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What are the symmetries of an equilateral triangle? In order to

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsWhat are the symmetries of an equilateral triangle?C A BIn order to answer this question precisely, we need to agree on what the word "symmetry" means.

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsWhat are the symmetries of an equilateral triangle?C A BFor 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

R1R2

FAFBFC

ID counting

Composition

GroupsSo, what are some symmetries? How can we describe them?

What is good notation for them?C

A B

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsRotate counterclockise, 120

about the centerO:O C A B

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsNote this is the following map (function):O

B O C A B CAWe can think of this as a function on the vertices:

A7!B;B7!C;C7!A:

We might denote this by:A B C

B C A

We also may denote this map byR120:

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsRotate counterclockise, 240

about the centerO:This is the map (function):O A O C A B BCWe can think of this as a function on the vertices:

A7!C;B7!A;C7!B:

We might denote this by:A B C

C A B

We also may denote this map byR240:

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsRe

ect about the perpendicular bisector ofAB:C A B

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsRe

ect about the perpendicular bisector ofAB;

This is the map (function):O

C O C A B BAWe can think of this as a function on the vertices:

A7!B;B7!A;C7!C:

We might denote this by:A B C

B A C We also may denote this map byFCto indicate the re ection is the one xingC:

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsRe

ect about the perpendicular bisector ofBC;

This is the map (function):O

B O C A B ACWe can think of this as a function on the vertices:

A7!A;B7!C;C7!B:

We might denote this by:A B C

A C B We also may denote this map byFAto indicate the re ection is the one xingA:

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsRe

ect about the perpendicular bisector ofAC;

This is the map (function):O

A O C A B CBWe can think of this as a function on the vertices:

A7!C;B7!B;C7!A:

We might denote this by:A B C

C B A We also may denote this map byFBto indicate the re ection is the one xingB:

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsThe identity map of the plane: (takes every point to itself).

This is the map (function):O

C O C A B ABWe can think of this as a function on the vertices:

A7!A;B7!B;C7!C:

We might denote this by:A B C

A B C

We also may denote this map byIdor 1:

Note, we might also denote this asR0;since it is a rotation through 0 :However { it isNOTa re ection. (WHY NOT??!!)

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsSo far we have 6 symmetries { 3 rotations,R0;R120;R240;and 3 re ections,FA;FB;FC:O C B

AAre there any more??

Why or why not??

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsIn fact these are all the symmetries of the triangle. We can see this from our notation in which we write each of these maps in the formA B C X Y Z :Note there are three choices forX(i.e.,Xcan be any ofA;B;C;). Having made a choice forXthere are two choices forY: ThenZis the remaining vertex. Thus there areat most

321 = 6 possible symmetries. Since we have seen each

possible rearrangement ofA;B:Cis indeed a symmetry, we see these are all the symmetries.

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsNotice these symmetries are maps, i.e., functions, from the plane to itself, i.e., each has the formf:R2!R2:Thus we can compose symmetries as functions: Iff1;f2are symmetries thenf2f1(x) =f2(f1(x));is also a rigid motion. Notice, the composition must also be a symmetry of the triangle.

For example,R120FC=?? It must be one of our 6

symmetries. Can we tell, without computing whether it is a rotation or re ection?? Why?? What about the composition of two re ections?

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsR

120FC;we can view this composition as follows:R

120
C F O A O C O C O C A B BA BA

BCSo,R120FC=FB:

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsWe use our other notation:

R

120FC=A B C

B C A A B C B A C =A B C C B A =FB

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsIsR120FC=FC=R120? Let's look:FCR120:R

120
C F O B O B O B O C A B CA C A

CASoFCR120=FA6=FB=R120FC:

Symmetries of

an Equilateral

Triangle

R1R2

FAFBFC

ID counting

Composition

GroupsSo on our set of symmetriesS=fR0;R120;R240;FA;FB;FCg; we get a way of combining any two to create a third, i.e., we get anoperationonS:(Just like addition is an operation on the integers.) We will call this operationmultiplication onS: We can make a multiplication table, orCayley Table. So far we have:R 0R 120R
240F
AF BF CR 0R 0R 120R
240F
AF BF CR 120R
120F
BR 240R
quotesdbs_dbs29.pdfusesText_35
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