"Tell me more about proportion." says Angelina as she looks up at a beautiful tessellation on the wall. "Well," says Sarah, "Proportion is basically the relationship between things or parts of things. It refers to the way the parts 'fit' together. Proportion is essential in architecture and art. The search for most exquisite proportion in nature was the search for the key to beauty. He points to the picture on the wall.

"One of the fathers of geometry, Pythagoras, discovered and articulated the Golden Mean as the 'Divine Proportion' because it was so exquisite and found most frequently in nature. Leonardo da Vinci actually called it the Golden Section and it refers to a certain length that is divided in such a way that the ratio of the longer part to the whole is the same as the ratio of the shorter part to the longer part."

"You can see the 'divine proportion' in the spiral of the nautilus shell, pine cones and pinapples, even in the shape of a strand of DNA. And of course you can see it in the natural objects with fivefold symmetry that we looked at earlier - the buckyball, quasicrystals and viruses. One of the most useful ways to look at proportion is to study the symmetries of tesselations -- polygons that fill space on the plane. She points to a large moving java tessellation from the web pages of Athanassios Economou at UCLA.

Different kinds of symmetry - rotational, translational and reflection symmetry - define the types of polygons that can fill space without gaps and the beautiful designs they make. Islamic tilings show that a high degree of geometric and mathematical understanding in the East goes back for many centuries before the West began to rediscover these important rules.

"In fact, Pythagoras most likely got most of his mathematical information from his extensive trips to the East early in his life." says Sarah.

"So if the artists, scientists and architects of the Renaissance used geometry for their work, what do we use it for now?" Angelina asks. Sarah turns to the stairway and motions her upward. "Geometry informs everything from biology to chemistry to astrophysics! But now scientists and artists are looking not just at the design of nature, but at the processes of nature's design. These are found in new explorations of complexity, emergence and dynamical systems."

"A few geometers stand out in this century for their experiments with dynamic or transformational geometries -- in other words, geometries that can grow and change." Sarah points to another wall in the Symmetry wing of the gallery before heading up the stairs.

"In 1927 Buckminster Fuller, the inventor of the geodesic dome, also looked to find the design principles of nature. But he wanted to look at how she does what she does, not just to understand her designs."

"Fuller's understanding of geometry was sparked by the work of Einstein in the 1920s, who redefined space to have curvature. While Einstein had to turn away from Euclidian geometry and to the newer spherical geometries of Riemann and Gauss, Fuller thought a new "energetic" geometry must map the angles and curvature of space to find the dynamics of form." By the late 1940s, he had fully articulated a different type of dynamic or energetic geometry called Synergetic Geometry. It showed how shapes transform. He started at the center of the sphere and defined a structural system based on the tetrahedron rather than the cube."

"Fuller's experiments with synergetic geometry formed the basis of his geodesic math. It was this geodesic math that made large lightweight spherical structures possible. He wanted to figure out the interangling of forces inside structure that make it grow, make it strong and efficient. His goal was to ultimately use these principles in archictectural design. Using spacefilling, Fuller found that tetrahedra and octahedra combine to form a very strong stable matrix in which lightweight materials can be used to make strong structures. This all came about through dispensing with points lines and planes and working with 60 degree angles and the triangles rather than the cube or square.

Sarah pointed to the cluster of moving animations on the walls of the second floor of the gallery. "Contemporary computer artists and others such as Gerald de Jong, Richard Hawkins, Karl Erickson, Kirby Urner, Russel Chu, Scott Childs, Dave Chako, Rick Bono have begun to further explore and animate some of the structures on which Fuller based his geodesic math. Their combined works are resulting in a whole new generation of synergetic geometries which model nature's dynamic properties."

They are very entrancing - elegant and beautiful - but what are these geometries useful for?" Angelina asks as she marvels at the large moving shapes around her.

"The more that chemists, biotechnologists and astrophysicists find out about and try to model structure on the largest and smallest scale, the more these scientists find that they have to model the dynamics or processes of these structures rather than merely to mimic their shapes. These structures are never static. They are always transforming! They also rarely find cubic shapes on the smallest and largest scales -- the angles are far more often based on the triangle and the tetrahedron."

"Triangles triangles triangles! You won't find many 90 degree angles in synergetic geometry -- or nature for that matter. Here is a tetrahelical moebius strip by Gerald de Jong" Sarah and Angelina walk inside of it and look up as it glows in different colors, showing the innovative structures possible using de Jong's elastic interval, STRUCK geometry.

Angelina looks at the large triangle on the wall behind the sculpture. "What is that?" she asks.