captivating symmetry by rndr daniela richtarikova slovak technical university (slovakia)

8/8/2019 Captivating Symmetry by RNDr Daniela Richtarikova Slovak Technical University (Slovakia)

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Captivating Symmetry

Daniela RichtrikovMechanical Engineering Faculty STU

Bratislava


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Symmetries

mathematics

transformations leaving an original invariant.

from geometric point of view

congruent geometric mappings - translation,

rotation, axial symmetry, and central symmetry -forming a group with respect to composition.

SCIENAR - Scientific Scenarios and ArtAmsterdam 2010

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History Survey

Ancient Greece

Philosophical category expressing a sort of regularity,

proportionality and harmony.

The first classifications of regular solids:

- discovery that there exist only five regular polyhedra,known as the Platonic solids - tetrahedron, cube,octahedron, dodecahedron, and icosahedron,

- their generalised truncated forms - Archimedean

semi-regular solids- Euclids geometric polyhedra description


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http://images.google.co.uk/imgres?imgurl=http://www.sewingupmath.com/multicolor%20platonic%20solids%203.jpg&imgrefurl=http://www.sewingupmath.com/poly%20kits.html&usg=__1Fp0M2v4T5Nkp1lZr3NWUCerktw=&h=687&w=1012&sz=38&hl=en&start=7&um=1&tbnid=FLm_znp7LCD8LM:&tbnh=102&tbnw=150&prev=/images?q=Platonic+solids&hl=en&sa=X&um=1http://www.ul.ie/~cahird/polyhedronmode/rhombi.htmhttp://www.ul.ie/~cahird/polyhedronmode/truncete.htm


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History Survey

Renaissance

masterpieces of Leonardo da Vinci and Albrecht Drer

Vitruvian Man (1487) De symmetria partium in rectis

Canon of Proportions formis humanorum corporum(1528)


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History Survey

16. 17. century

Johannes Keppler derived orbits of then known planets from the shape of platonic

solids,

and later he published three planetary motion laws.

discovered regular non-convex polyhedra In 1611 his paper on hexagonal symmetry of snow crystal was

released.


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http://www.its.caltech.edu/~atomic/snowcrystals/myths/w050118b108.jpghttp://home.connexus.net.au/~robandfi/GreatStelDod.html


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History Survey

18. 19. century

Symmetry development was tightly associated withdevelopment in crystallography and its strive to classify crystals

by groups according to their symmetries.

There exist 230 types. They are derived from combinations of32 crystallographic point groups with 14 Bravais lattices which

belong to one of 7 lattice systems.


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http://en.wikipedia.org/wiki/File:Hexagonal_lattice.svghttp://en.wikipedia.org/wiki/File:Orthorhombic-face-centered.svghttp://en.wikipedia.org/wiki/File:Orthorhombic-body-centered.svghttp://en.wikipedia.org/wiki/File:Orthorhombic-base-centered.svghttp://en.wikipedia.org/wiki/File:Orthorhombic.svghttp://en.wikipedia.org/wiki/File:Monoclinic-base-centered.svghttp://en.wikipedia.org/wiki/File:Monoclinic.svghttp://en.wikipedia.org/wiki/File:Triclinic.svg


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History Survey

1860

Luis Pasteurdiscovered the dependence between optical activity

and enantiomorphic molecular structure oftartaric acid.

Substance chemical structure theory started to be formed,explaining existence of isomers, substances of identicalchemical composition but of different physical and chemical

properties. Later, after the World War II thanks to x-ray, thistheory was confirmed.

levotartaric acid (D () tartaric acid) dextrotartaric acid (L (+) tartaric acid)


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History Survey

1832

Evariste Galois writes his work on permutation studies of

algebraic equations resulting into formation of the new branch,group theory. Group theory enabled to use symmetry as one ofthe main classification criteria in several science disciplines.

1872

Felix Klein proposed Erlangen program where geometry is

classified bysystem of invariants (properties which remainunchanged) with respect to geometric transformation group.

1890-91

Using Group theory, A. S. Fedorov and A. M. Schnfliesindependently characterised 17 two-dimensional and 230 three-dimensional crystallography groups.


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History Survey

20th century

Symmetry theory elaborating, its extention also to motion,

consequently contributed to the quantum theory formation. Using orbital symmetry L. Pauling and J. C. Slater built valence

bond theory in chemistry.

dz2 orbital fz3 orbital

Symmetry became one of the main principles and enabled to findcommon basis for microworld particles behaviour description.


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History Survey

... Regularity, rhythm has attracted humans attention forages. It has been reflected in human performance and

conserved in works. Remains of distinct cultures are the

evidence.

Surprising is the fact that long time before all possibilitiesof regular layout in circle, strip or plain were generated,

people used them in art. Ornamental art history shows that

first objects of rosette ornaments, all 7 types of frieze

ornaments, even signs of conformal and similarity

symmetry were used in middle and late Palaeolithic. All 17wallpaper patterns occurred in wall decorations of

Neolithic and early Middle Ages.


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Symmetry is Natural

Homogeneity, invariance are considered to be the state of

absolute symmetry. Interactions cause symmetry breaking

and patterns are produced. Although, patterns are not

symmetric in ideal way one can reveal wherever: regularmolecular structure, orbital symmetry, organism shape,

and periodic performances: seasons, Moon phases, heart

beat, biological rhythms, ... many substances exist in two

spatially symmetric forms from which only one form is

active. Even chaotic state presents elements of symmetry.

Nature is a self-regulating complex system keeping

balance signified just by means of symmetries.


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Symmetry

Ancient Greece

represents order

and regularity

Fractal Geometry

1970s

representsprobability

and chaos


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Symmetry & Chaos

Since beginning fractal geometry has been successfully

used for description and modelling ofnatural shapes andevens. It manages to join random influence with system of

simple rules resulting in production of deterministic object

repeating its structure on any scale.


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Nature


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Nature


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Nature


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Nature


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Fractals

iteration: xn+1=f(xn) selfsimilarity

non-integer Hausdorf dimension

a) self-similarity dimensionD,

m is a number of self-similar parts andris a contraction factor.

b) box-counting dimension d,

N(s) is a number of grid cellscovering the fractal ,

s is a distance between grid lines


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r

mD

1log

log=

( )

s

sNd

1log

log=


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Fractal Modelling

Fractals of Complex Plane

Mandelbrot set

zn+1 =zn2 + c

Colours represent the number of iterations


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Fractal Modelling

Fractals of Complex Plane

Julia sets

zn+1

= zn

3 - 1 zn+1

= zn

4 - 1

Colours represent the values to which the iteration rule

converges.

SCIENAR - Scientific Scenarios and ArtAmsterdam 2010 22


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Fractal Modelling


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IFS Fractals

IFS (Iterated Function Systems) are given by the finite set

of affine contraction mappings {W1

, W2, ... W

n} defined

on the complete metric space.

One step of iteration :

Ai+1= W(Ai) = W1(Ai) W2(Ai) ... Wn(Ai)

or Ri+1 = Wi (Ri)

The final object A

(A= W(A

) orA

= Ri ) represents

an attractor that from certain step becomes invariant.


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Fractal Modelling


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IFS Fractals

In following pictures one step of iteration was composed of

basic affine mapping,

main additional transformation which was picked at random

or under control out of the set 1, and

optional transformation which can also be picked at random

or under control out of the set 2.


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Fractal Modelling


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Rosettes

set 1: class of rotations set 2: axial symmetry


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Fractal Modelling


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Rosettes

set 1: class of rotations set 2: homothety


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Fractal Modelling


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Rosettes

set 1: class of rotations set 2: class of homotheties


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Fractal Modelling


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Translations

set 1: class of translations along a curve

set 2: class of homotheties


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Fractal Modelling


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Translations

set 1: class of rotations

set 2: class of translations along a curve


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Fractal Modelling


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Circle Inversions

set 1: circle inversions with symmetrical layout


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Fractal Modelling


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Circle Inversions

set 1: circle inversions

set 2: non linear mapping


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Fractal Modelling


Circle Inversions

set 1: circle inversions

set 2: non linear mapping


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34SCIENAR - Scientific Scenarios and ArtAmsterdam 2010

WebMathematica Applets

http://www.webmathematica.eu/Scienar1/index.php/


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Fractal Geometry

Being able to describe the regularity of items as well as

chaotic behaviour, producing naturally looking objects as

well as objects with surprising and interesting structure, it

sustainedly became a part of expressing art tools.



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Student Artworks



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Student Artworks



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Student Artworks



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Student Artworks



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Student Artworks



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Student Artworks



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Student Artworks

captivating symmetry by rndr daniela richtarikova slovak technical university (slovakia)

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