Star Polygons - Application of Cyclic Group
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The document discusses the mathematical concepts of groups, cyclic groups, and dihedral groups, particularly in relation to star polygons. It explores the properties of star polygons, their rotational symmetry, and their representation as cyclic group structures. Additionally, it highlights the significance of star polygons in art and culture, presenting historical contexts and examples.
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Star Polygons - Application of Cyclic Group
- 1. STAR POLYGONS As Cyclic Group Structures Presented by - Divyansh Verma M.Sc. Applied Mathematics SAU/AM(M)/2014/14 Email : itsmedv91@gmail.com
- 2. CONTENTS • What is a Group ? • What is a Cyclic Group ? • Dihedral Group Dn • Cyclic Group Cn • Star Polygons • Rotational Symmetry in Star Polygon • Star Polygon Cyclic Structure • Importance of Star Polygons in Art & Culture • Conclusion • Refrences
- 3. What is a Group ? A group consists of a set G together with a binary operation ' ' which satisfies∗ (1) Closure Property : g ∗ h G for all g, h G;∈ ∈ (2) Associative Property : g ∗ (h ∗ k) = (g ∗ h) ∗ k for all g, h, k G;∈ (3) Existence of Identity : there is an element 'e' in G which satisfies g ∗ e = e ∗ g = g for all g G ;∈ we call 'e' the identity of G; (4) Existence of Inverse : for each g G there is an element g∈ - 1 G satisfying g g∈ ∗ -1 = g-1 g = e ;∗ we call g-1 the inverse of g.
- 4. What is a Cyclic Group ? A group G is called cyclic if there exists an element a G such that G=<a>={a∈ n : n is an interger} i.e. there exists an element in group such that it can generate all the elements of the group. Also, if |G|= n, then an = e, where e is identity. Examples : (1) Z is cyclic; Z = <1> = <−1>. (2) Zn is cyclic; Zn = <1>. (3) The subgroup {(1), (123), (132)} is cyclic; it equals <(132)>.
- 5. Dihedral Group Dn A Dihedral Group Dn is a group of symmetries of a regular polygon with 'n' vertices defined for n≥3. Dihedral group deals with rotations and mirror reflections. It is not a cyclic group. Consider a polygon with 'n' vertices on the unit circle, with vertices labeled 0, 1, . . . , n -1 starting at 0 and proceeding counter-- clockwise at angles in multiples of 360 /n degrees, i.e. 2π/n⁰ radians. There are two types of symmetries of the polygon, each one giving rise to n elements in the group Dn : Rotations r0 , r1 , r2 ,..., rn-1 , where ri is rotation of angle 2πi/n. Reflections sr0 , sr1 , sr2 ,..., srn-1 where sri is reflection about the line through the origin and making an angle of πk/n with the horizontal axis. Order of Dn is '2n' as it has 'n' reflections and 'n' rotations.
- 6. s r s r 2 s s r s r 2 s r 3 s s r s r 2 s r 3 s r 4 s s r s r 2 sr 3 sr 4 s r 5 s D 4D 3 D 6D 5 1 r 1 1r 2 r 3 1 r r 2 r 3 r 4 1 rr 2 r 3 r 4 r 5D 4D 3 D 6D 5r 2 r 1 • The dotted lines are lines of reflection, and reflection of the polygon across each line brings the polygon back to itself.
- 7. 1 r 1 1r 2 r 3 1 r r 2 r 3 r 4 1 rr 2 r 3 r 4 r 5C 4C 3 C 6C 5r 2 r 1 Cyclic Group Cn A Cylic Group Cn is a group of symmetries with respect to rotations only. It has one type of symmetry i.e. 'n' rotations only which give rise to 'n' element of the group. It is therefore a subgroup of Dihedral Group Dn. Order of Cn is 'n'. Cn = { r0 , r1 , r2 ,..., rn-1 ; where ri is rotation of angle 2πi/n. }
- 8. Star Polygons Star Polygons are derived from Regular Polygons. Consider Regular Polygon with equally spaced 'n' vertices on the unit circle and connected every kth point for all choices of k. Regular Polygon with 'n' vertices connected with every kth point is referred as (n,k) Star Polygon. Interesting facts : The pattern include the fact that an (n, k) arrangement and an (n, n – k) arrangement are identical. All points on the circle will be connected only when n and
- 9. 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 S t a r P o ly g o n ( 5 ,2 ) , [ g c d ( 5 ,2 ) = 1 ] c r e a t e d b y c o n n e c t in g e v e r y 2 p o in t o f 5 e q u a lly s p a c e d p o in t s o n t h e c ir c le n d
- 10. ( 7 ,2 ) ( 7 ,3 ) ( 8 ,3 ) ( 1 0 ,3 ) ( 9 ,4 ) ( 1 1 ,4 ) ( 1 1 ,5 ) ( 1 2 ,5 ) O th e r e x a m p le s o f S t a r P o ly g o n ( n ,k ) w h e r e g c d ( n ,k ) = 1 , i.e . n a n d k a r e r e la tiv e ly p r im e
- 11. ( 6 ,2 ) , d e r iv e d fr o m 3 - g o n ( 9 ,3 ) , d e r iv e d fr o m 3 - g o n ( 1 0 ,4 ) , d e r iv e d fr o m S t a r P o ly g o n ( 5 ,2 ) ( 1 2 ,3 ) , d e r iv e d fr o m 4 - g o n E x a m p le s o f S t a r P o ly g o n ( n ,k ) w h e r e g c d ( n ,k ) 1 , i.e . n a n d k a r e n o t r e la t iv e ly p r im e , t h e s e s t a r p o ly g o n s a r e d e r iv e d fr o m r e g u la r p o ly g o n o r s ta r p o ly g o n s ( n ,k ) w h e r e g c d ( n ,k ) = 1 .
- 12. Rotational Symmetry in Star Polygons Star Polygons (n,k) when rotated about center at angles in multiples of 360 /n degrees, i.e. 2π/n radians illustrates⁰ rotational symmetry in there structure. For Example : Star polygon (5,2) when rotated 5 times at 360 /5=72 comes back to its original shape, which illustrates⁰ ⁰ its rotational symmetry. 7 2 ° 7 2 ° 7 2 ° 7 2 ° I II III 7 2 ° IV V V I
- 13. Similarly, all Star Polygons (n,k) satisfy the rotational symmetry, i.e. when they are rotated 'n' times at an angle 360 /n comes back to its original shape.⁰ So, we can say that all Star Polygons (n,k) are symmetric with respect to rotation. All above examples of Star Polygons (n,k) have rotational symmetry. ( 7 ,2 ) ( 7 ,3 ) ( 1 0 ,3 )( 8 ,3 ) ( 9 ,4 ) ( 1 1 ,4 ) ( 1 1 ,5 ) ( 1 2 ,5 ) ( 6 ,4 ) ( 1 0 ,4 ) ( 1 2 ,3 )( 9 ,3 )
- 14. Star Polygon Cyclic Structure Star Polygons (n,k) with gcd(n,k)=1 form a Cyclic Group Structure with respect to rotational symmetry and it is isomorphic to Cyclic Group Cn. Both have rotational symmetry defined for rotation at angle 360 /n.⁰ Star Polygons (n,k) with gcd(n,k)≠1 which are derived form a Cyclic Group Structures - Regular Polygons or Star Polygons (n,k) with gcd(n,k)=1 are isomorphic to a Subgroup of Cyclic Group Cn. Sub-Structure with 'm' vertics from which these Star Polygons are derived are isomorphic to some subgroup Cm of Cn.c
- 15. Importance of Star Polygons in Art & Culture The star polygons, derived from the regular polygons, were investigated in the High Middle Ages, specially well-known are the pentagram made from the pentagon as a secret sign of the Pythagoreans, and the Star of David made from the regular hexagon. Along with the regular symmetry figures in the plane, the symmetrical bodies of space have fascinated human beings from of old. In pre-Greek times some of these bodies already had cultic and religious symbolic value because of their regular construction and their crystalline structure.
- 16. Religious Symbols derived from Star Polygons secret sign of the Pythagoreans star of david islamic symbol star and cresent wheel of dharma bahai symbol
- 17. Symmetry in Human Body w.r.t Star Polygon A human body has a close resemblance to a five pointed star.
- 18. Star Polygons Designs in Architecture star fort in netherlands english cathedral architecture
- 19. Islamic Art Derived From Star Polygons window and door designs using star polygon
- 20. Star Polygons designs in nature star fish different types of flowers in star shape
- 21. Conclusion After this Study we can conclude that, for creating structures like Star Polygons of 'n' equally spaced vertices spread across a unit circle, we can always find 'kth ' point such that every kth point is connected for all choices of k (or in words) we can always find a one cyclic component (isomorphic to subgroup Cm of Cn) from which Star Polygons can be derived . These Star Polygons have importance in Art and Culture. And we can find many Patterns and Structures which are derived from them.
- 22. References Using Star Polygons to Understand Cyclic Group Structure, Sandy Spitzer, Bridges 2012. Symmetry and Complexity in Mathematics, Symmetry & Complexity, Klaus Mainzer, 2005. Aspects of Symmetry, Marlos Vaina, 2013 www.hyperflight.com/pentagon-construct.htm http://en.wikipedia.org/wiki/Cyclic_group
- 23. Thank You