Year 7 – Topology and Networks
Part 1 – Topology
Topology is the study of the effect of distorting a shape.
Shapes can be bent, twisted and flipped but must not be torn and no two points may be joined
together. The diagram below shows some topological transformations of a pin-man.
All of the pin-men above are topologically equivalent.
This is not equivalent to the shapes
above as a new join has been made.
This pin-man is not equivalent as
A line has been broken.
Part 1 - questions
1) A simple closed curve is any topological transformation of a circle. All the shapes below left are
simple closed curves. Copy the shapes on the right that are simple closed curves. Underneath
them right a sentence explaining why they are all simple closed curves.
Simple closed curves
Which are simple closed curves?
BCDP
2) From the eight shapes below pick out pairs of shapes that are topologically equivalent. Draw
them next to each other. Underneath the four pairs explain why they are topologically
equivalent.
3) Find out about Möbius Strip. Go to http://www.dhsbmaths.co.uk follow KS3, and Enrichment
projects, Networks.
Year 7 – Topology and Networks
Part 2 – Nodes, arcs and regions
A network is a collection of lines joined together at various points. The example below gives some
of the words we use to describe networks.
A 1 node

Not a node
An arc

A 3 node
A 4 node

Notice. Points are called nodes, Lines connecting nodes are called arcs. The number of arcs leaving
a node gives it a descriptive number, its order, e.g. a 3 node has three arcs leaving it. There cannot
be a 2 node, as this would be topologically equivalent to a single arc.
The final word we use is region. This is an area contained within arcs. The network above has three
regions, (labelled ) notice we count the outside as a region.
The example network has 3 nodes, 4 arcs and 3 regions.
Part 2 - questions
1) (a) Copy the eight letters shown below, making each one three lines high.
BCDE
FKOQ
A
3
1
3
1
(b) By each node write its order, as shown in the example.
2) Copy the networks. Write by each network the number of nodes, regions and arcs, as shown in
the example.
a)
b)
c)
d)
e)
Example
Nodes 4
Regions 4
Arcs 6
3) Copy the table and fill in the answers for the diagrams in question 2. Then add together the
number of nodes and regions for each diagram and compare the total with the number of arcs.
Write down any connection you find.
Year 7 – Topology and Networks
(a)
(b)
etc.
Nodes (N)
5
Regions (R)
5
Arcs (A)
8
4) Copy the nodes below. Make them about twice as far apart on your paper. Then draw the
networks with the given number of arcs.
(i) Four arcs
(ii) Five arcs
(iii) Four arcs
(iv) Six arcs
Does N + R = A + 2 for each diagram?
5) Find out about relationship between the number of faces, edges and vertices of threedimensional solids. Go to http://www.dhsbmaths.co.uk follow KS3, and Enrichment projects,
Networks for instructions.
Year 7 – Topology and Networks
Part 3 – Traversable networks
A traversable network is one which can be drawn with one pencil stroke without lifting the pencil
or going over any line twice.
Part 3 – Questions
1) Which of these networks are traversable?
(a)
(b)
(c)
(d)
(e)
2) Copy the incomplete networks below. Draw traversable networks using the given nodes. The
numbers tell you how many arcs are to leave that node. Be careful not to make any extra nodes.
(a)
(b)
3
2
(c)
4
4
4
4
3
3
1
3) Copy the networks below. By each node write its order. Which of the networks are traversable?
(a)
(b)
(c)
(d)
4) Copy the networks in Question 1. By each node write its order.
5) (a) Copy the table and fill in your answers to questions 1 and 3. You will need 10 rows in the
column including the headings. An odd node has order 1, 3, 5 etc. and an even node has order
4, 6, 8 etc.
1(a)
1(b)
etc.
Number of odd nodes
2
Number of even nodes
0
Traversable?
Yes
(b) Can you say which networks are traversable by the number of odd nodes that they have?
6) The Bridges of Könisberg problem. As before follow http://www.dhsbmaths.co.uk KS3 and
networks to get the instructions.
Year 7 – Topology and Networks
Part 4 – Route matrix
A matrix is a table of numbers. A route matrix describes a network. The plural of matrix is
matrices.
To
From
C
D
A
0
1
1
1
A
B
C
D
B
1
0
1
1
C
1
1
0
1
The loop from A
counts as 2.
(clockwise and
anticlockwise)
D
1
1
1
0
B
To
From
B
A
A
A
B
A
2
4
B
4
0
Part 4 – Questions
1) Write the route matrix for each diagram below.
(a)
A
B
A
(b)
(c)
A
B
(d)
B
A
B
C
D
C
C
D
C
2) Copy the two sets of nodal points below, making them about twice as far apart. Draw the
networks given by the route matrices, take care not to make any extra nodes.
A
A 0
B 2
C 1
To
B
2
0
1
A
B
C
1
1
0
(b)
From
From
(a)
C
To
A B C D
A 0 1 2 1
B 1 0 2 1
C 2 2 0 1
D 1 1 1 0
A
B
C
D
3) Copy the three sets of nodal points below, making them about twice as far apart. Draw each
network and write its route matrix. The figures show the order of each node. No loops are
allowed in this question.
(a)
A2
B3
(b)
A3
B3
(c)
A1
B3
D2
C2
E4
C3
C3
D3
Year 7 – Topology and Networks
4) A new pop group based in Taunton, have to arrange a tour to promote themselves. What is the
shortest route they can take to visit all the places in the topological map below. The route must
start and finish in Taunton.
153
Cardiff
56
49
64
109
69
Bristol
106
Oxford
36
43
Colchester
43
75
116
74 Taunton
146
91
67
65
116
London
77
78
63
139
Southampton
62
74
Brighton
Plymouth
72
Dover
Year 7 – Topology and Networks
Part 5 – One-way networks
A
B
A
B
This means ‘both ways’
This means form A to B only
Part 5 – Questions
1)
Write route matrices for the networks below.
(a)
(b)
B
A
(c)
B
A
B
A
D
C
C
2) Draw networks for the matrices given below.
A
A 2
B 1
C 0
To
B
2
2
1
(b)
C
1
0
2
(b)
B
C
To
B
1
2
2
A
A 0
B 0
C 1
A
From
From
(a)
A
C
2
0
1
B
C
3) As question 2. This time the column and row headings have been left out. This is the usual way
we write matrices.
(a)  1 1 2 


1 2 0
0 1 2


B
A
C
(b)  0 2 0 


1 0 1
 2 1 0


A
B
(c)  2 1 1 


 2 0 2
1 1 0


B
A
C
C
4) Write matrices, like question 3, for the networks below.
(a)
(b)
A
B
(c)
B
A
A
D
B
D
C
C