A non-empty plane set
Motif
Repetitions of a motif in the plane
Pattern
4 types of Transformation Isometries
Rotation, translation, reflection, dilation
Turns a figure about a certain point in a plane; an isometry
Rotation
Slides a figure in any particular direction or a distance; an isometry
Translation
Mirrors a figure of a line known as the axis of symmetry; isometry
Reflection
Shrinks or expands a figure by some scaling factor; not an isometry
Dilation
A combination of reflection and a translation
Glide reflection
The original figure and the resulting figure are congruent
Isometry
The composition of isometries in the plane; mapping the pattern in the plane back itself
Symmetry
Has only reflections and rotations; has a center point
Rosette patterns
Dihedral group; a symmetric group involving seven reflections and 7 rotations
D7
A symmetry of twelve rotations
D12
Has reflections and rotations; also has translations and glide reflection but only along one line; the distance of translation is minimum/fixed
Frieze pattern
Only involves translation
Hop
Glide reflection symmetry; translation and reflection
Step
Translation and vertical reflection
Sidle
Translation and rotation (120*)
Spinning hop
Translation, glide reflection, & rotation at 180*
Spinning sidle
Translation and horizontal reflection
Jump
Contains all symmetries
Spinning jump
Translation symmetry is added in a second, independent direction; has 17 groups; patterns extend infinitely in all directions
Wallpaper groups
A pattern covering a plane by fitting together replicas of the same basic shape
Tesselation
The father of tessellation
Maurits Cornellis Escher
A tessellation made up of congruent regular polygons (triangles, square, hexagon)
Regular tessellation
Also known as Archimedean tessellation; regular tessellations of two or more different polygons around a vertex; each vertex has the same arrangement of polygons
Semi-regular tesselation
An edge-to-edge tessellation, but the order or arrangement of polygons at each vertex is not the same
Demi-regular tessellation
A geometric pattern that is repeated at a smaller scales to produce smaller shapes and surfaces that cannot be represented by classical geometry
Fractals
Repeating a process or function
Iterations
The method for generating fractals
Iterative function system
Comes out of a rule goes back into the rule for the next iteration
Recursion
A fraction used to specify the distance from one plotted point to the next relative distance
Scaling factor
The number of distance measurement needed to specify the size of an object in the eudidean world
Dimension