10
POLYHEDRA AND NETWORKS
World is made up of solids of different shapes, sizes and materials. Among the
important subsets of the solids there are solids, which are bounded by plane surfaces
only. These solids are called Polyhedra.
You will come across many solids such as, a sweet box, a hexagonal prism, a
cubical shaped eraser, an instrument box, a glass prism etc., all these have polygonal
faces and are examples of polyhedra.
You are already familiar with Polygon. Let us revise and learn some more related
terms and definitions.
1. Polygon :
A closed figure bounded by straight line segments.
The region bounded by a polygon is polygonal region.
2. Regular Polygon :
It is a polygon having equal sides and equal angles.
All regular polygons are cyclic.
Think!
A polyhedron has
only area, but no
volume.
3. Polyhedron :
A closed figure in the space bounded by polygonal faces is a polyhedron. A
polyhedron divides the space into two regions, within the polyhedron and outside it.
Polyhedra is the plural of polyhedron.
4. Polyhedral solid :
A solid bounded by a polyhedron is a polyhedral solid. For a polyhedron, we denote
the number of faces by F, edges by E and vertices by V.
5. Regular Polyhedra :
A polyhedron is called a regular polyhedron, if its faces are congruent regular
polygons. There are only five types of regular polyhedra. They are
1.
Tetrahedron
2.
Hexahedron
3.
Octahedron
4.
Dodecahedron and
5. Icosahedron
These five polyhedra are known as ‘Platonic Solids’.
Know this :
Name of these solids are based on the number of sides of the faces.
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It is believed that Plato associated the Tetrahedron, Octahedron, Cube and
Icosahedron, with the four elements, fire, air, earth and water and saw that
the Dodecahedron as a symbol of the universe as a whole.
Suggested Activity
Regular Polyhedra and Polyhedral graphs are shown below. Draw the graph of
each Polyhedra on a cardboard, as shown in the table. Prepare Polyhedron using
cellophone tape.
Name
Polyhedral solid
Graph
Tetrahedron
Shape of
each face
Equilateral triangle
Hexahedron
Square
Octahedron
Equilateral triangle
Dodecahedron
Regular Pentagon
Icosahedron
Equilateral triangle
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Activity : Study the polyhedral solids given below and prepare graphs
Square based
Pyramid
Triangular
based prism
Square
based prism.
Tabulate the number vertices, faces and edges of the following polyhedral solids.
Sl.
Name of the
No. of
No. of
edges
F+V
E+2
No.
Polyhedra
faces
vertices
F
V
E
1.
Tetrahedron
2.
Hexahedron
3.
Octahedron
4.
Dodecahedron
5.
Icosahedron
6.
Triangular Prism
7. Square based prism
8. Triangular Pyramid
Study the data you have recorded. Find the relation
between F + V and E
You can observe that F + V = E + 2. This formula
is known as Eulers’ formula.
This formula was proved by
Leonard Euler in 1735. This is
true for all polyhedral solids
Exercise : 10.1
1) Find the number of faces, vertices and edges in each of these polyhedral solids
and verify Euler’s formula.
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Plato
Plato a Greek mathematician
and a great philosopher,
believed that mathematics and
philosophy were very closely
related. He founded the Academy
over the entrance to the Academy
was inscribed : “Let no one enter
who is ignorant of geometry.”
2) Find the names of some polyhedral crystals.
3) Find F, V, and E for the following solids and verify Euler’s formula (i) Pentagonal
based prism (ii) Hexagonal based prism (iii) Octogonal Pyramid.
4) Using Cardboard construct the following polyhedra :
(i) A square based prism (ii) Square based pyramid (iii) Hexagonal based pyramid.
GRAPHS
Some important tourist spots located at Bangalore and the roads linking them is
shown below.
Bangalore Palace
Railway station
Vidhana Soudha
Leonard Euler
Leonard Euler (1707-1783) a Swiss
mathematician, introduced the idea
of using circles for representing sets.
He also gave a solution to the
traversablity of graphs.
Lalbagh
Bull temple
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In the same way, hand shakes mutually exchanged by four friends A, B, C and
D can also be represented by using points and line segments as given below.
Such a diagramatic representation of joining points by line segments is an example
of a graph or a network.
The different places covered by a Bus in a city, the pipe connections to water
taps, the electric circuit connecting the lighting points etc., can be represented by networks.
Graphs are studied in ‘Graph Theory’ which is a branch of mathematics.
Graph theory has its application in various fields such as electronics
electrical engineering, network analysis and graphs etc.,
In this chapter some basic concepts related to graphs are discussed.
1. Graph : A set of points together with line segments joining the points in pairs is
a graph or network.
2. Nodes in a graph : A point is a node if there is atleast one path (line) starting
from it or reaching it.
Nodes are named by capital letters of English alphabet. However the number of nodes
is denoted by ‘N’.
3. Arcs in a graph : The line segment (path) joining two nodes is an arc. Number
of arcs of a graph is denoted by ‘A’.
4. Region : An area bounded by arcs
(including outside) is called a region. Number
of regions is denoted by ‘R’
Note : In a network an arc may
be a straight line or a curved line.
Worked Examples :
1)
Find the number of nodes, arcs and regions in the following networks.
Solution : In this graph, A, B, C, D and E are nodes.
AB, BC, CD, DE, AE and AC are arcs.
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The regions are, the area within ABC, area
within ACDE and the area outside ABDE.
The number of nodes = N = 5
The number of arcs = A = 6
The number of regions = R = 3
2)
Find N, A and R of the following graph
In the network nodes are A, B, C and D
The Arcs are 1, 2, 3, 4, 5 and 6
The regions are a, b, c and d.
The number of nodes = N = 4
The number of Arcs = A = 6
The number of regions = R = 4
3)
Find the number of nodes and arcs in the
graph given below
Solution : The nodes are A, B and C. The
arcs are AB, AC and the arc joining A to A.
∴ The number of nodes = N = 3
The number of arcs = A = 3.
4)
Four friends meet in a party and shake their
hands mutually. Draw a graph. Write the
number of nodes and arcs in it.
Let A, B, C and D be four friends.
The line segments joining A, B, C and D
represent the mutual handshakes.
Here the nodes are A, B, C and D and the
arcs are AB, BC, CD, DA, AC and BD
∴ N = 4 and A = 6
Note : In the figure, The point of intersection of AC and BD is not taken
as a node because A shakes hand with C and B shakes hand with D and
it is represented by arcs AC and BD.
Therefore in a network, a point which is not named, is not taken as a
node. It implies that no path is starting from it or reaching it.
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Exercise : 10.2
1) Find the number of nodes, arcs and regions in the following networks
(i)
(ii)
(iii)
(iv)
2) Draw a network, showing the location of your house, school, post office and a
hospital and the roads connecting them.
TRAVERSABLE GRAPHS
Activity : Draw the following graphs, in one sweep without lifting the pencil from the
paper and without tracing any arc from start to finish. Mark the starting point as ‘S’.
(5)
(1)
(2)
(3)
(6)
(4)
(8)
(9)
(7)
Which of the above graphs can be drawn in one sweep? You observe that the
graphs, 1, 2, 3, 4, 5 and 7 can be drawn in one sweep, without lifting pencil from
the paper.
Rest of the graphs 6, 8 and 9 can not be drawn as directed.
1. Condition for Traversable networks :
A graph is said to be traversable if it can be drawn with one sweep, without
lifting the pencil from the paper and tracing the same arc twice. It can pass through
the nodes several times.
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An Ancient problem of seven bridges.
The problem is about the seven bridges of Konigsberg,
a small German university town, which was at the
mouth of Pregal river. In the river there were two
islands, linked to the mainland and to each other
by seven bridges as shown below.
People of Konigsberg tried to cross all the seven bridges in one walk without
crossing any bridge more than once. They could not, no matter how many times they
tried. Repeated trials led to the belief that it was impossible to cross the bridges, satisfying
the conditions. This problem was submitted to Euler, a Swiss mathematician. He proved
mathematically that it was impossible to cross the bridges.
Euler simplified the problem by replacing the
land by points and the bridges by lines connecting
these points, as shown in the figure.
Is it possible to trace the entire diagram without lifting the pencil and without tracing
any path twice? In other words, is it traversable?
To solve such problems of graphs, you require the knowledge of graphs and related
concepts.
2. The order of a node in a graph
The order of a node is the number of paths starting from it or reaching it.
Consider the graph of seven bridges, you have.
Node
A
B
C
D
Order of the node
3
5
3
3
Loop : A single arc which connects a node to itself is a loop.
3. The order of a node with a loop
Ex. 1 A loop as shown in the adjoining graph,
can be traced in the clockwise as well as
in the anticlockwise directions. Therefore
the order of a node with a loop is 2.
Ex. 2
In the adjoining figure, the order of the
node P is 4. Why?
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4. Even and Odd nodes :
A node is called an even node if its order is an even number.
A node is called an odd node if its order is an odd number.
If the order of a node is 1, then it is called 1-node, if the order is 2, then it
is 2-node. If the order of a node is ‘n’ then, it is n-node.
5. Conditions of Traversibility :
Activity : Draw the following graphs and record your observations in the table.
Graph
number of
number of
traversable or
odd nodes
even nodes
not traversable
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Study the observations you have recorded in the table.
When do you say that a graph is traversable?
You can see that, a network is traversable.
When (i)
(ii)
It has only even nodes
It has only two odd nodes.
6. Euler’s solution for traversability of a graph
Euler discovered that, a graph is
Euler’s analysis of seven bridges
(i) traversable, if it has only even nodes.
problem was the first hint for a new
(ii) traversable, if it has only two odd nodes.
branch of mathematics Topology,
which reached its highest
(iii) not traversable if it has more than two
development in the twentieth century.
odd nodes.
Exercise : 10.3
1) For each graph given below, find the order and type of nodes.
(i)
(ii)
(iv)
(iv)
(iii)
(v)
(vi)
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2) Which of the following networks are traversable? Give reasons.
(ii)
(i)
(iii)
(vi)
(iv)
(v)
3) Draw 3 traversable networks of your own choice, showing the starting point and
the end point.
4) From the Alphabet, pick out one alphabet which has (i) one-3 node (ii) two-3 nodes
(iii) one 4-node.
5) Draw the graph with (i) four-4 nodes (ii) two-2 nodes
7. Matrix of a graph :
The information regarding the number of arcs, connecting the nodes can be
displayed by a matrix as illustrated below.
In the graph, there are three nodes, A,
B and C and these are connected
mutually by arcs.
In
*
*
*
the
A
B
C
above network,
is connected to B by 3 arcs and not connected to C.
is connected to A by 3 arcs and connected by C by 2 Arcs.
is connected to B by 2 arcs and not connected to A.
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This information can be tabulated as follows
A
A
0
B
3
C
0
B
3
0
2
C
0
2
0
Note
If a node is not connected to itself
or to any other node then it is
indicated as Zero
 0 3 0


3 0 2

The matrix representation of the above table will be
 0 2 0


This is the matrix of the graph or network
Activity : Construct the matrix for each of the following networks. Record your
observations in the table
(i)
(ii)
(iii)
(v)
Figures
(iv)
(vi)
Sum of the order of the nodes
(i)
(ii)
(iii)
(iv)
(v)
(vi)
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No. of Arcs
From the observations you have recorded, what is your conclusion about the
relationship between sum of the orders of the nodes and the number of arcs?
It can be seen that,
The sum of the elements in the matrix is the sum of the orders of
the nodes, which is equal to twice the total number of Arcs in the
graph.
8. To draw a graph from a given matrix
Ex. Consider the matrix
 0 1 2


 1 0 3
 2 3 0


A
B
C
A
0
1
2
B
1
0
3
C
2
3
0
Since it is a 3x3 matrix, the graph represented by the matrix has 3 nodes. Let
the nodes be A, B and C.
Step 1 : Mark three points A, B and C
Step 2 : From the matrix, you see that from A to
B there is only one arc. From A to C there
are 2 arcs. Therefore draw one arc from A
to B and two arcs from A to C. Similarly,
draw three arcs from B to C and complete
the graph as shown.
9. Euler’s formula for graphs
Activity : Study the graphs and tabulate the number of nodes, Ares and Regions as
shown below
Number of
Graph
N
265
R
A
N+R
A + 2
Number of
Graph
N
R
A
N+R
A + 2
In the above table, the last two columns are identical and we can observe that
N + R = A + 2.
This formula N + R = A + 2 is known as Euler’s formula, which holds
good for all graphs
Exercise : 10.4
1) Construct the matrix for each of the following graphs.
(ii)
(i)
(iv)
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(iii)
2) Draw the graphs for the following matricies.
(i)
 0 2 2


 2 0 1
 2 1 0


(iv)  0 1 2 


1 0 1
 2 0 0


(ii)
 0 2


 2 0
(iii)
(v)  2 1 0 
1 4 1
 0 1 2


 0 1 0


 1 0 1
 0 1 0


(vi)  2
0
0

1

0 0 1

2 0 1
0 2 1

1 1 2 
3) Verify Euler’s formula for the following graphs
(ii)
(i)
(iii)
(iv)
(v)
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(vi)