Final

Question 1

group

Answer 1

A group is a collection of elements and an operation · such that:

1) performing the operation on all ordered pairs of elements generates a new element in the group
2) there is an identity element e∈G such that e·g=g·e=g for all g∈G
3) for all elements g∈G there is a corresponding element g^−1 such that g·g^-1=g^-1·g=e
4) the operation is associative: (g·h)·j=g·(h·j)

Question 2

subgroup

Answer 2

A subgroup H of a group G is a subset of the group that is also a group under the operation from G. This means that:

1) the identity element is in H
2) every inverse of an element in H is in H
3) H is closed under multiplication, which means that all products of elements in H are also in H.

Question 3

generating set

Answer 3

Let G be a group and let A = {g_1, g_2, …, g_n} be a collection of distinct elements of G. We say A is a generating set for G if every element of G is a product of elements from A and their inverses.

Question 4

prototile

Answer 4

notebook?

Question 5

protoset

Answer 5

notebook?

Question 6

tiling

Answer 6

A tiling T is a collection of tiles and finite collection of prototiles such that:

1) each prototile is topologically equivalent to a disk
2) each tile is congruent to a prototile
3) T is a covering
4) T is a packing

Question 7

vertices and edges of a tiling

Answer 7

A vertex of a tiling is a point where three or more tiles intersect
An edge of a tiling is a subset of the boundary of a tile that contains no vertices in its interior and that is bounded by two vertices

Question 8

valence/valency of a vertex

Answer 8

The valence of a vertex is the number of edges that share (or meet at) that vertex

Question 9

isometry

Answer 9

A distance-preserving motion on points:

d(x,y) = d(σ(x),σ(y))

Question 10

direct and indirect isometry

Answer 10

A direct isometry sends a triangle with edges oriented clockwise to a triangle with edges oriented clockwise
An indirect isometry sends a triangle with edges oriented clockwise to a triangle with edges oriented counterclockwise

Question 11

rotation

Answer 11

A direct isometry that fixes a point p and rotates the entire plane about p by a counterclockwise angle θ (to rotate clockwise, take θ to be negative)

Question 12

translation

Answer 12

A direct isometry that has no fixed points and moves every point in the plane by a vector v

Question 13

reflection

Answer 13

An indirect isometry

Question 14

edge to edge tiling

Answer 14

A tiling where all the corners and sides of the polygons coincide with the vertices and edges of the tiling

Question 15

How many tiles can a disk meet?

Answer 15

A disk D ⊂ E^2 meets finitely many tiles.

Question 16

How many vertices can a tile (in a tiling) have?

Answer 16

A tile in a tiling has finitely many vertices.

Question 17

How many reflections at most needed for an isometry?

Answer 17

An isometry is the PRODUCT of at most 3 reflections.

Question 18

How many point determine an isometry (of E^2)?

Answer 18

An isometry of E^2 is determined by its action on 3 points OR 2 points plus the knowledge of whether it is direct or indirect

Question 19

What are the 4 classifications of isometries?

Answer 19

The classification of isometries as rotations, translations, reflections, and glide reflections.

Question 20

The isometries of E^2 form what? The direct isometries form what?

Answer 20

The isometries of E2 form a group. The direct isometries form a subgroup.

Question 21

The symmetry group of a tiling is finite

Answer 21

we did this in class?

Question 22

covering

Answer 22

A covering covers the whole plane (the union of all the tiles covers the whole plane)

Question 23

packing

Answer 23

Any pair of tiles do not interest

Question 24

identity isometry (aka the trivial isometry)

Answer 24

the isometry that does not move any points