Pure Mathematics B1: Symmetry and Groups

78 flashcards · Mathematics · by lakesare

What's a plane figure?
Give 1 example of bounded plane figure, and unbounded plane figure
bounded plane figure - a heart Can be surrounded by a circle (of finite radius).unbounded plane figure - infinitely long straight line
What's an isometry function f: R2 → R2?
Any function that preserves distances: for all points X, Y ∈ R2, distance(X, Y) is equal to distance(f(X), f(Y)).
What's a symmetry function f: R2 → R2?
Isometry function that maps F to itself: f(F) = F.
What type of object is symmetry?
Symmetry is a function f: R2 → R2.
Name 2 possible symmetries of a bounded figure.
rotation (including the identity transformation e!) reflection
What's the connection between: the centre of rotational symmetry, and the axis of the reflectional symmetry.
All axes of reflection pass through the centre of rotations.
[Regular polygons] Picture regular polygons.
[Regular polygons] What is this figure trying to convey?
Regular polygons with n edges has 2n symmetries: n rotations n reflections
[Regular polygons] What are the reflectional axes of symmetry for polygons with: odd even n of edges?
Reflectional axes in polygons: with odd n of edges - vertex-midpoint with even n of edges - vertex-vertex, midpoint-midpoint
What does S(F) denote?
Set of all symmetries of a figure F.
What does e denote? S(F) = { e, ... }
e is identity symmetry (the 0-degree rotation).
[4 properties of S(F)] What does it mean that S(F) is closed under the composition of functions?
List the elements of S(F).
S(F) = { e, (identity symmetry) a, b, c, (rotational symmetries) r, s, t, u (reflectional symmetries) }
What symmetry do you have to combine with itself to get the e symmetry?
Compose any reflection with itself.
Where is the 6th symmetry of this triangle? Regular n-gons always have 2n symmetries.
It's e. In each case the identity symmetry is not shown but is denoted by e, as usual.
[4 properties of S(F)] Fundamentally why are the elements of S(F) associative under composition?
Because all of these mean: do f then do g then do h.
[4 properties of S(F)] Which element of S(F) is an identity element?
e (identity symmetry).
[4 properties of S(F)] Which element of S(F) an inverse to a?
c.
[4 properties of S(F)] Which of the element of S(F) are self-inverses?
e and all reflectional symmetries.
What do these mean when we're labelling the symmetries of the disk: rθ qθ Standard labelling for the symmetries of S(◻)
rθ - rotation through angle θ qθ - reflection in the axis at angle θ Standard labelling for the symmetries of S(◯)
What's the number of symmetries for n-gon VS for the disk?
n-gon - 2n symmetries disk - ∞ symmetries
Explain what's rθ and qθ?
rθ - rotational symmetries qθ - reflectional symmetries
What does S+(F) denote?
All direct symmetries of a figure F.
Direct VS indirect symmetries?
direct symmetries - demonstrated without lifting the piece of paper. indirect symmetries - the piece of paper needs to be lifted and turned over.
How would you show it?
What does this image denote?
What symmetries are For bounded figures: direct - ______ indirect - ______ A bounded figure For unbounded figures: direct - ______ indirect - ______ Infinite triangular grid
For bounded figures: direct - rotation indirection - reflection For unbounded figures direct - rotation, translation indirect - reflection, glide-reflection
What's a "two-line notation"?
It's a way to denote symmetries.
How do you read the two-line notation?
The vertex that used to be at location 1 is now at location 2.
What's symmetry r in two-line notation?
Two-line symbol VS two-line notation?
two-line symbol - matrix representing the symmetry. two-line notation - notation that uses two-line symbol.
How would you find what location 1 maps to? Use two-line symbols to find the composite r ◦ a in S(□).
What's the inverse of this two-line symbol?
What kind of object can be considered: a Group (provided it fits the group axioms) a Field (provided it fits the field axioms)
Group - (G, ◦) Field - (F, +, *)
What's meant by "Group axioms"?
Axioms that need to hold for (G, ◦) to be a group.
Abelian group VS non-abelian group? Niels Henrik Abel (1802–1829), Norway
abelian group - group + commutativity non-abelian group - group – commutativity
What's meant by the order of group (G, ◦)?
|G| - number of elements in G.
Name all group axioms.
(CIA) Closure Associativity Identity Inverses
[Cayley table] What does Cayley table display? Arthur Cayley (1821–1895)Arthur Cayley was the leading British algebraist of the nineteenth century. Prior to his appointment in 1863 as the first professor of pure mathematics at the University of Cambridge, he spent fourteen years as a lawyer during which time he produced over three hundred mathematical papers.
To display binary operations.
What's the alternative way to say (G, ◦) is a group?
G is a group under ◦.
What's the important difference between identity element and inverse element?
identity element - one element for the entire group. inverse element - a separate el for each el in a group.
[Cayley table] What's the main diagonal of the Cayley table? Main diagonal is also called the leading diagonal.
[Cayley table] Using Cayley table - how to check (G, ◦) for G1. Closure?
We should check whether all elements ∈ G. (All elements in the body of the Cayley table)
[Cayley table] Using Cayley table - how to check (G, ◦) for G2. Associativity?
No easy way to do it - check all combinations of elements.
[Cayley table] Using Cayley table - how to check (G, ◦) for G3. Identity?
Find an element such that: its row repeats the border, its column repeats the border.
[Cayley table] Using Cayley table - how to check (G, ◦) for G4. Inverse?
Given identity element e, make sure that each element has e in its row and it's symmetric across the main diagonal.
What's the big thing G2. Associativity tells us?
We can bracket any number of letters however we want, because these are all equivalent! In fact axiom G2 tells us more: it tells us that we can write a composite of any finite number of group elements with no brackets, without there being any ambiguity about the meaning of the expression.
[Symmetry in R3] When is a 3d figure considered solid?
Figure with non-zero height, non-zero width, and non-zero depth.
[Symmetry in R3] Polygon VS polyhedron?
polygon - finite number of straight line segments connected to form a closed polygonal chain. many anglespolyhedron - bounded solids whose faces are polygons. many faces
[Symmetry in R3] What do polygon VS polyhedron derive from (in Greek)?
polygon - many angles. πολύς (polús) 'much', 'many' and γωνία (gōnía) 'corner' or 'angle'polyhedron - many faces. poly- (stem of πολύς, "many") + -hedron (form of ἕδρα, "base" or "seat").
[Symmetry in R3] What's plural for polyhedron?
Polyhedra The plural of polyhedron is polyhedra or simply polyhedrons.
[Symmetry in R3] Name 2 ways to name these figures.
regular polyhedra Platonic solids
[Symmetry in R3] How many Platonic solids/regular polyhedra are there? Regular polyhedron - polyhedron in which all faces are congruent regular polygons.
There are only 5 Platonic solids.
[Symmetry in R3] Why are Platonic solids called so?
Because Plato associated them with 4 elements (+ aether).
[Symmetry in R3] Name 3 types of transformations that are potential symmetries in R3.
rotation (isometry) reflection (isometry) composite of rotations and reflections (not an isometry)
[Symmetry in R3] How many rotational symmetries does the tetrahedron have? Platonic solids
12 rotational symmetries.
[Symmetry in R3] How many indirect symmetries does the tetrahedron have given: it has 12 rotational symmetries has at least one indirect symmetry Platonic solids
24 symmetries
[Symmetry in R3] What's the reason there exist only 5 Platonic solids? Minimum 3 faces must meet at each vertex to form the vertex. Platonic solids
(3) Consider equilateral triangles. There could be 3, 4, or 5 faces meeting at the vertex. 6 faces would lie flat. (1) Consider squares. There could be 3 faces meeting at the vertex. 4 faces would lie flat. (1) Consider 5-gons. There could be 3 faces meeting at the vertex. 4 faces would make us concave.
[Symmetry in R3] What would it look like for 3, 4 or 5 equilateral triangle faces to meet at a single vertex? Minimum 3 faces must meet at each vertex to form the vertex.
Platonic solids
What's a group table?
Cayley table for the group.
What's the identity element in this group table?
The identity element is w. The 3rd row and the 3rd column repeat the borders of the table.
[Group properties] What's the name of this group property?
Left Cancellation Law.
[Group properties] Name 2 facts about the uniqueness of Identity element and Inverse element?
Identity element - is unique for the entire group! Proof: consider the Cayley table with 2 identity els.Inverse element - is unique per el! Proof: consider i1 ⚬ el ⚬ i2.
[Group properties] What's a good analogy for this group property?
[Composite action] Put your socks on; then put your shoes on. [Its inverse] Take your shoes off; then take your socks off.
[Group properties] How would we check that this property is true?
Just make sure y–1⚬x–1 fits the definition of the inverse.
[Group properties] What's this equal to?
[Group properties] In what way are Groups similar to Sudoku?
Each element of the Group occurs: exactly once horizontally exactly once vertically.
[Group properties] Why is the identity element always either: on the diagonal, or symmetric with respect to the diagonal?
Because the element is always either: self-inverse (g ⚬ g = e), or g ⚬ h = e = h ⚬ g
[Group properties] How can we tell whether the group is abelian from its group table?
The group is abelian iff it's symmetric (with respect to the main diagonal).
What does the set Un denote? The U in the notation Un is for units, which means elements that have multiplicative inverses.
All integers in ℤn that are coprime to n.
What's U5?
U5 = { 1, 2, 3, 4 } All numbers in ℤn coprime to 5.
[Groups in modular arithmetic] What prevents all ℤn to be groups under + or ×?
Existence of inverse elements. All (ℤn, +n) are groups All (Un, ×n) are groups All (ℤ*prime, ×prime) are groups
Indirect symmetries: in R2 in R3?
in R2 - when we need to lift the paper to turn it over. in R3 - when we need the 4th dimension.
How does the one connect to the other?
[Counting the symmetries of a regular polyhedron] How many symmetries does the octahedron have?
48.
[Counting the symmetries of a non-regular polyhedron] How many symmetries does this figure have?