Finite triangular grids

8/7/2019 Finite triangular grids

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FINITE TRIANGULAR GRIDS


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The shape below is a finite triangular grid of

order 1 because the least number of arcs used in

path from node A to Nodes (1,2) is 1.

A

1 2


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The shape below is a finite triangular grid of

order 3 because the least number of arcs used in

path from node A to Nodes (1,2,3,4,5,6) is 3.

A

1 2 3 4


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How many small triangles are there in a finite

triangular grid of order 4?

16 small triangles

1

3

2

97

6

4

5

8

10

5

12

11 13

14

15

16


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What About an finite triangular grid of order 5?


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To make it easy on our selves we can use this law

Number of small triangles = n2

Where n is the finite triangle order number

n Nas we can see a triangular grid of order 1

has only one triangle, which is 12 =1, atriangular grid of order 2 has 4 small triangles

which is 22 =4.

a triangular grid of order 3 has 9 smalltriangles which is 32=9, and so on ..


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Also for the number of nodes we dont want to

bother ourselves counting them ,and checking

our answers, consequently


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We can find out the No. of nodes by two ways1st one which is a recursive formula:

Un = Un-1 + (n+1), Ex: n=3

U3 = 6+(3+1)=6+4=10

or by the given formula : (an2+bn+c)= No. of nodes.

Where a=1, b=3, c=2

which were found from the following solution :


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a (1)2+b(1)+c=6

a+b+c=6 1

a (2)2+2b+c=124a+2b+c=12 2

9a+3b+c=20 3

4a+2b+c=12 2

a+b+c=6 1

3a+b=6 4


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9a+3b+c=20 3 b=3

4a+2b+c=12 2 b and a in 15a+b=8 5 1+3+c=6

5a+b=8 c=2

3a+b=6

2a=2a=1 in 5

5(1)+b=8


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Regarding the nodes, it is very difficult to

determine them manually. Thus this law is very

useful to us when counting nodes.

Number of nodes= (n2+3n+2), where n is the

order of the finite triangular grid.


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Testing this on finite triangular grid of order 5.

(52+3(5)+2)=21


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Now well prove that the rule we conclude is

equal to the rule given ( (n2+3n+2))

So by mathematical induction.

Let p(n)=1+2+3+4+.+n+(n+1) = ( (n2+3n+2)),n is the order number, n N .

P(n) is true for n=1.

e.g. 1+(1+1)=( (12+3+2)) = 3

Assume that p(n) is true for all n=k ,k N .e.g. p(k)=1+2+3+4+.+k+(k+1) = ( (k2+3k+2))


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Testing the validity of n=k+1

e.g. p(k+1)=1+2+3+4+.+k+(k+1)+(k+2) = (

(k2+5k+6)).

L.H.S= (k2+3k+2) +(K+2)

= (k2+5k+6).

p(n) is true for all n N .


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Maybe we wonder about the ratio or the

proportion between number of small triangles

and those of nodes, as each triangle has 3 nodes

but it may share them with other triangles. To

stop this sophisticated ideas lets together

observe the ratio between them.


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Grid order of the

infinite triangular

grid

Number of small

triangles

Number of nodes

1 1 3

2 4 6

3 9 10

4 16 15

5 25 21

6 36 28

7 49 36

8 64 45

9 81 55

10 100 66


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To get the ratio we will number of nodes on

number of small triangles for each infinite

triangular grid ,and then getting the mean.

Mean= 0.5623

So the ratio is 0.6 approximately

Which means that the number of nodes is half the

number of the small triangles for almost all


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FINITE TRIANGULAR GRIDSDone by:

Ahmed Samaka.

Zinone Nabil.

Yaacoub Jasim

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Finite triangular grids