Symmetry in Numbers

David Marshall Monmouth University

April 13, 2005

808

√ √ s s −2 3 −169 1007 3 −169 1007 + + + − 3 54 18 54 18

– Typeset by FoilTEX – Symmetry

One of the guiding principles in mathematics today is that in order to fully understand a mathematical structure one should investigate its group of symmetries.

-Paul Yale, in Geometry and Symmetry

2. Due or just proportion; harmony of parts with each other and the whole; ﬁtting, regular, or balanced arrangement and relation of parts or elements; the condition or quality of being well-proportioned or well- balanced. In stricter use (approaching or passing into 3b): Exact correspondence in size and position of opposite parts; equable distribution of parts about a dividing line or centre. (As an attribute either of the whole, or of the parts composing it.)

-O.E.D. Online

– Typeset by FoilTEX –1 Symmetry in Euclid’s Plane

• The Euclidean plane is a metric space ( a space with a notion of “distance”)

• The symmetries are the bijections of the plane that preserve distances; called isometries.

• Four types: – translations (along a vector) – rotations (through an angle, about a point) – reﬂections (across a line) – glide reﬂections

– Typeset by FoilTEX –2 Symmetries of objects in the plane

– Typeset by FoilTEX –3 Symmetries: First Attempt

• Let A be an object in some ambient space S.

• The symmetries of S are the bijections (rearrangements, permutations) of S which preserve its structure.

• The symmetries of A are those symmetries of S which ﬁx A (as a set, not pointwise).

– Typeset by FoilTEX –4 Symmetry

3b.(a) Sci. Exact correspondence in position of the several points or parts of a figure or body with reference to a dividing line, plane, or point (or a number of lines or planes); arrangement of all the points of a figure or system in pairs (or sets) so that those of each pair (or set) are at equal distances on opposite sides of such line, plane, or point. More widely, a property by virtue of which something is effectively unchanged by a particular operation; an operation or set of operations that leaves something effectively unchanged; in Physics, a property that is conserved (cf. symmetry operation, sense 4 below). 4. attrib. and Comb., as property; symmetry- breaking ppl. a. and vbl. n. Physics, (causing) the absence of manifest symmetry in a situation despite its presence in the laws of nature underlying it; symmetry group, a group whose elements are all the symmetry operations of a particular entity; symmetry operation Physics, an operation or transformation that leaves something effectively unchanged.

– Typeset by FoilTEX –5 Symmetry: Problem with ﬁrst attempt; the letter “I”

• As written, it has the same symmetries as a tall, thin rectangle . . . not a problem.

• What happens if the “I” becomes so thin as to have no thickness at all. . . that is, it becomes a line segment?

• What are the symmetries of a line segment? – There are 4 symmetries of the plane which ﬁx the line – But they pair up in terms of what they do to the line.

• My answer: the line has two symmetries.

– Typeset by FoilTEX –6 Symmetries: Second attempt

• Let A be an object in some ambient space S.

• The symmetries of S are the bijections (rearrangements, permutations) of S which preserve its structure.

• For the symmetries of A: – First take all the symmetries of S which ﬁx A (as a set) – Then equate those which treat A the same pointwise.

• Group Theory Version: Sym(A)=G/H, where G is the subgroup of Sym(S) which ﬁxes A, and H is the normal subgroup of G which ﬁxes A pointwise.

– Typeset by FoilTEX –7 Symmetries of a Structureless Set

If X is simply a set, with no additional structure, then every bijection yields a symmetry.

• When X is ﬁnite of order n, the group of symmetries (typically denoted Sn) is called the symmetric group on n objects.

• For sets of its size, it has the most symmetry possible.

• Example: X = {1, 2, 3, 4, 5}, A = {1, 2}. The permutations which ﬁx the set A either – ﬁx A pointwise (there are six of these) – swap 1 and 2 (there are six of these) So A has just 2 symmetries (but we already new that).

– Typeset by FoilTEX –8 Symmetries of a number

For us, “number” will mean complex number, and we denote the set of all complex numbers by

C = {a + bi|a, b ∈ R}

For example

√ √ √ s s −2 1+ −3 3 −169 1007 3 −169 1007! + + + − 3 2 54 18 54 18

This provides us with an ambient space to begin our investigation.

– Typeset by FoilTEX –9 Symmetries of C

What is the “structure” of C? There are many possible answers:

• C as a 2-dimensional real vector space (so C ' R2).

• Sym(C)=invertible linear operators on R2, or just the 2 × 2 invertible matrices.

0 • Let A be the line spanned by ~v = (the y-axis). 1

• What are the symmetries of A? The matrices that ﬁx A are those with eigenvector ~v , so those of the form a 0 ,aλ=06 cλ

• Two such matrices with the same λ are then equated.

– Typeset by FoilTEX –10 Symmetries of C

1. C as a 1-dimensional complex vector space.

2. C as an aﬃne algebraic variety.

3. C as a complex or real manifold

4. C as a topological space (you choose the topology)

5. C as an additive group

6. C as a ﬁeld

– Typeset by FoilTEX –11 Objects in C

With the complex numbers as our ambient space, we can begin to talk about symmetries of numbers.

• Numbers come in packets (of arithmetically indistinguishable elements). For example: √ i

√ √ {± i, ± −i}

• These are just the roots of x4 +1.

– Typeset by FoilTEX –12 Algebraic vs.√ Transcedental ( 2 vs. π)

• algebraic = roots of polynomials with integer coeﬃcients.

• transcendental = not.

• Algebraic numbers have ﬁnite packets of conjugates.

• Transcendentals are put into one BIG packet.

• This partitions C into a countably inﬁnite number of ﬁnite packets, and one uncountable packet.

The symmetries of a number are just the symmetries of its packet.

– Typeset by FoilTEX –13 The Good, the Bad, and the Ugly

• the Good: we’re ready to compute symmetries of numbers.

• the Bad: as a first step we must determine the symmetries of the field of complex numbers, called field automorphisms, denoted Aut(C).

• the Ugly: Aut(C) is one of the least understood groups in mathematics. For example: Aut(C) is uncountably inﬁnite, but we can only describe two of its elements.

– Typeset by FoilTEX –14 The Ugly

Fact: The only automorphisms that ﬁx R (as a set) are the identity and complex conjugation (so R,asa ﬁeld, has no symmetries).

Fact: Other than the identity and complex conjugation, every automorphism

• is discontinuous.

• ﬁxes a dense subset of R.

• maps all of R to a dense subset of C.

Yale, Paul. Automorphisms of the Complex Numbers, Mathematics Magazine, V.39, No.3, 1966.

– Typeset by FoilTEX –15 Symmetries of a Number

So how are we to compute the symmetries of numbers? √ √ Ex: A = {± i, ± −i}

Q: Which automorphisms of C ﬁx A as a set?

A: They all do.

So how do I know that A has 4 symmetries?

I can put the numbers in A into a much smaller and well behaved ambient space, whose symmetries are much more easily computed.

That would be a course in Galois Theory.

– Typeset by FoilTEX –16 What the Symmetry Group Tells Us

Let α be a complex number, A its packet, and GA its symmetry group.

• GA is ﬁnite if and only if α is algebraic.

• GA = {1} if and only if α ∈ Q. √ • GA = C2 if and only if α = a + b n where a, b, and n are rational.

– Typeset by FoilTEX –17 What the Symmetry Group Tells Us

More advanced:

• α is solvable in radicals iﬀ GA is solvable.

• α is constructible iﬀ GA is solvable, having elementary 2-groups as the quotients of the composition series.

• α can be written as a ﬁnite Fourier series

2πikj/nj α = X qje j

if and only if GA is abelian. For example √ 5=e2πi/5 − e4πi/5 − e6πi/5 + e8πi/5

– Typeset by FoilTEX –18 Questions and Problems

1. Given a number (or polynomial), determine its symmetry group. √ 4 • ex: i (or x +1) has symmetry group C2 × C2.

2. Construct polynomials with prescribed symmetry group.

• ex: ﬁnd a polynomial with Galois group PGL2(7). • Here’s one: x8 − 2x7 − 35x6 + 308x4 + 308x3 − 462x2 − 556x +6

3. What ﬁnite groups occur as the symmetry group of an algebraic number? • all ﬁnite abelian groups occur. • the Monster group of order 808017424794512875886459904961710757005754368000000000 does too.

– Typeset by FoilTEX –19