Final Flashcards
group
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)
subgroup
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.
generating set
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.
prototile
notebook?
protoset
notebook?
tiling
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
vertices and edges of a tiling
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
valence/valency of a vertex
The valence of a vertex is the number of edges that share (or meet at) that vertex
isometry
A distance-preserving motion on points:
d(x,y) = d(σ(x),σ(y))
direct and indirect isometry
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
rotation
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)
translation
A direct isometry that has no fixed points and moves every point in the plane by a vector v
reflection
An indirect isometry
edge to edge tiling
A tiling where all the corners and sides of the polygons coincide with the vertices and edges of the tiling
How many tiles can a disk meet?
A disk D ⊂ E^2 meets finitely many tiles.
How many vertices can a tile (in a tiling) have?
A tile in a tiling has finitely many vertices.
How many reflections at most needed for an isometry?
An isometry is the PRODUCT of at most 3 reflections.
How many point determine an isometry (of E^2)?
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
What are the 4 classifications of isometries?
The classification of isometries as rotations, translations, reflections, and glide reflections.
The isometries of E^2 form what? The direct isometries form what?
The isometries of E2 form a group. The direct isometries form a subgroup.
The symmetry group of a tiling is finite
we did this in class?
covering
A covering covers the whole plane (the union of all the tiles covers the whole plane)
packing
Any pair of tiles do not interest
identity isometry (aka the trivial isometry)
the isometry that does not move any points
