Our universe is full of unexplored beauties of nature which hide different symmetries and patterns in it. Scientists are continuously defining new bounds to the existing knowledge of numbers and figures and their relationships. Graphical methods were first used in the theory of groups in

and

A circuit of type

We now divide all the circuits of type

Let

Let

Let

The Shortest number of edges linking two ambiguous numbers, say

It has been able to establish the connection between reduced numbers of above three genres.

Similar can be done for

Similarly

It is clear that

(ii) Straightforward from proof of (i

(iii) Again from [

(i)

(ii)

(iii)

also

Proof: It is obvious from [

Following theorem will elaborate some properties of genre C.

(a) As a real number the whole part of

(b) The numeric part of continued fraction of

(c)

(d) For any non-zero positive integer

First suppose

Secondly suppose

[

Sr. No | Number in Coset diagram | Continued fraction |
---|---|---|

1 | 1+132 | [2;3-] |

2 | ξ3=3+132 | [3;3-] |

3 | 5+132 | [4;3-] |

4 | 1+136 | [0;1,3-] |

5 | 7+136 | [1;1,3-] |

6 | 5+136 | [1;2,3-] |

On contrary suppose that

Then

Gives

Similarly suppose on contrary

Gives

This shows that for any particular type

Similarly suppose

Now it can be generalized that our result regarding the location of reduced quadratic irrational number occurring in the closed-circuit of length 2 on real line for genre A.

Now it will be possible to derive a general result.

Then

Gives

Also, if

Therefore, conclusion can be derived that for any type of length two the reduced number has specific limits as a real number.

Now we take a specific interval of length, one on real line, it becomes interesting to know how many distinct reduced numbers can have in this particular unit the interval.

It is clear from the definition of reduced number that every reduced number lies between

In this section, group of permutation is used to classify G-circuits therefore, square brackets represent the type of G-circuit and round brackets to represent permutations.

We now formally define the notion of equivalent, cyclically equivalent and similar circuits in G-orbits of

If

We have proved that there are exactly four classes of equivalent circuits of length 4 namely

Now reversion towards the application and physical interpretation in this area of research with a consideration of finding non-equivalent circuits of length

Given the integer

Some others are

The number of non-equivalent circuits of length two are only 2. For the circuit of length 2,

It is interesting to note that a circuit of length 2 corresponding to the partition

Proof: Every time in determining all non-equivalent circuits of length

Since we have such an explicit description of determining the non-equivalent circuits of length

1.

2.

3.

4.

Hence, all the circuits of length 4 are

For the number of possibilities of selecting r distinct objects from n objects, where the order of arrangements is considered, are

The number of different arrangements of n objects of which

Therefore, for

We claim that all above twenty-four circuits are divided in three sub classes of equivalent circuits called classes of cyclically equivalent circuits. This claim is proved in the following theorem.

We prove the relationship between

θ∈D4 | [aθ(1),aθ(2),aθ(3),aθ(4)] |

(1)(2)(3)(4) | [a1,a2,a3,a4] |

(1234) | [a2,a3,a4,a1] |

(13)(24) | [a3,a4,a1,a2] |

(1432) | [a4,a1,a2,a3] |

(12)(34) | [a2,a1,a4,a3] |

(14)(23) | [a4,a3,a2,a1] |

(24) | [a1,a4,a3,a2] |

(13) | [a3,a2,a1,a4] |

θ∈(23)D4 | [aθ(1),aθ(2),aθ(3),aθ(4)] | θ∈(12)D4 | [aθ(1),aθ(2),aθ(3),aθ(4)] |

(23) | a1,a3,a2,a4 | (12) | a2,a1,a3,a4 |

(134) | a3,a2,a4,a1 | (234) | a1,a3,a4,a2 |

(1243) | a2,a4,a1,a3 | (1324) | a3,a4,a2,a1 |

(142) | a4,a1,a3,a2 | (143) | a4,a2,a1,a3 |

(14) | a4,a2,a3,a1 | (1423) | a4,a3,a1,a2 |

(123) | a2,a3,a1,a4 | (132) | a3,a1,a2,a4 |

(1342) | a3,a1,a4,a2 | (34) | a1,a2,a4,a3 |

(243) | [a1,a4,a2,a3] | (124) | [a2,a4,a3,a1] |

It is interesting to see that in this case

These twenty-four circuits in 3 classes can be written depicted by

Such that

Next corollary will classify G-orbits of

All the circuits in

We put

Similarly, we can prove it for

Moreover, all the circuits in

under all the

Now we explore three more possible circuits of length four in

Following results can be easily deduced from subsection 3.2.

Hence

[Table 4] shows the complete classification of circuits of length four.

This is now clear from [Table 4] that for all 44 circuits of length 4, we obtain only 7 corresponding values of

Equivalence Class E[_,_,_,_] | E[_,_,_,_] | Cyclically Equivalence class Ec[_,_,_,_] | Ec[_,_,_,_] | Distinction of orbits ∈Q*(_) |

E[a1,a2,a3,a4] | 24 | Ec[a1,a2,a3,a4] | 8 | γG,γ-G,-γG,-γ-G ∈Q*(n2) |

Ec[a1,a3,a2,a4] | 8 | γG,γ-G,-γG,-γ-G ∈Q*(n1) | ||

Ec[a2,a1,a3,a4] | 8 | γG,γ-G,-γG,-γ-G ∈Q*(n3) | ||

E[a1,a1,a2,a3] | 12 | Ec[a1,a1,a2,a3] | 8 | γG,γ-G,-γG,-γ-G∈Q*(n4) |

Ec[a1,a2,a1,a3] | 4 | γG=-γ-G, -γG=γ-G∈Q*(n5) | ||

E[a1,a1,a1,a2] | 4 | Ec[a1,a1,a1,a2] | 4 | γG=γ-G, -γG=-γ-G∈Q*(n6) |

E[a1,a1,a2,a2] | 4 | Ec[a1,a1,a2,a2] | 4 | -γG=γ-G, -γ-G=γG∈Q*(n7) |

It is clear from [Table 3] that there are only two classes of equivalent circuits which corresponds to more than one orbits of

From Corollary 3.2.6 we have,

Now consider

Now using above substitutions we have,

after simplification, we have,

Since

Similarly, we can prove

Since

Thus

Discussion about the different properties of reduced numbers and distributed types of length 2 in three different categories leads to the reader to understand more deeply the role of reduced numbers in the coset diagram with the help of continued fractions. Distance in coset diagram is defined and new results are derived in this sense. General results are obtained by considering these reduced numbers on real line which makes easy for us to understand their behavior. This work can be used to generalize the properties of types of length 4 and so on. Reduced numbers have upper and lower bounds as real numbers in a specific type of length 2. Classification has been done for all the circuits of length four under the action of modular group into different classes and sub classes of circuits. These provide answer to the question that for given four positive integers which need not to be distinct, how many circuits of length four exist? Moreover, how many to these circuits are contained in the same orbits of