[GEMATMW] Module 6: Mathematics in Natural Sciences and Arts

Transformation

process which shifts points of the plane to possibly new locations in the plane

Isometries

transformations which leave the dimensions of the object and its image unchanged

Translation

moves a shape in a given direction

Reflection

getting a mirror image

Rotation

has a point and an angle that tells where and how far the movement is made

Dilation

changes the size of an object

Glide Reflection

combination of reflection and translation

design

a figure with at least one non-trivial symmetry

pattern

a design that has a translation symmetry

Rosette

finite groups of isometries which can contain just rotations and reflections

Cyclic

has n-fold rotational symmetry and no reflectional symmetry

Dihedral

has n-fold rotational symmetry and reflectional symmetry

Frieze

an infinitely long strip imprinted with a design given by a repeating motif

Frieze

has translational symmetry in one direction

Frieze

has at least the basic unit and a copy of it by translation

Wallpaper

covers the plane and can be mapped onto itself by translations in more than one direction

Wallpaper

must have at least the basic unit, one copy by translation, and a copy of these two by translation in a second direction

Wallpaper

must be at least two rows, each one at least two units long

Basic unit

smallest region of the plane such that the set of its images under translations of the pattern generates the whole pattern

Fundamental region

smallest region in the basic unit whose images under the full symmetry group of the pattern cover the plane

Motif

subset of a fundamental region which has no symmetry but which generates the whole pattern under the symmetry group of the pattern

Leonardo of Pisa

invented the Fibonacci sequence

Fibonacci sequence

outcome of a mathematical problem about rabbit breeding

Fibonacci sequence

the first two numbers are 1 and 1, and each subsequent number is the sum of the previous two

Golden ratio

Shapes proportioned according to this have long been considered aesthetically pleasing

Fractals

a never-ending pattern

Fractals

infinitely complex patterns that are self-similar across different scales

Fractals

created by repeating a simple process over and over in an ongoing feedback loop

Fractals

involves symmetry of magnification

Benoit Mandelbrot

coined the word fractal