patterns and symmetries Flashcards
it is ideal to stat with the concept of motif
transformation and isometries
a motif is “any non-empty plane set”
according to Grunbaum and Shephard, 1987
a pattern can be described as “____”
repetitions of a motif in the plane
each rotation of a figure is a ____
isometry
“the image of the basic motif under the additional no. of rotation is a pattern”
renee scott 2008
this is not a symmetry as it either shrinks or enlarges a figure
dilation
what is the difference between transformation and isometries
transformation:
changes the size, shape, position of a figure = create a new figure
geometry transoformation. two types are rigid (isometry) and nonrigid
isometry
change the rotation, translation and reflection = does not change the size or shape of figure
what are the types of transformations
translation
rotation
reflection
dilation
initial image =
transformed object =
initial image = pre-image
transformed object = image
briefly explain the different type of transformations
translation: moves object a certain distance (object is not altered)
reflection: mirror image
rotation: turns figure abt a fixed point called the center of rotation (clockwise and counterclockwise)
dilation: changes the SIZE of figure (smaller/ larger) but the SHAPE remains the same
naming of frieze patterns is attributed to ___
john conway, an english mathematician
John Conway is active in ____, ____, ____, ____ and____
finite theory
knot theory
number theory
combinational game theory
coding theory
these are patterns that repeat in a straight vertical or horizontal line
frieze pattern
where can frieze patterns be found
architecture
fabrics
wallpaper boarder
what are the 7 frieze patterns
hop
step
sidle
spinning hop
spinning sidle
jump
spinning jump
hop
translation only
step
translation and reflection (gidle reflection)
sidle
translation and vertical reflection
spinning hop
translation and rotation
spinning sidle
translation, gidle, reflection and rotation
jump
translation
horizontal reflection symmetry
spinning jump
all symmetries (translation, horizontal and vertical reflection and rotation )
how many different plane symmetry groups in wallpaper groups
17
symmetria means
to measure tgt
it is widely used in the study of geometry
symmetry
how does 2 object become symmetrical
have same size and shape
with one object having a diff orientation from the first
an object is not symmetrical
asymmetric
symmetry and patterns appear in ____ and ____
chemistry
biology
briefly explain the 2 types of symmetry
bilateral symmetry: object that has 2 sides that are mirror images of e/o
radial symmetry: center point and numerous lines of symmetry could be drawn (example: spiderweb)
Patterns in living things are explained by the …. (example)
Biological process of natural and sexual selection (bacterial population growth)
A pattern covering a plane by fitting tgt replicas of the same basic shape
Tessellation
The Latin word tessera means
A square table or die used in gambling
Semi-regular tessellations is aka
Archimedean tessellations
Briefly explain the different types of tessellations
- Regular tessellations
Made up of congruent regular polygons (tessellations must tile the floor w/o overlapping & and must be the same regular polygons) - Semi-regular tessellations
2 or more diff polygons arnd a vertex which has the same arrangement of polygons
3.Demi-regular tessellations
Edge to edge tessellations. order of arrangement polygons at each vertex is not the same
The function which iterates a figure to make it smaller and smaller or
bigger and bigger using a scaling factor
Fractals
means repeating a process over and over
Iteration
What is the special kind of iteration
Recursion
Explain Iterative function system
It is a method for generating fractals involving a large no.of calculations of a simple formula.
