lectures: mathematics of symmetry spring...

1-DIMENSIONALSYMMETRIES


CONTENTS

LECTURES: MATHEMATICS OF

SYMMETRYSPRING 2020

Copyright 2020 by James Bernhard



TUESDAY, JANUARY 21

Our first object of study is the real number line, so it’s usefulto establish some notation for it.

DEFINITION

We denote the real number line by R.



TUESDAY, JANUARY 21

Mathematicians think of symmetries as being active: theyare something that you do to a figure.

The mathematical language for this is expressed throughfunctions, so some more notation is useful here.

DEFINITION

We use the notation f : A→ B (read “f is from A to B”) todenote that f is a function whose input is from A andwhose output is in B.

Less technically, “f eats things in A and spits out things inB”.



TUESDAY, JANUARY 21

If we have two functions with the right input and outputpossibilities, there’s an easy way to get a third: composethem.

DEFINITION

If f : A→ B and g : B→ C, then we denote thecomposition of these functions by gf , so for all x in A:

gf (x) = g(f (x)).

Note: in gf , the function f is applied first, and then g isapplied. That is, we apply functions from right to left.



TUESDAY, JANUARY 21

Two functions f : A→ B and g : A→ B are considered tobe equivalent (or intuitively, “the same”) when f (x) = g(x)for every x in A.

That is, two functions are equivalent when they both sendeach point in A to exactly the same point in B.

Concept Whenever we define a new mathematical object,we should also define what it means for two of those objectsto be equivalent (intuitively “the same”).



TUESDAY, JANUARY 21

For example, suppose f : R→ R and g : R→ R are givenby

f (x) = |x| and g(x) =√

x2.

These two functions look different, and they accomplishthings differently, but they are considered equivalent becausethey both send each point in R to the same place.

In other words, these are considered to be two differentdescriptions of the same function.



TUESDAY, JANUARY 21

If a function has the same input and output space, we canthink of it as transforming that space.

DEFINITION

A transformation is a function from a space to itself.



TUESDAY, JANUARY 21

DEFINITION

A rigid transformation (or isometry) of R is a functionf : R→ R that preserves distances:

Dist(p, q) = Dist(f (p), f (q))

for all points p, q in R

EXAMPLEFind a rigid transformation of R, and explain how you knowthat it is one.



TUESDAY, JANUARY 21

DEFINITION

The identity transformation on A, denoted by I : A→ A isdefined by

I(x) = x for all x in A.

Here is what I : R→ R does to some points:

−3 −2 −1 0 1 2 3xoxo xo

Each point x is indicated with an x, and I(x) is indicatedwith a circle o

Since this is the identity transformation, the x’s and o’s areall on top of each other



TUESDAY, JANUARY 21

THEOREM

The identity transformation of R is a rigid transformation ofR.

It’s good practice to think about why this is true.



TUESDAY, JANUARY 21

DEFINITION

For any point p in R, the reflection (or flip) of R across pis the function F : R→ R that sends a point x to the pointthat is the same distance from p but on the other side

−3 −2 −1 0 1 2 3

p xF(x)

If F is reflection across p, then the red and blue distances areequal



TUESDAY, JANUARY 21

THEOREM

All reflections of R are rigid transformations of R.




TUESDAY, JANUARY 21

DEFINITION

A vector or directed line segment in R is a line segmentin R with one of its endpoints designated as the initial pointand the other endpoint designated as the final point

We draw a directed line segment as an arrow whose tail is atthe initial point and whose head is at the final point of theline segment



TUESDAY, JANUARY 21

Two directed line segments are considered the same if theyare of the same length and have the same orientation(meaning both are pointing in a negative direction or bothare pointing in a positive direction along the real number lineR)




TUESDAY, JANUARY 21

DEFINITION

For any directed line segment~s, the translation of R by~s isthe function T : R→ R that sends each point in R to thefinal point of a copy of~s placed with its initial point at theoriginal point.

−3 −2 −1 0 1 2 3

x1 x2 x3T(x1) T(x2) T(x3)

If T is translation of R by~s, then each red arrow in thepicture depicts~s



TUESDAY, JANUARY 21

THEOREM

All translations of R are rigid transformations of R.




TUESDAY, JANUARY 21

We now know three types of rigid transformations of R:

1 the identity transformation

2 reflections

3 translations.

What others are there?



TUESDAY, JANUARY 21

There aren’t any others:

THEOREM

Any rigid transformation of R is either: the identity, areflection, or a translation.

This isn’t so straightforward to prove, but it should at leastbe believable.



FRIDAY, JANUARY 24

THEOREM

The composition of two rigid transformations is again a rigidtransformation

How can we show this?



FRIDAY, JANUARY 24

To see why this theorem is true, suppose f , g : R→ R arerigid transformations. Then since f is a rigid transformation:


Since f (p), f (q) are also points in the real line, and since g isa rigid transformation:

Dist(f (p), f (q)) = Dist(g(f (p)), g(f (q))).



FRIDAY, JANUARY 24

Since the composition of two rigid transformations is again arigid transformation, and since all rigid transformations of R

are either the identity, a reflection, or a translation:

THEOREM

Composing reflections and/or translations of R yields eitherthe identity transformation, a reflection, or a translation ofR.



FRIDAY, JANUARY 24

EXAMPLELet T1, T2 : R→ R be translations by directed line segments~s1,~s2. Which rigid transformation is T1T2? Which rigidtransformation is T2T1?

Note that “which rigid transformation” includes “which typeof rigid transformation” and “which specific transformationof that type”.



FRIDAY, JANUARY 24

DEFINITION

A figure in R is a subset of R.

DEFINITION

Two figures are called congruent if there is a rigidtransformation taking one to the other.




FRIDAY, JANUARY 24

DEFINITION

A symmetry of a figure in R is a rigid transformation of R

that sends the figure to itself. That is, the figure isindistinguishable before and after the rigid transformation isapplied to R.

This doesn’t mean that the points don’t move — theymight. But they have to move to other places in the figure.



FRIDAY, JANUARY 24

EXAMPLEWhat are all the symmetries of the figure below?

−3 −2 −1 0 1 2 3



FRIDAY, JANUARY 24

EXAMPLEWhat are all the symmetries of the figure below?

−3 −2 −1 0 1 2 3. . . . . .

The “. . . ” on both sides indicate that the pattern continuesinfinitely in both directions.



FRIDAY, JANUARY 24

For practice, come up with more figures in R and find theirsymmetries.

Something to think about: if f , g : R→ R are bothsymmetries of a figure in R, is fg : R→ R also a symmetryof the figure?

(The possible answers are: never, sometimes, and always. Ifthe answer is “sometimes”, the next question is: When?)



TUESDAY, JANUARY 28

Discussion of Livio Chapter 3 Reading Questions



TUESDAY, JANUARY 28

For an exploration related to Livio Ch 3, let’s look at howTartaglia revealed the solution to the cubic. It was in apoem:

https://www.maa.org/press/periodicals/convergence/

how-tartaglia-solved-the-cubic-equation-tartaglias-original-poem

https://www.maa.org/press/periodicals/convergence/how-tartaglia-solved-the-cubic-equation-tartaglias-original-poem

https://www.maa.org/press/periodicals/convergence/how-tartaglia-solved-the-cubic-equation-tartaglias-original-poem



TUESDAY, JANUARY 28

Here’s a small portion of it, in translation:When the cube and the things togetherAre equal to some discrete number,Find two other numbers differing in this one.Then... their product should always be equalExactly to the cube of a third of the things.The remainder then as a general ruleOf their cube roots subtractedWill be equal to your principal thing.

What does this mean, in modern symbols?



TUESDAY, JANUARY 28

Cube: x3

The things: hard to say, maybe something times x or aconstant

Some discrete number: ah, that must be a constant, so “thethings” is probably something times x

So we start with: x3 + ax = b.



TUESDAY, JANUARY 28

Two other numbers differing in this one: u− v = b

Product equal to a third of the things: uv = (a/3)3

Remainder of the cube roots subtracted: 3√

u− 3√

v

Your principal thing: x.



TUESDAY, JANUARY 28

So we’re looking for a solution to the equation

x3 + ax = b,

and Tartaglia states that it is

x = 3√

u− 3√

v,

where u, v are chosen to satisfy u− v = b and uv = (a/3)3.



TUESDAY, JANUARY 28

For example, to solve

x3 − 3x = −2,

we find u, v satisfying u− v = −2 and uv = (−3/3)3.

Simplifying, we have u + 1/u = −2, which givesu2 + 2u + 1 = 0.

The quadratic formula gives u = −1, so v = 1, andTartaglia’s method gives x = 3

√−1− 3

√1 = −2.

And sure enough, x = −2 is a solution of the originalequation.



TUESDAY, JANUARY 28

For another exploration related to Livio Ch 3, let’s look atDiophantus and the notation that he introduced, in thearticle “The Symbolic and Mathematical Influence ofDiophantus’s Arithmetica”, by Cyrus Hettle (Journal ofHumanistic Mathematics, Vol 5, No 1, Jan 2015):

https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1190&context=jhm

https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1190&context=jhm



FRIDAY, JANUARY 31

DEFINITION

We denote the Euclidean plane by R2.



FRIDAY, JANUARY 31

DEFINITION

A directed line segment in R2 is a line segment in R2 withone of its endpoints designated as the initial point and theother endpoint designated as the final point.

We draw a directed line segment as an arrow whose tail is atthe initial point and whose head is at the final point of theline segment



FRIDAY, JANUARY 31

Two directed line segments are considered the same if theypoint in the same direction along parallel lines and have thesame length

The black arrows represent the same directed line segmentsbelow



FRIDAY, JANUARY 31

DEFINITION

For any directed line segment~s, the translation of R2 by~sis the function T : R2 → R2 that sends each point in R2 tothe final point of a copy of~s placed with its initial point atthe original point.



FRIDAY, JANUARY 31

x1

x2

x3

T(x1)

T(x2)

T(x3)

Notice that all points are moved by the same amount in thesame direction

The translation by directed line segment~s is shown in red



FRIDAY, JANUARY 31

DEFINITION

For any line L in R2, the reflection of R2 across L is thefunction F : R2 → R2 that sends each point x in R2 to theunique point that:

• lies on the unique line through x that is perpendicularto L,

• is the same distance (along this perpendicular line) as xis from L• is on the other side of L.



FRIDAY, JANUARY 31

x1x2

x3

F(x1)

F(x2)

F(x3)

In each case the distances on the two sides of the reflectionline are equal, and the red line segments are perpendicular tothe reflection line.



FRIDAY, JANUARY 31

DEFINITION

A glide reflection of R2 is a function G : R2 → R2 that is atranslation of R2 followed by a reflection of R2 across a linethat is parallel to the direction of translation.

This might seem strange, since it is just a specific type ofcombination of translation and reflection.

We’ll see later why we make this definition.



FRIDAY, JANUARY 31

DEFINITION

For any point p in R2 and any angle a, the rotation of R2

about p by a is the function R : R2 → R2 that sends eachpoint x in R2 to the unique point that:

• lies on the unique ray making an angle a with the rayemanating from p and passing through x, and

• is the same distance from p as x is.

If clockwise or counterclockwise is not specified for theangle, we take counterclockwise to correspond to positiveangles, and clockwise to correspond to negative angles.



FRIDAY, JANUARY 31

c

x1

R(x1)

x2

R(x2)

x3

R(x3)

Notice that the distance to the center of rotation ispreserved: the red line segments at the two ends of the bluecircular arcs have the same length



FRIDAY, JANUARY 31

DEFINITION

A rigid transformation (or isometry) of R2 is a functionf : R2 → R2 that preserves distances:


for all points p, q in R2

What type of rigid transformations of R2 have we met sofar?



FRIDAY, JANUARY 31

THEOREM

Any rigid transformation of R2 is either

• a translation,

• a reflection,

• a glide reflection,

• a rotation, or

• the identity (which could be considered either a rotationby 0 degrees or a translation by a directed line segmentwith length 0)

We won’t prove this theorem in this course, but we will makeuse of it.



FRIDAY, JANUARY 31

THEOREM

The composition of two rigid transformations is again a rigidtransformation.

Why is this theorem true?



FRIDAY, JANUARY 31

To see why this theorem holds, let f , g : R2 → R2 be rigidtransformations. Then since f is a rigid transformation:


Since f (p), f (q) are also points in the real line, and since g isa rigid transformation:

Dist(f (p), f (q)) = Dist(g(f (p)), g(f (q))).



FRIDAY, JANUARY 31

Since the composition of two rigid transformations is again arigid transformation, and since all rigid transformations ofR2 are either the identity, a translation, a reflection, a glidereflection, or a rotation:

In R2, composing translations, reflections, glide reflections,and/or rotations must yield either the identity, a translation,

a reflection, a glide reflection, or a rotation

You’ll investigate this on the next homework assignment



FRIDAY, JANUARY 31

EXAMPLELet R1, R2 be rotations with centers p1, p2 by angles θ1, θ2.What transformation is R1R2? What about R2R1?



FRIDAY, JANUARY 31

EXAMPLELet R be a rotation about p by angle θ, and let F be areflection across a line `. What transformation is RF? Whatabout FR?



TUESDAY, FEBRUARY 4

Discussion of Chapter 4 of Livio.



TUESDAY, FEBRUARY 4

Link to the original memoir by Abel:

https://www.abelprize.no/c54178/binfil/download.

php?tid=53608

Link to the Abel Prize website:

https://www.abelprize.no/

https://www.abelprize.no/c54178/binfil/download.php?tid=53608

https://www.abelprize.no/c54178/binfil/download.php?tid=53608

https://www.abelprize.no/



FRIDAY, FEBRUARY 7

Summary so far: a rigid transformation of R2 is a functionf : R2 → R2 that preserves distances (and also angles, infact)

Every rigid transformation of R2 is one of the followingtypes:

1 translation

2 rotation

3 reflection

4 glide reflection

The identity transformation can be considered a translation(by a directed line segment of length 0) or a rotation (aboutany center through angle 0)



FRIDAY, FEBRUARY 7

For any function f : R2 → R2, a fixed point is a point p forwhich f (p) = p, meaning p doesn’t go anywhere when f isapplied

We can classify rigid transformations to some extent by theirfixed points:

• Every point in R2 is a fixed point of the identitytransformation

• Reflections in R2 have a line of fixed points (the line ofreflection)

• Rotations of R2 have exactly one fixed point (the centerof rotation)

• Nonzero translations and glide-reflections have no fixedpoints



FRIDAY, FEBRUARY 7

A rigid transformation of R2 is orientation-preserving if itdoes not change the clockwise/counterclockwise sense ofangles; otherwise it is called orientation-reversing

Orientation-reversing rigid transformations send a forwards“R” to a backwards “R”, and vice versa

We can classify rigid transformations of R2 by what they doto the orientation of R2:

• The identity, translations, and rotations areorientation-preserving

• Reflections and glide-reflections are orientation-reversing



FRIDAY, FEBRUARY 7

We can now begin to use these rigid transformations

A major concept in modern mathematics is that whenworking with “things” (elements of sets), it is important todefine precisely when two such things are considered “thesame,” or in mathematical terms, equivalent

In geometry, we often use the term congruent rather thanequivalent, but how do we define this?

Recall that a figure in R2 is a subset of R2

We define two figures to be equivalent (or congruent) ifthere is an orientation-preserving rigid transformation takingone to the other



FRIDAY, FEBRUARY 7

A symmetry of a figure in R2 is a rigid transformation ofR2 that leaves the figure indistinguishable before and afterthe transformation is applied

A simple but important observation: the composition of twosymmetries of a figure is again a symmetry of the figure

This means that composition can be considered anoperation that is defined on the set of symmetries of a figure

Slightly more precisely, an operation is a way to combine toelements of a set to get an element of the set (here we“combine” two symmetries to get a symmetry)



FRIDAY, FEBRUARY 7

Let’s list the symmetries of a capital X:



FRIDAY, FEBRUARY 7

They are: identity I, reflection across the horizontal Fh,reflection across the vertical Fv, rotation by 180 degrees R1/2



FRIDAY, FEBRUARY 7

What are the symmetries of this square with noses?



FRIDAY, FEBRUARY 7

Rotations by multiples of 90 degrees: I = R0, R1/4, R1/2, R3/4



FRIDAY, FEBRUARY 7

Are these two figures symmetric in “the same way”?

They have the same number of symmetries, but still itdoesn’t look like they are

For this we need a further concept. . .



FRIDAY, FEBRUARY 7

Some other observations about composing symmetries:

• Composition is associative: f (gh) = (fg)h for allsymmetries f , g, h• The identity transformation I has the property that

If = f and f I = f for all symmetries f• Every symmetry f has an inverse symmetry f−1 that

“undoes” what it “does”, meaning ff−1 = I andf−1f = I

Also worth noting is that composing symmetries is notcommuntative: fg might be different from gf



FRIDAY, FEBRUARY 7

In mathematics, a group is a set with an operation(generically called “multiplication”) that satisfies threeproperties:

1 The operation is associative

2 There is an identity element i in the set with theproperty that if = f and fi = f for all f in the set

3 Every element f in the set has an inverse f−1 in the set,which has the property that ff−1 = i and f−1f = i

If the operation is commutative, then the group is calledabelian.



FRIDAY, FEBRUARY 7

For a finite group, we can write out a multiplication table,which tells how to compose all pairs of symmetries

We write them as we read them: the leftmost column entryindicates the element composed on the left (meaningsecond), and the top row entry indicates the elementcomposed on the right (meaning first)



TUESDAY, FEBRUARY 11





A bit of mathematical humor (admittedly it’s not that good):

Stefanie (The Ballad of Galois)https://www.youtube.com/watch?v=l3hNtBbh_E0

https://www.youtube.com/watch?v=l3hNtBbh_E0




EXAMPLEWhat is the multiplication table for the symmetry group of asquare with noses?




EXAMPLEWhat is the multiplication table for the symmetry group of acapital X?



FRIDAY, FEBRUARY 14

Recall that a group is a set together with an operation(generically called multiplication) satisfying:

1 multiplication is associative,

2 there is an identity element, and

3 every element has an inverse.

The set of symmetries of a figure, together with theoperation of composition, forms a group.

DEFINITION

The group that consists of the set of symmetries of a figure,together with the operation of composition, is called thesymmetry group of the figure.



FRIDAY, FEBRUARY 14

As we have discussed, the idea of when are twomathematical objects to be considered “the same” is animportant one in contemporary mathematics.

When are two symmetry groups to be considered “thesame”?

This leads us to a new concept. . .



FRIDAY, FEBRUARY 14

DEFINITION

Two groups are isomorphic if their multiplication tables arethe same except for possibly a relabeling of the elements inthe group.

DEFINITION

We say that two figures have the same symmetry type iftheir symmetry groups are isomorphic.



FRIDAY, FEBRUARY 14

EXAMPLEWhat are the symmetry groups of T and Z? Do these twofigures have the same symmetry type?



FRIDAY, FEBRUARY 14

The symmetry groups of T and Z hav the followingmultiplication tables:

T I FvI I Fv

Fv Fv I

Z I R1/2I I R1/2

R1/2 R1/2 I

These two symmetry groups are isomorphic, as can be seenby relabeling Fv as R1/2 (no reordering is needed).

This means that these two figures have the same symmetrytype.



FRIDAY, FEBRUARY 14

An observation: if two groups are isomorphic, then they havethe same number of elements.

DEFINITION

For any element g of a group, its order is the smallestpositive n for which gn = i (where i is the identity). If nosuch n exists, then we say that g has infinite order.

The order of an element can indeed be infinite, if the grouphas infinitely many elements.



FRIDAY, FEBRUARY 14

The orders of some familiar transformations:

• The order of I is always 1.

• The order of a reflection is always 2.

• The order of a rotation could be anything: for example,the order of R1/2 is 2, the order of R1/3 is 3, the orderof R3/4 is 4, etc.

• The order of a nonzero translation or glide reflection isinfinite.

An observation: if two groups are isomorphic, then theircorresponding elements (under a relabeling making themisomorphic) have the same orders.



FRIDAY, FEBRUARY 14

EXAMPLEDoes the square with noses have the same symmetry type ascapital X?



FRIDAY, FEBRUARY 14

The square with noses and the letter X do not have thesame symmetry type, since the letter X has no symmetries oforder 4, but the square with noses does.



FRIDAY, FEBRUARY 14

EXAMPLEWhat are the symmetries of the following figure (shown inwhite; the black is not part of the figure)?



FRIDAY, FEBRUARY 14

What are the symmetries of the following figure (ignoring allcolors)?



FRIDAY, FEBRUARY 14

Classification theoremfor figures in R2 with no translational symmetries

The symmetry group of such a figure satisfies exactly one ofthe following:

• It consists of the identity I and nothing else

• It consists of exactly 2 elements, in which case thefigure’s symmetry type can be called either C2 or D1(these are the same symmetry type)

• It consists of exactly n ≥ 3 rotations (incuding I), inwhich case the figure’s symmetry type is Cn

• It consists of exactly n ≥ 3 rotations (including I) and nreflections, in which case the figure’s symmetry type isDn

• It contains an infinite number of symmetries



FRIDAY, FEBRUARY 14

DEFINITION

The Cn symmetry types (and their corresponding symmetrygroups) are called cyclic, and the Dn symmetry types (andtheir corresponding symmetry groups) are called dihedral

Notice that Cn consists of n elements, while Dn consists of2n elements.



FRIDAY, FEBRUARY 14

For example, the “square with noses” has symmetry type C4,while a square has symmetry type D4.

A circle has an infinite symmetry group (as do some othersimilar but more complicated bounded figures.)

Notice that the Dn symmetry group has 2n symmetries,while the Cn symmetry group has n symmetries.




Key ideas in Galois’ proof:

• “symmetry profile” of a group: permutations of roots

• normal subgroups

• solvable groups




Roots of x2 − 2 are ±√

2

If we extend the rational numbers by these, we are workingin a field whose elements look like a + b

√2

We can transform that field by the identity function and bythe function

f (a + b√

2) = a− b√

2

This preserves the real numbers

The Galois group of this polynomial is C2




A subgroup is a group within a group

Theorem The order of a subgroup divides the order of agroup

The quotient of the order of the big over the order of thesmall is called the index of the small

A normal subgroup is a group whose elements almostcommute with the whole group

That is, H is normal in G if for each h ∈ H there exists andh′ ∈ H such that gh = h’g




In an abelian group, all subgroups are normal

In D3, {I, R, R2} is normal, but {I, F} is not (since FR 6= RFand FR 6= IF)




Maximal (proper) normal: normal, and isn’t contained in anylarger normal proper subgroups

Any maximal normal subgroup can be extended in a chaindown to the identity subgroup

S2 : 2, 1

S3 : 6, 3, 1

S4 : 24, 12, 4, 2, 1

S5 : 120, 60, 1

A group is solvable if it contains a chain of maximal normalsubgroups for which every index is a prime number




This associates an algebraic object with each object of study(polynomial) and then answers questions about thepolynomial by answering questions about the algebraic object



FRIDAY, FEBRUARY 21

How do we know that the symmetry groups for symmetrytypes Cn and Dn are not isomorphic (except C2 and D1,where they are isomorphic)?

We can describe the multiplication tables in terms ofgenerators and relations



FRIDAY, FEBRUARY 21

A set of transformations generates a group if the entiregroup can be obtained by composing these transformationswith each other (as many times as you like)

In this case, the transformations in a set that generates thegroup are called generators of the group

These generators are not unique



FRIDAY, FEBRUARY 21

For example, the cyclic group Cn is generated by a singleelement (although the generating element is not unique)

On the other hand, the dihedral group Dn is generated bytwo elements (again not unique)



FRIDAY, FEBRUARY 21

Relations are equations satisfied by generators of a groupthat completely determine the multiplication table of thegroup

A listing of the generators and relations describing a group Gis called a presentation of the group, often written as:

G = 〈generators | relations〉.



FRIDAY, FEBRUARY 21

For example, the cyclic group Cn is generated by R subjectto the relation Rn = I, so

Cn = 〈R | Rn = I〉.

Note that this group is abelian

It is not hard to show that the symmetry group for Dn isnonabelian

On the homework, you will find a presentation for thesymmetry group for Dn



FRIDAY, FEBRUARY 21

A frieze pattern is a figure in R2 whose translationalsymmetries are generated by a single translation

A wallpaper pattern is a figure in R2 whose translationalsymmetries are generated by two translations (but not one)

In a frieze pattern, what symmetries can occur? How manysymmetry types are there?



FRIDAY, FEBRUARY 21

When translational symmetries are present, other symmetriesmultiply like rabbits!

For example, if translational symmetries are generated by Tand if a single vertical reflection F is present, then so are TFand T2F, etc.



FRIDAY, FEBRUARY 21

A useful concept is that of a fundamental domain

Suppose we have a frieze pattern F, and that X is some partof F

The X is a fundamental domain if the following conditionshold:

• The set of all T(X) (where T varies over all thetranslational symmetries of F) is all of F• The set of all T(X) (where T varies over all the

translational symmetries of F) do not overlap at all

As an example, consider the frieze pattern that consists of allthe integers on the real line

It is possible to define fundamental domains related to notonly translations, but also other symmetries, but we won’tdo so in this class



FRIDAY, FEBRUARY 21

Given a fundamental domain, fundamental symmetry is asymmetry satisfying one of the following three conditions:

1 It is a rotation and its rotocenter is in the givenfundamental domain.

2 It is a reflection and its line of reflection intersects thegiven fundamental domain.

3 It is an irreducible glide reflection which generates theirreducible glide reflections across that line of reflection,and whose line of reflection intersects the givenfundamental domain.

Also, since any irreducible glide reflection whose line ofreflection intersects the given fundamental domain has aninverse with the same property, only one of those two shouldbe counted as a fundamental symmetry (you choose whichone)

The term intersects means has at least one point in commonwith



FRIDAY, FEBRUARY 21

Let’s try to figure out all the symmetry types of friezepatterns. . .



FRIDAY, FEBRUARY 21

The following table gives a classification of the symmetrytypes of frieze patterns (parentheses means that it isautomatic if the non-parentheses items hold):

Symmetry Reflection across Reflection across 180 degree IrreducibleType horizontal? vertical? rotation? glide reflection?p111 no no no nop1a1 no no no yesp112 no no yes (no)pm11 no yes no (no)pma2 no yes yes (yes)p1m1 yes no (no) (no)pmm2 yes yes (yes) (no)



FRIDAY, FEBRUARY 21

This has a reflection symmetry across the horizontal and areflection symmetry across the vertical, so its symmetry typeis pmm2.



FRIDAY, FEBRUARY 21



FRIDAY, FEBRUARY 21

This has a reflection symmetry across the horizontal butdoes not have a reflection symmetry across the vertical, soits symmetry type is p1m1.



FRIDAY, FEBRUARY 21

Important: With this classification table, finding thesymmetry type is easy. What’s more important is:

• Being able to notice and describe all the symmetries ofthe figure, and

• Understanding what exactly is meant by the symmetrytype of a figure.

So don’t get lulled into complacency by just using the table.I want you to be able to spot the symmetries! The more youpractice at that, the more automatic it will become.

lectures: mathematics of symmetry spring...

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