That is a Sierpinski triangle -- one of the simplest fractals we know. It can be generated in numerous ways. Fractals are intricate patterns repeating at different size scales. Similar structures can be seen on every scale of a fractal. Hence, they demonstrate the principles of self-similarity, emergence and self assembly. They are the beautiful geometries that emerge when simple algorithms reiterate and are generated by the computer. They are a visual representation of mathematics in computer feedback loops.

"The modern study of fractals dates back to 1975 when Benoit Mandelbrot discovered the Mandelbrot set fractal in his revitalization of the studies done by Koch, and Gaston Julia earlier this century." "Yes," says Angelina, "I do know what a mandelbrot fractal looks like but I never quite knew what it meant."

"Well, Mandelbrot was the first mathematician to actually apply the study of fractals to a better understanding of natural phenomena. The mandelbrot set, unlike the Sierpinski triangle, is one of the most infinitely complex fractals known and is generated by complex numbers."

"What is this strange object? Is it a fractal too?" asks Angelina as the move away from the alcove to get a better look.

"Actually this is a strange attractor. They represent a stable factor in the complex dynamics of chaos that fractals represent. They sort of embody in a simpler form, the general trend of dynamics of the whole system." Sarah clicks on the kiosk and continues. "See this page on the WWW? Archeologist Roger Grace of Norway is applying the study of strange attractors to the natural evolutionary aspects of culture such as the development of early paleolithic stone tool technology"

"And here is the Chelsea High School physics class investigation of the fractal characteristics of natural phenomena such as coastlines." says Angelina as she clicks on the other kiosk. "So complexity seems to be just as important as symmetry."

"Yes," answers Sarah, "both symmetry and complexity are fundamental concepts in all of the natural sciences -- they are essential in our understanding of the structure, dynamics and processes of life. Artists and computer artists, however, can often provide excellent visualizations that can be used by scientists to better understand these important characteristics.

Here is another good example. Like fractals and synergetic geometry, cellular automata are also a simple class of mathematical systems which can be useful as model for nature's processes -- showing how parts are related to one another in the growth process - showing how new aspects of form emerge into a system." Angelina looks at the blue and red tiling on the wall -- "hey, isn't that a tiling like the one we saw below -- a Penrose tiling?" Actually it is cellular automata on a quasicrystal. Physicist Eric Weeks at the University of Pennsylvania is using this combination to better understand nonlinear dynamics!"

"So physicists, chemists, biologists... all kinds of scientists can use these visualizations as tools to study all kinds of things?" asks Angelina.

"Yes," Sarah motions her over to another section on nature's growth patterns where exotic flowers seem to be blooming right out of the wall. "Geometry -- especially linked to computer visualizations -- provide us with a way to simulate certain kinds of design behaviors and system dynamics. Here is another way to simulate the complexity of nature's growth patterns - it is called an L-System"

"Yes, they look like plants, flowers and trees!" notes Angelina. "But why, if they are similar to fractals, do they look so much more natural?"

"Because they are based on the Fibonacci sequence. The fibonacci sequence was discovered by Italian Mathematician Leonardo Fibonacci in the 13th century. It is the number sequence in which every new number is formed from the two that preceded it: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. The Fibonacci sequence is the number series from which one can derive the Golden Mean"

Angelina remembers the Nautilus shell in the Proportion section of the gallery on the first floor. "You mean the Divine Proportion!"

"Exactly!" says Sarah, "the same characteristic you see in flowers, pinecones, starfish...."

"and leaves, ferns, trees!" Angelina finishes Sarah's sentence. "Now I get it, I understand why geometry is so important. It is the language that nature uses. And if we understand its properties... "

"Like Leonardo and Durer did to make their artworks as realistic as possible..." Sarah continues, "We can simulate the growth patterns of nature and replicate its forms in order to understand so many things about the world."

Sarah takes Angelina over to one last section of the alcove. She continues "And musician John Dunn is also using nature's language -- the DNA code. He produces each piece of music from the primary and secondary structure of protein sequences in the code of various lifeforms such as bats, starfish, sea urchins & scorpions." James clicks on the kiosk and they listen to the sound of a Starfish's DNA music

"It is undulating and beautiful -- sounds like it is under the water! -- Are you sure it is based on a natural code?" asks Angelina.

"Yes, In fact Mary Anne Clark, the botanist who worked with John Dunn in creating this "algorithmic DNA music" said that she felt that the amino acid sequences would have the right balance of complexity and patterning to generate musical combinations that are both aesthetically interesting and biologically informative."

They turn on the synthetic HIV Real Audio and a slow rhythmic cascade of low tones come out -- the beginning and the end of the piece has a strange, solemn reminiscence to a section of Mozart's Requiem.

Sarah and Angelina teleport outside to look underneath the alcoves to see the gallery from below through the beautiful glass floors.

"Hmmm, Sarah, I think I am beginning to see a lot of similarities here between all of these things. Geometry, music, art, nature. They all seem sort of linked -- the rules seem to be more about pattern, about beauty than about hard concepts that hurt my brain. It seems easier to learn this language visually so you can understand the relationships with your eyes rather than by sitting in front of a bunch of numbers on a page."

"Yes," says Sarah, "many artists, scientists and musicians have gotten great inspiration out of their close inspection of nature. The easiest way is to start by looking closely at her patterns and discover the intricate and delicate ways that nature forms around us."

Sarah and Angelina now fly above the gallery to look down through its glass ceiling. She continues, "When we try to duplicate nature's rules we begin to create, and to stumble on these forms that tell us we are on the right track. Buckminster Fuller's geodesic 'Expo 67' dome, like a transparent bubble on the landscape attests to what he said of his quest to use nature's principles for architecture "When I am working on a problem, I never think about beauty. I think only of how to solve a problem" , "But when I have finished, if the solution is not beautiful, I know it must be wrong."