to this, children will also need to know that
the sum of the angles of any triangle is 180° (see earlier section ‘Preparatory Knowledge for
Understanding the Tessellation of Triangles and Quadrilaterals’).
If it can be established that, for any triangle, the sum of the angles around any common vertex
is always 360° this will ‘prove’ that all triangles tessellate.
A
This can be done simply with children by asking them to draw with a ruler any
triangle with angles labelled A, B and C.
B
C
C
A
B
B
C
A
C
A
A
C B
B C
A
B
C
A
A
C B
B C
A
B
C
A
B
A
C B
B C
A
B
A
C B
A
C B
We know that
A + B + C = 180°
The Mathematics of Tessellation
A
C B
A
C
B
This triangle is then replicated and the triangles
arranged so that around any common vertex
there will be from the various triangles involved
two of each of the angles A, B and C.
A
C
(because the sum of the angles in any triangle is 180° or,
alternatively, because A, B and C form a straight line)
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© 2000 Andrew Harris
and around any common vertex
the sum of the angles
= (A + B + C) + (A + B + C)
= 180° + 180°
= 360°
Thus, any triangle can be used as a repeating unit for tessellating.
(b) Tessellating with Quadrilaterals
A similar exploration of the tessellating properties of different types of quadrilaterals can be
undertaken.
As a result of the properties of various quadrilaterals, there may be more than one way in which
a particular quadrilateral may tessellate. For example, there are many ways of tessellating with
rectangles. These are just a few:
Brickwork patterns are often a rich source of everyday examples of such tessellations.
Initial explorations involving quadrilaterals suggest that, like triangles, most quadrilaterals can
be used as a repeating unit with which to tessellate:
Tessellating with a rhombus
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© 2000 Andrew Harris
Some other possible tessellations using a rhombus
When tessellating with a parallelogram, children are likely to discover that they can produce
similar tessellations to the first two rhombus tessellations given above.
A trapezium or kite will also tessellate:
Tessellating with a trapezium
Tessellating with a Kite
It is also worth asking children to investigate quadrilaterals which have re-entrant angles such
as the ‘dart’ which also tessellates:
tessellating with a ‘dart’
After some exploration, children should begin to form the hypothesis that ‘all quadrilaterals can
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© 2000 Andrew Harris
be used as a repeating unit with which to tessellate’.
This hypothesis can be ‘proved’ geometrically in a similar way to that for triangles.
Again, it relies on the fact that a whole turn around any point on the surface to be tessellated is
360°. This time, however, children will also need to know that the sum of the angles in any
quadrilateral = 360° (see earlier section ‘Preparatory Knowledge for Understanding the
Tessellation of Triangles and Quadrilaterals’).
If it can be established that, for any quadrilateral, the sum of the angles around any common
vertex is always 360° this will ‘prove’ that all quadrilaterals can be used to tessellate.
B
C
Children are asked to draw any quadrilateral
using a ruler and pencil.
A
D
This is then used to tessellate in such a way that around any common vertex there will be (from
the 4 quadrilaterals which meet there) one of each of the angles A, B, C and D as shown below:
A B
D
C D
A B
C
C
B
B A
C D
A
D C
A B
D
C
B
A
D C
A B
B A
C D
B A
C D
B A
C D
D
A
D C
A B
D C
A B
B
C
B A
C D
D
A
D C
B A
D C
B
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© 2000 Andrew Harris
Given that we know the angle sum of any quadrilateral to be 360° then
A + B + C + D = 360°
Since the angles around any common vertex are precisely A, B, C and D and so must total 360°,
we can then state that ‘all quadrilaterals can be used as a repeating unit with which to
tessellate’.
A tessellation using a combination of two quadrilaterials.
The repeating unit contains 3 kites and 1 rhombus.
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© 2000 Andrew Harris
4. Tessellation of Irregular Shapes obtained by Transformation of Other
Tessellating Shapes
It is possible to produce some very irregular shapes which will tessellate by transforming other
shapes which are known to tessellate. These irregular shapes include shapes bounded by curved
lines (up until now only shapes with straight sides have been considered).
Squares, rectangles, equilateral triangles and hexagons are suitable shapes from which to start.
By translating or rotating about the mid-point of any side sections of the starting shape (or a
combination of both) a new, irregular shape can be made which will also tessellate. Some
examples of this are given below:
Translating a Section of a Square
The original
square
A section of the original
square is moved by
translating (sliding) it
to the opposite side of the
square.
The resulting tessellation
Rotating Sections of a Square
The original
square
A section of the original
square is moved by
rotating it about the midpoint
of the side of the square.
A similar rotational process
is applied to another side
of the square.
The resulting tessellation
The same methods can be applied to rectangles, equilateral triangles and regular hexagons.
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To explain why these irregular shapes tessellate is more difficult than for previous cases
encountered in Sections 1 - 3 and is usually considered to be beyond the level expected of
primary school children. The following explanation is included for subject knowledge purposes.
The argument that, around any common vertex within the tessellation, the sum of the angles is
360° is still true but this is now complicated by the fact that the boundary of the repeating unit
may include curved lines. Where curved lines are involved, we use the tangents to the curves to
define the limits of the angles:
a
d
c
b
The dotted lines are the tangents to the curves at the
common vertex. The angles a, b, c and d are the
angles between the tangents (dotted lines) and as
before
a + b + c + d = 360°
Practical Work with Children
The designs of M.C. Escher owe much to this idea of transformation of shapes to create new,
irregular, tessellating shapes. Examples of Escher’s work are useful resources for inspiring
children to create their own designs in a similar style.
Escher’s
Lizard
Tessellation:
A hexagonal grid has
been superimposed
on this tessellation to
show how it has been
created. It uses a
replicating Lizard tile
made from a regular
hexagon which has
undergone
several
mathematical
transformations.
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© 2000 Andrew Harris
Left: Escher’s Flying
Horse tessellation
Right: Escher’s Birds
tessellation
The simplest way in which to do this type of work with children is to use a gummed paper shape as
the initial shape. Sections of this gummed paper shape can then be cut out and either translated or
rotated appropriately into their new positions. All of the pieces of gummed paper can then be stuck
to cardboard. Cutting around the outline of the transformed gummed paper shape will create a
template. This can then be used as the repeating unit for a tessellation.
Effective displays of children’s work can be created if
colour is used to emphasise the different arrangements of
the repeating unit within the tessellation.
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© 2000 Andrew Harris
5. Tessellations involving other irregular shapes
At this stage, it becomes sensible to widen the range of shapes to include all irregular shapes.
Some examples of activities might be ...
... investigating irregular polygons:
e.g. irregular pentagons
or
or
(This is useful to consider with children in order to avoid/confront the misconception that
‘pentagons don’t tessellate’ which arises when children over-generalise the non-tessellation of
regular pentagons and thereby assume that it is impossible to tessellate with any pentagon).
There are, currently, 14 known irregular pentagons which tessellate but no-one, to date, is
certain if any more exist.
... using letters of the alphabet:
H-shapes
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Y-shapes
E-shapes
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© 2000 Andrew Harris
... or shapes derived from circles:
e.g. using
to produce
... or to investigate the tessellating possibilities offered by the various sets of polyominoes
(extension of the idea of the domino):
The set of triominoes
The set of tetrominoes
Shapes composed of
3 squares
Shapes composed of
4 squares
All of the triominoes and the tetrominoes can be used as a repeating unit with which to
tessellate.
Shapes composed
of 5 squares. Only
some of these can
be used as repeating
unit with which to
tessellate.
The set of 12 pentominoes
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Hexominoes are composed of 6 squares (there are 35 of these). Each of these will tessellate.
Similar tessellation investigations can be carried out with polyiamonds (similar to polyominoes
but made of equilateral triangles instead of squares):
etc.
The set of triamonds
(just one!)
The set of tetriamonds
Use of ICT in Teaching Tessellation
There are several ICT software packages which address aspects of tessellation.
Most common are dedicated ‘tiling’ packages which allow children to select from a range of
different polygons and use them to tessellate, thereby using the computer screen as the surface
to be covered.
The advantage of using such a software package to do this is that it is relatively easy for a child
to obtain a paper-based record of his or her work via a printer and it bypasses the tedious and
time-consuming aspects of tessellating (drawing round templates and colouring in). There are
obvious advantages in relation to presentation of children’s work as well. If the computer is
linked to a large monitor, a large TV or a data-projector then a software package may be used
as a resource for direct teaching and discussion (perhaps within the mental/oral starter or
plenary of the daily mathematics lesson).
However, there are also some disadvantages involved with using ICT packages for this purpose.
One is that it is often difficult to measure and/or angles within shapes on a computer screen. the
screen size and resolution may also be a problem. Most packages limit the types of shapes
available to children (often to just regular polygons and perhaps a few common irregular
shapes such as rectangles). In some packages the emphasis is on making patterns (i.e. the
application of tessellation to design) rather than on the mathematical aspects of tessellation.
Both of these areas of knowledge have their value but it is important to identify which of these
aspects a software package addresses.
The best tiling packages allow the user to design their own ‘tile’ or repeating unit of shape(s).
The very best of these allow the user to translate or rotate portions of tessellating shapes so as to
form new, irregular, tessellating shapes (by the procedures outlined previously in the section
‘Tessellation of Irregular Shapes obtained by Transformation of Other Tessellating Shapes’)
which can themselves be used to tessellate across a surface.
Logo packages are also useful tools for exploring tessellation. When using Logo the onus is on
the child:
• to consider the interior and exterior angles of the shapes they are using (thus
developing their understanding of the mathematical reasons for tessellations being
feasible or infeasible),
• to identify what constitutes the repeating unit for their tessellation,
and
• to recognise how the repeating unit is replicated within that tessellation.
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The usual approach to tessellation with Logo is to encourage children to build up towards a
completed tessellation as follows:
1 Create one or more sub-procedures which generate each of the shapes involved in the
replicating unit;
2 Write a further sub-procedure which uses the shape sub-procedures to draw the unit
of shape(s) that will be repeated;
3 Create a final procedure which:
a) calls the sub-procedure that draws the repeating unit of shape(s);
b) repositions the turtle to draw the next repeating unit of shape(s);
c) repeats steps (a) and (b) until the desired surface area has been covered.
Just as for any other area of mathematics in which ICT is used, it is important to consider
whether the use of ICT is the most suitable tool/resource for achieving the desired learning or
teaching objectives and thereby use ICT as a means of teaching or learning only when it is
appropriate.
Resources
The photocopiable sheets of tiling mats which follow can be photocopied onto card or paper
and cut up to provide a selection of shapes for tiling large areas.
Typical activities for using these with children are:
• creating tessellations which show different repeating patterns. How many patterns are
possible?
• creating tessellations with closed or open patterns
• creating tessellations so that the pattern has the maximum number of regions possible
• is it possible to make pattern, make a larger version of the pattern which encloses the
first, and an even larger version enclosing that.... and so on?
• make a design with as many ‘triangles’ as possible.
• create a pattern and investigate what happens if you slide one row of mats sideways
• How many mats are needed to make the smallest possible square? And how many are
required for a slightly larger square? And the next square?
• creating tessellations which spell particular letters of the alphabet, words or numbers
• creating tessellations which have different kinds of symmetry (reflective, rotational,
translational)
• design your own tiling mat by drawing or using the computer. How versatile is it?
Use of these tiling mats adds an additional dimension to tessellation activities since, in addition
to considering the position of each regular polygon within the tessellation, the children also
have to think about the orientation of each tile and its contribution towards the pattern created
as well. Tiling mats are also good for encouraging collaborative work and for developing the
use and understanding of mathematical language.
They also have uses in the teaching of area: for example, estimating and calculating how many
tiling mats are needed to cover a given area (table-top, home corner, hall floor etc.).
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Hexagonal Tiling Mats
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Square Tiling Mats - 1
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Square Tiling Mats - 2
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Square Tiling Mats - 3
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Useful References: Tessellation
Deboys, M.
& Pitt, E.
Wells, D
1979
Lines of Development in Primary Mathematics Blackstaff Press
1991
The Penguin Dictionary of Curious and Interesting
Geometry
Penguin
Some Web site URLs
http://www.shodor.org/interactivate/activities/tessellate/index.html
http://users.erols.com/ziring/escher.htm
http://www.djmurphy.demon.co.uk/escher.htm
http://www.camosun.bc.ca/~jbritton/jbaraki.htm
http://www.camosun.bc.ca/~jbritton/jbsymteslk.htm
http://www.etropolis.com/escher/
http://www.iproject.com/escher/escher100.html
http://www.uvm.edu/~mstorer/escher/artgallery.html
http://www.WorldOfEscher.com/
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