Unit 12.5 Applying Geometry in Papua
New Guinean Arts
Topic 3: Application of patterns and shapes
in traditional arts
As part of your assessment in Grade 12 General Mathematics you will be required to
study an object of art from your local community. You will need to apply your knowledge
of shapes and patterns to analyse this object and use mathematical terms to describe it.
The examples and exercises in this Topic should help you and give you ideas.
ES
Having covered polyhedra and tessellations in Topics 1 and 2, Topic 3 looks at how
mathematical shapes are applied in the traditional patterns and measurement in PNG and
uses mathematical language to describe them.
Patterns and shapes in traditional arts
G
Gather some examples of art from your home or your local community. Apply your knowledge
of shapes and patterns to analyse these objects and use mathematical terms to describe them.
PA
For example, you can look at the decorative pattern of a work of art and describe it in terms
of polyominoes, poliamonds or other tessellations. You can identify lines of symmetry and
reflections or rotations of the basic pattern. If the work of art is three-dimensional you can
investigate volume or total surface area.
Example A
SA
M
PL
Analysis of pattern
E
Graph paper or isometric paper can be used to help with illustrations of patterns.
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334 Unit 12.5 Applying Geometry in Papua New Guinean Arts
The main pattern on the bag is reproduced below:
A monomino
A triomino
PA
G
ES
The pattern is a combination of polyominoes:
A pentomino
A triomino
The lighter-shaded triomino, illustrated above, is a rotation of the darker-shaded triomino, so
is the same trionimo.
PL
E
The pattern combination of these polyominoes could not be used to tile a rectangle but can
be used to tile the plane. This means that reproductions of the basic pattern, joined together,
will always have ragged edges.
The basic unit of pattern is given below. There is only one axis of symmetry for the basic unit
of pattern and, to tile the plane, reflections of the basic unit of pattern are required. This is
illustrated on the diagram below, at right.
SA
M
Basic units
Axis of symmetry
Basic unit of pattern
Reflections of basic units
Further work could be done on the other patterns on the bag and handle.The capacity
(volume) of the bag and the total surface area could also be investigated.
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Topic 3: Application of patterns and shapes in traditional arts 335
Example B
E
PA
G
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Analysis of shape
PL
The basic shape of this basket is that of a
truncated right cone. A truncated right cone
has the centre of the upper circular edge
vertically above the centre of the base circle.
SA
M
Truncated means cut off, so this cone is
inverted with the top cut off parallel to the
base of the cone (this becomes the open
top of the basket.)
top of
cone that
is removed
base of basket
open top of basket
32 cm
Measuring the basket gives the following
dimensions:
Diameter of top circular edge: 32 cm.
Diameter of base circle: 24 cm.
Height of the basket: 20 cm.
20 cm
24 cm
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336 Unit 12.5 Applying Geometry in Papua New Guinean Arts
The capacity (volume) of the basket can be found using the formula:
1
3
Volume = (B + B’ + BB’) × h where:
B is the area of the lower base.
B’ is the area of the upper base.
h is the vertical height of the truncated solid.
In this case:
1.The radius of the lower base is 12 cm, so the area of the lower base is
π ×122 = 452.39 cm2 so B = 452.39.
2.The radius of the upper base is 16 cm, so the area of the lower base is
π ×162 = 804.25 cm2 so B’ = 804.25.
3. h = 20 cm.
1
(452.39 + 804.25 + 452.39 × 804.25) × 20 = 12398.82cm3
3
ES
Volume of the basket =
The capacity of the basket is 12.398 or approximately 12.4 litres.
(There are 1 000 cm3 in one litre)
G
The total surface area (TSA) of a cone is given by the formula:
TSA = πrs + πr 2
s
PA
where:
r
πrs is the TSA of the curved surface of the cone and πr2 is the area of
the circular base.
The basket that we are studying has a net given in the diagram at right:
20
h
=
4
16
Giving: h =
PL
(tan θ =)
E
Using similar triangles and the ‘whole cone’:
20 ×16
= 80 cm
4
For the whole cone s = 162 + 802 = 81.58 cm .
Subtract from this 81.58 (the length of the sloping
edge of the basket) gives 81.58 – 20.40 = 61.18 cm.
M
16 cm
4 cm
θ
SA
The TSA of the curved surface of the whole cone
= πrs
20 cm
= π × 16 × 81.58
= 4 100.66 cm2
12 cm
The TSA of the discarded part of the cone:
= πrs
= π × 12 × 61.18
x = 202 + 42 = 20.40
h cm
= 2 306.43 cm2
Hence the TSA of the curved surface of the basket is:
4 100.66 – 2 306.43 = 1 794.23 cm2
61.18 cm
The area of the base of the basket = π × 122 =
452.39 cm2
Adding these two areas together gives:
The TSA of the basket = 2 246.62 cm2
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Topic 3: Application of patterns and shapes in traditional arts 337
Unit 12.5 Activity 3: Application of patterns and shapes in traditional arts
PA
G
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1. Apply your knowledge of shapes and patterns to analyse this bilum and use mathematical
terms to describe it.
SA
M
PL
E
2. Apply your knowledge of shapes and patterns to analyse the shape of this tray. Use
mathematical terms to describe it. The shape is part of a sphere where the diameter of the
outside circle of the tray (excluding the handles) is 25 cm and the greatest depth is 3 cm.
3. Find objects from your own home or local community and analyse them in mathematical
terms.
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