1 Algebra of operations Thm 1: Rotation about A thru angle  followed by translation T to axis = rotation thru same angle  about B on bisector of AA' at T/2 cot (  /2) from AA'

2 Algebra of operations Thm 1: Rotation about A thru angle  followed by translation T to axis = rotation thru same angle  about B on bisector of AA' at T/2 cot (  /2) from AA ' A T = B B A T = 1     

3 Algebra of operations Thm 1: Rotation about A thru angle  followed by translation T to axis = rotation thru same angle  about B on bisector of AA' at T/2 cot (  /2) from AA' A T = B B A T = 1 B A = T = T'       

4 Algebra of operations Thm 1: Rotation about A thru angle  followed by translation T to axis = rotation thru same angle  about B on bisector of AA' at T/2 cot (  /2) from AA' A T = B B A T = 1 B A = T = T'        A A' B

5 Algebra of operations Meaning of A B A: The operation B as transformed by A 2 B 3 (2B3) A (1B'4) 1 B' 4 1 A 2 2 B 3 3 A 4  A B B' A A A 

6 Algebra of operations Thm 2: Rotation about A thru angle  followed by translation T to axis = screw. Rotation & translation operations permutable A T = T A = A   

7 Algebra of operations Thm 2: Rotation about A thru angle  followed by translation T to axis = screw. Rotation & translation operations permutable A T = T A = A Then: T = T + T A T = A T T = B T = B      

8 Algebra of operations Thm 2: Rotation about A thru angle  followed by translation T to axis = screw. Rotation & translation operations permutable A T = T A = A Then: T = T + T A T = A T T = B T = B       A A' B screw axis

9 Algebra of operations Intersecting mirrors: m m = A Parallel mirrors: m T = m    

10 Algebra of operations Inversion: i T = A m T = A m = i     

11 Algebra of operations Inversion: i T = A m T = A m = i Glides: Define: m = m t     

12 Algebra of operations Inversion: i T = A m T = A m = i Glides: Define: m = m t        Glide symbols axialdiagonaldiamond 1/8 ca b dn

13 Algebra of operations Inversion: i T = A m T = A m = i Glides: Define: m = m   m T = m T T  m T = m            

14 Algebra of operations Inversion: i T = A m T = A m = i Glides: Define: m = m   m T = m T T  m T = m            

15 Algebra of operations Glides:  m T = m T T  = m T T  = m T  = m          

16 Algebra of operations Glides:  m T = m T T  = m T T  = m T  = m          

17 Algebra of operations 2 screw & glide:  A m = A t m  = A m t'    t     

18 Algebra of operations 2 screw & glide:  A m = A t m  = A m t'  = i t'  = i t = i      t        

19 Algebra of operations 2 screw & glide:  A m = A t m  = A m t'  = i t'  = i t = i      t         t 1 t 1  i 1 i 2

20 Algebra of operations 2 screw & glide:  A m = A t m  = A m t'  = i t'  = i t = i      t         t 1 t 1  i 1 i 2

21 Algebra of operations 2 screw & glide:  A m = A t m  = A m t'  = i t'  = i t = i      t         t 1 t 1  i 1 i 2

22 Algebra of operations 2 screw & glide:  A m = A t m  = A m t'  = i t'  = i t = i      t         t 1 t 1  i 1 i 2

23 Algebra of operations 2 screw & glide:  A m = A t m  = A m t'  = i t'  = i t = i      t         t 1 t 1  i 1 i 2

24 Algebra of operations Two 2-fold axes at angle  apart: A B = C' (from Euler calc.)     2 2  C

25 Algebra of operations  C  In general: A B = C' If screw axes: A B = t A B t = t C' t  t C t = C'         t         C' C" B A t 2 t 1 t 1 t 2 +

26 Algebra of operations   C' C" B A t 2 t 1 t 1 t 2 + In general: A B = C' If screw axes: A B = t A B t = t C' t  t C t = C'  t C' t = C t t  = C"         t  C            

27 Algebra of operations   C' C" B A t 2 t 1 t 2 + In general: A B = C' If screw axes: A B = t A B t = t C' t  t C t = C'  t C' t = C t t  = C"         t  C      t 1       Not a screw axis - no t II C

28 Algebra of operations If screw axes: A B = t A B t = t C' t  t C t = C'  t C' t = C t t  = C"       t  C   t      C' C" B A t 2 t 1 t 1 t 2 +     If A, B do not intersect, but are separated in direction along C, C" has screw axis translation component