webMathematica - on the border between Science and Art by Dr Monika Kovacova, Slovak Technical University (Slovakia):

8/8/2019 webMathematica - on the border between Science and Art by Dr Monika Kovacova, Slovak Technical University (Slovakia):

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Vine and Table cloth - two geometries.


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The project SCIENAR explicitly aims in particular

to create an interactive environment for both Scientist and

Artists,

through this environment Scientists can explore the role that

Mathematics plays in understanding and making Art, as well

as produce mathematical objects that are useful in Art; while Artists can found mathematical structures and forms

that they can directly use, without needing the subtleties of

Mathematics, to inspire and produce their artworks.

ONE OF POSSIBLE WAYHOW TO FILL UP THIS AIM IS

webMATHEMATICA


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We will speak in this speech

1. about webMathematica as a possible tools for

producing artistic objects

2. about Lindenmayer systems, fractal plants and

Random walking as one of possible concepts

3. about our result achieved by the interaction between

science and art sphere in Slovakia

4. about the possibility - how to use dynamical webpages in creating Your own "piece of art"


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What is webMathematica?

WEBMATHEMATICA ENABLES THE CREATION

OF DYNAMICAL WEB SITES

THAT ALLOW USERS TO COMPUTE AND VISUALIZE

RESULTS DIRECTLY FROM A WEB BROWSER.

All of the computational power in Mathematica is available to

build special calculators and problem solvers that are

delivered over the web or over your corporate intranet to the

specific intranet site.

The development process is so simple that mostMathematica users can proceed through it without having

to go through long development cycles or needing the

services of dedicated developers.

In many cases, all that is required is adding the Mathematica

commands and a couple of simple tags to a web page.


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webMathematica technology

The web interaction of webMathematica is provided by a Java

web technology called Java servlets. Servlets are special Java

programs that run on a web server machine.

Support is provided by a separate program called a servlet

container(or sometimes a "servlet engine") that connects to theweb server. One of popular servlet container is Apache Tomcat.

Essentially all modern web servers support servlets natively

or through a plug-in servlet container.

Closely related to Java servlets are Java Server Pages (JSPs);

both servlets and JSPs integrate very closely with

webMathematica.

The computation and visualization engine for

webMathematica is Mathematica.


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And now for artists - simple and more useful

webMATHEMATICA allows a site to deliver HTML pages

that are enhanced by the addition ofMathematica

commands.

When a request is made for one of these pages, the

Mathematica commands are evaluated and the computed

result is inserted into the page and delivered to the client

browser.


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And now for artists - simple and more useful

How it works ?

1. Browsersent requests to webMathematica server.

2. webMathematica serveracquire Mathematica kernelfrom the

pool.

3. Mathematica kernel is initialized with input parameters, it

carries out calculations, hand returns result to server.

4. webMathematica serverreturns Mathematica kernelto the

pool.

5. webMathematica server returns result to browser.


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We can demonstrate the real applicationrunningon one part of Lindenmayer turtle graphics.


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1st step

Browser sends request to webMathematica server.

You can write to the web browser YOUR own Axiom,

Replacement rules, Number of iterations and Angle of

rotation.

How to choose these parameters is explained on all

created webMathematica pages.

How it works in real ?

explanation for artistson example of Lindenmayer systems


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2nd step

webMathematica server acquires Mathematica kernel from the

pool.

Don't understand? Never mind!

This server activity is realized automatically, withoutany needs from user.

Fractals - Lindenmayer Systems




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3rd step

Mathematica kernel is

initialized with input

parameters, it carries outcalculations, hand returns

result to server.

Don't understand? Nevermind!

This server activity is realized

automatically, without any

needs from user.

LSystemWithF[axiom_, (* initial sequence *)

rules_, (* replacement rules *)

iterations_, (* number of iterations *)\[Delta]_ (* angle of rotation *)] :=

Module[{minus, plus, fastRules, last, direction},

(* computation of the two rotation matrices, for "right" and "left" *)

minus = {{ Cos[ \[Delta]], Sin[ \[Delta]]}, {-Sin[ \[Delta]], Cos[ \[Delta]]}};plus = {{ Cos[-\[Delta]], Sin[-\[Delta]]}, {-Sin[-\[Delta]], Cos[-\[Delta]]}};

(* we rewrite the replacement rules in a form that is faster.*)fastRules = rules /. {(a_ -> b_) -> (a :> Sequence @@ b)};

(* Initial position and direction *)

last = {0, 0}; direction = {1, 0};

(* - multiple application of the replacement rules using Nest- interpretation of F, + and - using Which.

If "only" the direction direction is to be altered, the result is ...; , i.e.,

Null- add the initial position using Prepend

- sort out all "non - motions" - i.e. Null using Select *)

Select[Prepend[(Which[# == "F", last = last + direction,# == "+", direction = plus.direction;,

# == "-", direction = minus.direction;]& /@Nest[(# /. fastRules)&, axiom, iterations]),

{0, 0}], (* select all points *) # =!= Null&]]




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4rd step

webMathematica server returns Mathematica

kernel to the pool.

Don't understand? Never mind!

This server activity is realized

automatically, without any needs

from user.




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5th step

webMathematica server returns result to browser.

Do you want see it in practice?

Look at ...

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


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How it looks ?

real webMathematica pages


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Do you want see it in practice?

Look at ...



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Lindenmayer systems

Mathematics and beauty plants

Fractal geometry is appropriate for many natural forms.

Natural shapes which are well-approximated by fractals

include clouds, mountains, trees, bushes, rocks, dirt,

leaves, snow flakes, lightning, turbulent water, tree bark,rugged coastlines, brain convolutions, capillary beds,

bronchial tubes, and the distribution of galactic clusters.

To the artist, well-approximated means visually

convincing, while to the scientist it means descriptiveand/or predictive in a quantitatively meaningful way.


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The beauty of plants has attractedthe attention of mathematicians for centuries.

Geometric features such as the bilateral symmetry of leaves, the

rotational symmetry of flowers, and the helical arrangements of

scales in pine cones have been studied most extensively.

Beauty is bound up with symmetry.

In case we want to understand the beauty of flowers from the

mathematical point of view, it is need to analyze two separate

look in for that prolem.

The first is the elegance and relative simplicity

ofdevelopmental algorithms.

The second is self-similarity.When each piece of a shape is

geometrically similar to the whole, both the shape and the cascade

that generate it are called self-similar.

Lindenmayer systems



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Thus, self-similarity in plants is

a result of developmental processes.

The developmental processes are captured using theformalism ofmodeling of L-systems.

L-systems were introduced in 1968 by Lindenmayer as a

plants theoretical framework for studying the development of

simple multi-cellular organisms.

After then the plant models expressed using L-systems

became detailed enough to allow the use of computer graphics

for realistic visualization formal graphics languages but also for

visualization of plant structures and developmental processes.

Lindenmayer systems



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In 1968 a biologist, Aristid Lindenmayer, introduced a new

type ofstring-rewriting mechanism,

subsequently termed L-systems.

L-systems are applied in parallel and simultaneouslyreplace all letters in a given word.

The generated structures are one-dimensional chains of

rectangles, reflecting the sequence of symbols in the

corresponding strings.

In order to model higher plants, a more sophisticated

graphical interpretation of L-systems is needed

we will show it later.

Lindenmayer systems



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Lindenmayer systems



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Lindenmayer systems


How it looks ?webMathematica pages - in real


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Lindenmayer systems


"axiom " F - F - F - F

"replacement rule " F F F - F + F - F - FF

"axiom " F - F - F - F

"replacement rule " F F - F F - - F - F


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Lindenmayer systems


webMathematica pages - in realWe can create also automatic generator for L-systems.

It can produce also very attractive results.


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Lindenmayer systems



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Lindenmayer systems


Here are results produced by turtle walking (Lindenmayer systems)See also (gallery or create Your own)

http://www.webmathematica.eu/Scienar1/index.php/l-systems


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Lindenmayer systems


Here are results produced by turtle walking (Lindenmayer systems)See also (gallery or create Your own)



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Lindenmayer systems



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Lindenmayer systems


According to the rules presented before, the turtle interprets

a character string as a sequence of line segments.

Depending on the segment lengths and the angles between

them, the resulting line is self-intersecting or not, can be more

or less convoluted, and may have some segments drawn many

times and others made invisible, but it always remains just a

single line.

However, the plant kingdom is dominated by branching

structures; thus a mathematical description of tree-like

shapes and methods for generating them are needed formodeling purposes.

An axial tree complements the graph-theoretic notion of

a rooted tree with the botanically motivated notion of branch

axis.


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Lindenmayer systems


Root or base

In the biological

context, these

edges are referred

to as branch

segments.

Internode

A terminal

segment (with no

succeeding edges)

is called an apex

Plants are generally modeled with a special type of rooted

tree called an axial tree.


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Lindenmayer systems


Bracketed string representation

of an axial tree

F[+F][-F][-F]F]F[+F][-F]

Here are results produced by Fractal Plants application

See also (gallery or create Your own plants on)



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You can

create

Your own

plant

simple,

safety

and Your

plants do

not want

water for

long time.

http://www.webmathematica.eu/Scienar1/index.php/l-systems/

webmathematica/157-lindenmayers-plants


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Lindenmayer systems


"axiom " X

"replacement rules " X F X + X + F + F X - X " "

" " F F F" "

"axiom " X

"replacement rules " X F X + X + F + F X - X - X + F + FX " "

" " F F F" "


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Lindenmayer systems


And the real art application?

The following application were created by Slovak

students of School of Applied Art,

cat Birch tree


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cat Birch tree

Lindenmayer systems



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Other applications

Starting with such an image, and creating mirror images to

the left, below, and lower left, we get an image that is typical

for a kaleidoscope.

Art application inspired by

webMathematica application


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Other applications

We take a curve given in the form list

of circles and their parts and reflect

parts of this curve on some randomly

selected segments list of points.


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Other applications Penrose tilling


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Other applications

Thank You very much and

You are welcome onYou are welcome on


Altering server

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

Documents

webMathematica - on the border between Science and Art by Dr Monika Kovacova, Slovak Technical University (Slovakia):