Math and Art by audre WeirdArts.com

Math and Art

Math and Art

• creativity comes in many forms and sometimes scientists are artists, and artists are mathematicians!

–Scientists are creative

• It takes aesthetics and logic to be good at complex problem solving

• “Seeing” intricate relationships in numbers and data takes inner vision that is a creative function

–Learning how to be creative in the sciences is what differentiates a good scientist from a merely adequate one

–Mathematics is not just about formulas and logic, but about patterns, symmetry, structure, shape and beauty

Math and Art: Aesthetics

• Geometry affects the human psyche–Mathematical relationships affect our perceptions

• Wavelength - the mathematical property of light and sound – it’s around us everywhere

– Harmonics of colors – the basis of all color theory– Harmonics of sound – the basis of all music theory

• Shapes:– Pyramids - power, strength– Spirals - dizzy, vertigo, movement– Lines - sharpness, suddenness– The use of shapes (math/geometry) in art allows the artist to control

the viewing perspective, or ‘space’ of the viewer

• By taking advantage of our brain’s pre-wired concepts of our environment, artists selectively use (and abuse) geometry to force the viewer into a certain perspective

• Just because artist aren’t aware that they are using math doesn’t mean they aren’t.

Math and Art: Perspective

Before perspective drawing techniques were invented in the 16th century, artists often used ways to show depth and space in their art by overlapping objects,

showing large objects in foregroundand small objects in the background

Math and Art: Perspectives

Linear Perspectivea geometric method of representing on paper the way that objects appear to get smaller and closer

together, the further away they are.

Math and Art: Perspectives

One of the first perspective paintings

Math and Art: Playing with Optics

• ANAMORPHOSIS - Geometry is a very powerful artistic tool

• An anamorphosis is a deformed image that appears in its true shape when viewed in some "unconventional" way.

• Webster's 1913 Dictionary: A distorted or monstrous projection or representation of an image on a plane or curved surface, which, when viewed from a certain point, or as reflected from a curved mirror or through a polyhedron, appears regular and in proportion; a deformation of an image.

• the image must be viewed from a position that is very far from the usual in-front and straight-ahead position from which we normally expect images to be looked at.

• the image must be seen reflected in a distorting mirror (typical shapes being cylindrical, conical and pyramidal).

• Anamorphoses are based on precise mathematical and physical rules, the same rules that apply to the construction of all two-dimensional representations of three-dimensional objects, except that the rules are applied in ways that are a deliberate break from the usual and conventional

Math and Art: Perception and Geometry

The orientation of the picture is normally 90 degrees to the viewer. But there is no reason why the window cannot be turned…


On the banks of the River Wear in Sunderland… a project lead by Colin Wilbourne

Math and Art: Perception and GeometryWhen you view the wall from the viewing seat… something magical happens…

Math and Art: Holbein, The Ambassadors, 1533

Math and Art: M.C. Escher

For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art. – M.C. Escher

The “logic” of space is important to artists and represents those spatial relations among physical objects which are necessary, and which when violated result in visual paradoxes, sometimes called optical illusions. (e.g. corners in our dimension are 90 degrees)

This space has no 90 degree angles. As the fish move from the inside to the outside of their ‘universe’ they get smaller, but they don’t know it because their space is shrinking with them.

All artists are concerned with the logic of space, and many have explored, and broken, its rules quite deliberately.

Circle Limit III by M.C. Escher, 1959

Among the most important of Escher's works from a mathematical point of view are those dealing with the nature of space itself.

http://www.mathacademy.com/platonic_realms/minitext/eschfram.html


Math and Art: Tiling and Symmetry

Every culture has a preference for certain symmetry type of patterns

Patterns in Islamic Art

Math and Art: Tiling and Symmetry

Every culture has a preference for certain symmetry type of patterns

Math and Art: Kaleidoscope

Kaleidoscope-type, or Mandala, images have been around for centuries!

The Mandala is considered by many cultures to be Sacred

Geometry and ArtMandala by artist Paul Heussenstamm ©2005

Math and Art: Fractals

•Fractal appears to be a very modern term created by Benoit Mandelbrot in an article published in Scientific American magazine around 1975. Mandelbrot discussed his choice of names in The Fractal Geometry of Nature. In it he wrote "I coined fractal from the Latin adjective fractus.“

–The fractus is a derivative of frangere, for broken, which is also the root of fraction.

•Ancient peoples appreciated fractals too, even through they didn’t have a specific word or concept for them… things like sea shells, and plants like broccoli, pines, ferns too are naturally occurring fractals


•What is a fractal?–objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some piece of the object is a scaled down version of the previous iteration

–Recursion: With a recursion we are given starting information and a rule for how to use it to get new information. Then we repeat the rule using the new information as though it were the starting information.

–So we have a loop. What comes out of the rule goes back into the rule for the next iteration.

Math and Art: Fractals - basics

•Example of a recursion which cranks out a sequence of numbers called the Fibonacci Numbers:

We have starting information and a rule for generating a new value. The n increases by one each time, so we can ask questions like find the ninth fibonacci number. We are given two starting values since each new value is calculated from the two previous ones.


So the ninth number in this sequence would be calculated like this:

The first numbers are 1 and 1, as given.

The rule says to take the two previous numbers and add them to get the new number:

n = 3: 1 + 1 = 2n = 4: 1 + 2 = 3n = 5: 2 + 3 = 5n = 6: 3 + 5 = 8n = 7: 5 + 8 = 13n = 8: 8 + 13 = 21n = 9: 13 + 21 = 34

Math and Art: Fractals – more information

• Some more information about fractals:–The starting information is called the initiator–The rule for iterating is called the generator–Many fractals are self similar.

• Self-similarity means that two or more objects have the same characteristics.

• In fractals, the shapes of lines (or patterns) at different iterations look like smaller versions of the earlier shapes

Remember this one?

It’s a great example of a very artistic, hand-generated fractal!

This is Escher’s concept of a hyperbolic space.



•Other interesting things to ponder:–Things that work for finite sets may not work for infinite sets.

–An infinite amount of stuff doesn't always take an infinite amount of space.

–The sum of an infinite number of numbers can be finite.

• Think about the tortoise and hare race: The tortoise travels the following distances, one fraction for

each time step:1/2 + 1/4 + 1/8 + 1/16 + 1/32 + etc... This means he never gets to the end of the race because the

sum of the above sequence never gets past 1!

–Seemingly chaotic (random appearing) functions can ultimately create an ordered appearing system


• Let’s play the Chaos Game–Why is this game interesting?

• What happens when we spill out a bag of marbles onto the floor? After they stop rolling around, we get a pattern formed by the marbles - perhaps not a very interesting one. If we spill the same marbles out on the same floor again, do we expect to get the same pattern of marbles? No. The pattern is pretty random, and we expect that. The chaos game was proposed by Michael Barnsley in the mid-1980s as a way to see how patterns can result from certain random events.

–start with a set of dots on a page -- we'll call them vertices. The classic game starts with three vertices numbered 1, 2, and 3, and places them at the corners of an equilateral triangle:


• Now we choose a point at random on the page, and then roll a three-sided die. Move half way to the point whose number was rolled and draw a new point. Real life three-sided dice are hard to find (in fact impossible!) so to fake one, we will use a regular six-sided one, letting a roll of 1 or 2 move towards vertex 1, rolls of 3 or 4 towards vertex 2, and rolls of 4 or 5 moving towards vertex 3.

–For example, suppose we chose point P below and then rolled a 1, 6, 2, 2, and 4 in that order:

Math and Art: Fractals – Chaos Game

Recognize the pattern? This is what Barnsley wanted to demonstrate: Randomness can

generate a very precise pattern sometimes.

We continue plotting points in this way. After five hundred points a pattern starts to appear:

Math and Art: Fractals – Self Similarity

To build the original Koch curve, start with a line segment 1 unit long. (Iteration 0, or the initiator) – that would be the top-most image of the line.

Replace each line segment with the generator shown in the middle illustration.

Then take the line segment and replace it with four new segments, each a copy of the original generator.

Repeat this process on all line segments and you get the image shown at the bottom.

This process repeats itself infinitely in the Koch Curve so that zooming in on any part of the curve yields a copy of itself.

While iteration and recursion describe the process of repeating steps. Self-similarity is a property of the object, not of the

steps used to build the object.

The Koch Curve is a great example of this concept.

Math and Art: Fractals – Self Similarity, another example

This is a Van Koch fractal. It is based on a very simple shape.

This process is repeated again and again to create an infinitely complicated fractal. Every part of the fractal contains the original shape. We say that the fractal is self-similar.

To create the fractal, the flat lines are replaced by the entire shape itself.

Math and Art: Fractals – Self Similarity

Here is the process which yields the Koch Star

Math and Art: Julia sets

One of the most basic fractal types is the family of Julia sets, discovered by the French mathematician Gaston Julia during the first World War. Julia sets are created by a simple formula with one complex parameter

called c or seed. This parameter can be varied to create many variations. Here are a few examples.

Julia sets are also self-similar, as illustrated by the following zooms into the last image above. The first zoomed image shows the top of the original. Further zooms are illustrated by the small red rectangles in the

images.

The same spiral-like shape is repeated over and over again.

Math and Art: Mandelbrot set

The Mandelbrot set, discovered in 1980 by Benoit Mandelbrot, is probably the most famous fractal. Like Julia sets, it is generated by a very simple formula, but it is incredibly complex.

The Mandelbrot set is loosely self-similar: parts of the original fractal appear again when zooming in, but often deformed and with different ornaments. This is what makes it so rewarding to zoom into this fractal: you never know what you will see next.This is illustrated by the following short zoom, starting at the very left of the Mandelbrot set shown above. As you zoom in, you see copies of the original Mandelbrot set, but with different surroundings.

Another interesting aspect of the Mandelbrot set is that it's actually a map of all Julia sets. Each point corresponds to a Julia set. Points inside the Mandelbrot set (here shown as black) are connected Julia sets;

points outside the Mandelbrot set tend to give more disorganized Julia sets.

Math and Art: Fractal Popularity

• Why are fractals so popular?–Because of the many softwares available today, which allow

unprecedented ease-of-use in the creation of fractals with almost no math required by the user!

–The affordability of serious computing power—available in even the most modest home computer system—allows almost everyone to create magnificent images utilizing complex fractals, versatile coloring schemes and intricate transformations.

• Fractals have grown from a mathematical curiosity to a respected form of art.

–There are fractal exhibitions in museums and galleries all over the world.

–There is a large number of online galleries on the web, even some that offer prints and posters from various fractal artists. (hey, sounds like deviantART! Doh!)

–And because there are so many fractal programs available (some freeware, some for a modest shareware registration fee), there is bound to be one user interface that is right for just about every comfort and skill level.

• It’s no wonder that we are seeing a huge increase in the number of artists taking advantage of computer technology to explore fractals and their application in their own work.

Math and Art: Fractals Today — Ultra Fractal

• While Ultra Fractal is pretty good at exploring the classic fractal types discussed so far, it can do much more than that.

– There are many more fractal types to choose from– You can even write your own fractal formulas (or use formulas written by other,

very generous people, who make their formulas available for public download)– Most fractal types are variations on the Mandelbrot and Julia sets.

• Transformations can be added to distort the shape of the fractal. • Multiple layers are also available, allowing the user to combine different

fractals or different coloring methods to form the final image.• With the switch feature in Ultra Fractal, you can easily pick a point of a

Mandelbrot fractal to see the corresponding Julia set. This is the best way to discover interesting Julia sets.

• Ultra Fractal solves the fractal equation for a 2 dimensional solution. – That is, the values of the fractal at each point in the 2 dimensional space

represented by the equation are encoded with a particular color based on the calculated value of the fractal at that point.

– The colors that are assigned to fractal outputs cans be manipulated by the user so that the change in color from one fractal solution value to the next can be in steps of varying size.

• For example: – linear rates of change from one color to the next– logarithmic – sinusoidal (and the corresponding cosine, tangent, and hyper values)– Exponential

Math and Art: Fractals Today — Ultra Fractal

• Each fractal type can be combined with various coloring algorithms, each capable of coloring the fractal in a different way.

• Since fractals are functions whose values are derived from several iterations of calculations, the user is also in control of how many maximum iterations are allowed at each point.

• In some cases, the value of the fractal at a certain point is indeterminate because the fractal is somewhat chaotic at that position, under those initial conditions. In cases like these, the software reaches the maximum number of iterations before zeroing in on the value, so it applies the ‘default’ color as defined by the user.

Math and Art: Ultra Fractal Work Space

The basis for the fractal calculation begins with the

formula.

To the right is a tiny snip of what a typical Ultra Fractal formula

might look like

As you can see, it’s a fairly straight forward c-like language.

Every part of the fractal calculation process is controlled

by formulas.

There are three types of formulas: fractal formulas,

coloring algorithms, and transformations.

A formula can be seen as a small, specialized computer

program that is compiled and executed by Ultra Fractal.

By writing your own formulas, you can completely customize

how a fractal is calculated.

→

comment { Carr1800.ufm Version 1.1 Ultra Fractal conversions of Robert Carr's formulas Carr1800 - Carr1899 by Erik Reckase, March 10, 2000

History: Version 1.1 - Major optimization overhaul, cleanup

Version 1.0 - Initial Release Original formulas pulled from _rc1.frm

Carr1800 : divide by undefined variable Carr1809,1812,1847-1851, 1854-1855,1877-1878 : cosxx fix}

Carr1800 { ; Updated for UF2 by Erik Reckase, March 2000init: z=1/pixel, c=1/(pixel-sqr(z/z^z+z)/atan(pixel))loop: z = z^3 + cbailout: |real(z)| <= 10default: title = "Carr 1800" periodicity = 0 maxiter = 500 magn = .6 center = (0,0)}

Carr1801 { ; Updated for UF2 by Erik Reckase, March 2000 ; Mandelbrotinit: z=1/(pixel-conj(1/pixel)-flip(1/pixel)) c=1/(pixel-z-cos(sqr(z/(pixel-.124))))loop: z = z^2 + cbailout: |real(z)| <= 10default: title = "Carr 1801" periodicity = 0 maxiter = 500 magn = .6

… program continues …

Math and Art: Fractals – Ultra Fractals As Art

The following are examples of 2-dimensional renders of fractal equations created using Ultra Fractal

Math and Art: Fractals – As Art

Math and Art: Fractals – Another Dimension

• Fractals can exist in multiple dimensions. –Theoretically, they can exist in an infinite number of dimensions. –For our presentation, we’ll stick to just the 2nd and 3rd dimensions.– (Although, a 4th dimensional fractal is one that changes over time, so it

is possible to calculate fractals with respect to time and make some very engaging animations!)

• So far, we’ve only talked about viewing and calculating fractals in 2 dimensions. Working in 3 dimensions opens up, well, a whole new dimension in what can be done with fractals!

• The same concepts apply to a 3D fractal as a 2D, only for every point within your viewing window instead of just rendering the value at X an Y, we now have a Z to worry about.

• Having a Z-axis allows us to rotate our fractal’s frame of reference with respect to our rendering camera to obtain some very cool shapes.

Math and Art: Fractals – As Art

Math and Art: Fractals – 2D vs 3D

As you can see mathematics and fractals play a fundamental role in not only the geek side of

things, but also the art side of things. With math, we, as artists, can better control the

effect our art has on our viewer – and that’s what it’s all about!

Math is the foundation of it all!

Math and Art by audre WeirdArts.com

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