The Mathematics of Patterns and Symmetries Flashcards
A transformation changes the _____ , _____, or _________of a figure and creates a new figure.
size
shape
position
either rigid or non-rigid; another word for a rigid transformation is “isometry”.
geometry transformation
An isometry, such as a ____ , ______ ,
or _____ , does not change the size or shape of the figure.
rotation
translation
reflection
not an isometry since it either shrinks or enlarges a figure
A dilation
Types of Transformations
- Translation
- Reflection
- Rotation
- Dilation
The initial object to be transformed is
called
pre-image
the transformed object is called
image.
A mathematical term used in geometry to describe a function that moves an object a certain distance. The object is not altered in any other way.
Translation
It is a transformation in which the figure or object is mirror image of the other.
Reflection
It is a transformation that turns a figure about a fixed point called the center of rotation. Rotations can be done clockwise or counterclockwise.
Rotation
a transformation that changes the size of a figure. It can become larger or smaller, but the shape remains the same.
Dilation
can be combined to yield an effect shown
below.
Translation and reflection
Naming of__________ is attributed to ______, an English mathematician who is active in finite theory, knot theory, number
theory, combinatorial game theory and coding theory.
Frieze Patterns
John Conway
he died last April 11, 2020 due to COVID19.
John Conway
patterns that repeat in a straight vertical or
horizontal line.
Frieze patterns
are found in architecture, fabrics, and
wallpaper borders, just to name a few. There are seven types of frieze
patterns and we will discuss each type and how to identify them.
Frieze patterns
7 Frieze Patterns
- Hop (F1)
- Step (F2)
- Sidle (F3)
- Spinning Hop (F4)
- Spinning Sidle (F5)
- Jump (F6)
- Spinning Jump (F7)
- A pattern which only involves translation.
- Hop (F1)
- It is a combination of translation and reflection shown by the
following figure. Conway also called it glide reflection symmetry.
- Step (F2)
- The third consists of translation and vertical reflection symmetries.
- Sidle (F3)
- It contains translation and rotation (by half turn or rotation at 180o
angle) symmetries
- Spinning Hop (F4)
- It contains translation, glide, reflection and rotation (by a half-turn or
rotation at 180o angle) symmetries.
- Spinning Sidle (F5)
- It contains translation and horizontal reflection symmetries.
- Jump (F6)
- It contains all symmetries ( translation, horizontal and vertical reflection, and rotation)
- Spinning Jump (F7)
Symmetry comes from a Greek word _________ meaning _________ and is widely used in the study of geometry.
symmetria
‘to measure together’
means that one shape becomes exactly
like another when you move it in some way: turn, flip or slide. For two objects to be symmetrical, they must be the same size and shape, with one object having a different orientation from the first.
symmetry
Not all objects have symmetry;
if an object is not symmetrical, it is called _____
asymmetric
widely used in fields of mathematics and
arts, it also appears in chemistry and biology.
Symmetry and pattern
in which an object has two sides that are mirror images of each other.
Bilateral symmetry
this is where a center point and numerous lines of symmetry could be drawn.
Radial symmetry
A pattern covering a plane by fitting together replicas of the same basic shape.
Tessellation
The word tessellation comes from Latin word ______, which means a square tablet or die used in gambling.
tessera
3 TYPES Tessellations
- Regular Tessellation
- Semi-Regular Tessellations
- Demi-Regular Tessellations
A tessellation made up of congruent regular polygons which have the following properties:
- The tessellation must tile a floor (that goes on forever) with no overlaps or gaps.
- The tiles must be the same regular polygons.
- Regular Tessellation
Also known as ________ are regular tessellations of two or more different polygons around a vertex which has the same arrangement of polygons.
Archimedean Tessellations
Is an edge to edge tessellation, but the order or arrangement of polygons at each vertex is not the same.
- Demi-Regular Tessellations
Also known as Archimedean Tessellations are regular tessellations of two or more different polygons around a vertex which has the same arrangement of polygons.
- Semi-Regular Tessellations
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. In mathematics, iteration means repeating a function over and over.
Iteration
a special kind of iteration. With
recursion there is given starting information and a rule for how to use it to get new information.[
Recursion
