Symmetries of an Equilateral What are the symmetries of an equilateral triangle? Triangle C R1R2 FAFBFC ID counting Composition Groups A B In order to answer this question precisely, we need to agree on what the word ”symmetry” means.
What are the symmetries of an equilateral triangle? Symmetries of an Equilateral Triangle C R1R2 FAFBFC ID counting Composition Groups A B For our purposes, a symmetry of the triangle will be a rigid motion of the plane (i.e., a motion which preserves distances) which also maps the triangle to itself. Note, a symmetry can interchange some of the sides and vertices.
Symmetries of an Equilateral Triangle So, what are some symmetries? How can we describe them? What is good notation for them? R1R2 FAFBFC C ID counting Composition Groups A B
Symmetries of an Equilateral Triangle Rotate counterclockise, 120 ◦ about the center O . R1R2 C FAFBFC ID counting Composition Groups O A B
Symmetries of an Equilateral Note this is the following map (function): Triangle C B R1R2 FAFBFC ID counting O O Composition Groups A B C A We can think of this as a function on the vertices: A �→ B , B �→ C , C �→ A . � A � B C We might denote this by: B C A We also may denote this map by R 120 .
Symmetries of Rotate counterclockise, 240 ◦ about the center O . This is the an Equilateral Triangle map (function): R1R2 C A FAFBFC ID counting Composition O O Groups A B B C We can think of this as a function on the vertices: A �→ C , B �→ A , C �→ B . � A � B C We might denote this by: C A B We also may denote this map by R 240 .
Symmetries of Reflect about the perpendicular bisector of AB : an Equilateral Triangle C R1R2 FAFBFC ID counting Composition Groups A B
Symmetries of Reflect about the perpendicular bisector of AB , an Equilateral Triangle This is the map (function): C C R1R2 FAFBFC ID counting O O Composition Groups A B B A We can think of this as a function on the vertices: A �→ B , B �→ A , C �→ C . � A � B C We might denote this by: B A C We also may denote this map by F C to indicate the reflection is the one fixing C .
Symmetries of Reflect about the perpendicular bisector of BC , an Equilateral Triangle This is the map (function): C B R1R2 FAFBFC ID counting O O Composition Groups A B A C We can think of this as a function on the vertices: A �→ A , B �→ C , C �→ B . � A � B C We might denote this by: A C B We also may denote this map by F A to indicate the reflection is the one fixing A .
Symmetries of Reflect about the perpendicular bisector of AC , an Equilateral Triangle This is the map (function): C A R1R2 FAFBFC ID counting O O Composition Groups A B C B We can think of this as a function on the vertices: A �→ C , B �→ B , C �→ A . � A � B C We might denote this by: C B A We also may denote this map by F B to indicate the reflection is the one fixing B .
The identity map of the plane: (takes every point to itself). Symmetries of an Equilateral This is the map (function): Triangle C C R1R2 FAFBFC ID counting O O Composition Groups A B A B We can think of this as a function on the vertices: A �→ A , B �→ B , C �→ C . � A � B C We might denote this by: A B C We also may denote this map by Id or 1 . Note, we might also denote this as R 0 , since it is a rotation through 0 ◦ . However – it is NOT a reflection. (WHY NOT??!!)
So far we have 6 symmetries – 3 rotations, R 0 , R 120 , R 240 , and Symmetries of an Equilateral 3 reflections, F A , F B , F C . Triangle C R1R2 FAFBFC ID counting Composition Groups O A B Are there any more?? Why or why not??
Symmetries of an Equilateral Triangle In fact these are all the symmetries of the triangle. R1R2 We can see this from our notation in which we write each of FAFBFC � A � B C these maps in the form . Note there are three ID X Y Z counting choices for X (i.e., X can be any of A , B , C , ). Having made a Composition choice for X there are two choices for Y . Groups Then Z is the remaining vertex. Thus there are at most 3 · 2 · 1 = 6 possible symmetries. Since we have seen each possible rearrangement of A , B . C is indeed a symmetry, we see these are all the symmetries.
Symmetries of an Equilateral Triangle Notice these symmetries are maps, i.e., functions, from the R1R2 plane to itself, i.e., each has the form f : R 2 → R 2 . Thus we FAFBFC can compose symmetries as functions: If f 1 , f 2 are symmetries ID then f 2 ◦ f 1 ( x ) = f 2 ( f 1 ( x )) , is also a rigid motion. Notice, the counting composition must also be a symmetry of the triangle. Composition Groups For example, R 120 ◦ F C =?? It must be one of our 6 symmetries. Can we tell, without computing whether it is a rotation or reflection?? Why?? What about the composition of two reflections?
R 120 ◦ F C , we can view this composition as follows: Symmetries of an Equilateral Triangle C C R1R2 F C FAFBFC ID O O counting B A A B Composition Groups A C R 120 O O B A C B So, R 120 ◦ F C = F B .
Symmetries of an Equilateral Triangle R1R2 FAFBFC ID We use our other notation: counting � A � � A � � A � B C B C B C R 120 ◦ F C = ◦ = = F B Composition B C A B A C C B A Groups
Is R 120 ◦ F C = F C = R 120 ? Let’s look: F C ◦ R 120 : Symmetries of an Equilateral Triangle C B R 120 R1R2 FAFBFC O ID O counting C A A B Composition Groups B B F C O O C A C A So F C ◦ R 120 = F A � = F B = R 120 ◦ F C .
So on our set of symmetries S = { R 0 , R 120 , R 240 , F A , F B , F C } , Symmetries of an Equilateral we get a way of combining any two to create a third, i.e., we Triangle get an operation on S . (Just like addition is an operation on the integers.) We will call this operation multiplication on S . R1R2 We can make a multiplication table, or Cayley Table . So far FAFBFC we have: ID counting ◦ R 0 R 120 R 240 F A F B F C Composition R 0 R 0 R 120 R 240 F A F B F C Groups R 120 R 120 F B R 240 R 240 F A F A F B F B F C F C F A Notice we have already seen F C ◦ R 120 � = R 120 ◦ F C , so this operation is non-commutative .
Now we fill in the rest: (check) Symmetries of an Equilateral Triangle ◦ R 0 R 120 R 240 F A F B F C R 0 R 0 R 120 R 240 F A F B F C R1R2 R 120 R 120 R 240 R 0 F C F A F B FAFBFC R 240 R 240 R 0 R 120 F B F C F A ID F A F A F B F C R 0 R 120 R 240 counting F B F B F C F A R 240 R 0 R 120 Composition F C F C F A F B R 120 R 240 R 0 Groups We make note of several things about this table: (i) Every symmetry appears exactly once in each row and in each column; (ii) Every symmetry has an ”opposite” or ”inverse” symmetry; (iii) Less clear from the table: If f , g , h are symmetries of our triangle ( f ◦ g ) ◦ h = f ◦ ( g ◦ h ) . BUT THIS IS A FACT ABOUT FUNCTIONS (and we already know it!!).
Symmetries of an Equilateral Triangle R1R2 FAFBFC We learn in High School Algebra, and again in Calculus (and ID re-learned in Ch. 0) counting f ( g ( h ( x ))) = f ◦ ( g ◦ h )( x ) = ( f ◦ g ) ◦ h ( x ) . We call this Composition Groups ”Associativity”.
Symmetries of an Equilateral Triangle R1R2 FAFBFC ID counting Observations Composition Groups
Symmetries of an Equilateral Triangle So our set S of symmetries has the following property: R1R2 (i) There is a binary operation on S , i.e., a way to combine FAFBFC two members of S to get another one,(composition) we ID write ψϕ instead of ψ ◦ ϕ ; counting (ii) This operation is associative: ψ ( σϕ ) = ( ψσ ) ϕ. Composition (iii) There is an identity element for the operation, i.e., an Groups element σ so that σψ = ψσ = ψ, for all ψ ; (The identity is R 0 . ) (iv) Every element has an inverse – Given ψ ∈ S there is a σ ∈ S so that ψσ = σψ = R 0 .
Symmetries of an Equilateral Triangle There are other examples of sets, say, G satisfying (i)-(iv)– R1R2 The integers Z , with the operation + (i), is associative (ii), the FAFBFC integer 0 is the additive identity (iii) and for any n we have ID n + ( − n ) = 0 . (iv). counting Composition The positive real numbers R with multiplication · Groups GL (2 , R ) the set of all 2 × 2 invertible real matrices with the operation of matrix multiplication. A set G with a closed binary operation, · , satisfying (i)-(iv) is called a group .
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