Angelina enters the Gallery Show entitled, Geometry - The Language of Art and Science....

She remembers that she had a tough time in her geometry class last semester. All those numbers and formulae, points, lines and planes. Too many rules to remember -- yeesh! And what do they have to do with anything? But this is a show about geometry and art. Well that might be okay..... Inside, Angelina sees her classmate, Sarah who helped work on the gallery show.

"They are all very beautiful and intricate, but what is fivefold symmetry?" asks Angelina. She clicks on the kiosk and gets to a web page that explains that fivefold symmetry or pentagonal symmetry is very common in living forms around us - flowers, starfish and sand dollars. "Fivefold symmetry is most clearly seen in the dodecahedron - a polyhedron with twelve pentagon faces and the icosahedron with 20 triangular faces." says Sarah as she shows Angelina the icosahedron on the wall. "Wow it sort of looks like a sand dollar or a starfish!" she remarks. They click to VRML polyhedra on Vladimir Bulatov's polyhedra web page and look more closely at the icosahedron and dodecahedron.

"Fivefold symmetry is the property that makes some of the natural structures we know extremely strong and flexible. That is why viruses are so hard to fight -- their structure is based on the icosahedron - scientists did not even know the structure of viruses until they were able to view them with an electron microscope in 1962." Angelina looks at the large tomato bushy stunt virus on the wall -- "Wow - it is so beautiful!" Sarah asks her to click on the kiosk to get to a web page on virus structure from London University's biology department

They move down to another image on the wall - an X-ray diffraction image of a quasicrystal. Sarah continues to explain, "It was not until the mid- 1980s that scientists discovered that fivefold symmetry can be found in what we consider to be inanimate matter. With the discovery of both buckminsterfullerene and quasicrystals in about 1985, we found fivefold symmetrical forms of our most common element, carbon and in crystals!" They click to a page from the University of Japan on quasicrystals to find out more.

Sarah continues. "In order to study the structure of quasicrystals and how they grow, we can look at the work of mathematician Roger Penrose, who discovered a way of tiling the plane with fivefold symmetry -- they have a very important mathematical property -- they are based on the Golden Mean"

"Wow, if you look from back here, it looks like pentagons within pentagons within pentagons!" says Angelina. It reminds me a little of MC Escher's work!" Yes, even though Escher was not a mathematician, he drew tessellations that even mathematicians learned from years later. We enjoy them because they are beautiful but mathematicians enjoy them because they tell us a lot about important mathematical concepts such as nonperiodicity and hyperbolic space.

"Many other artists and architects in the Renaissance used polyhedra and tessellations as tools to make their art more realistic. George Hart of Hofstra University has a great collection information and images from the 15th century onward about polyhedral artists -- they were often people we are already familiar with, such as artists Leonardo da Vinci and Albrecht Durer and scientist, Johannes Kepler." She points to a large sketch of polyhedra on the wall. "They were some of the first Europeans to return to the geometries of ancient Greece -- works by Euclid, Plato and Pythagoras. These works had been kept hidden for almost 1000 years!"

"Oh, that polyhedron on the wall sort of looks like the buckminsterfullerene we saw before." says Angelina. "Yes, it actually is the same shape - an icosahedron with all of its tips cut off -- a truncated icosahedron," answers Sarah, "In fact it is the same structure that is shared by clathrin, a protein found in your brain. But Leonardo da Vinci sketched this polyhedron, an archimedean solid, and many others as open cage structures in 1509 for Fra Luca Pacioli's mathematical treatise, "Da Divina Proportione." Pacioli was the father of accounting and one of the greatest Mathematicians of his time!"

"Did you know that in the last ten years of his life, da Vinci devoted most of his time to the study of polyhedra and geometry? He was trying to understand the design principles of nature and he almost stopped painting altogether to focus on his mathematical studies." says Sarah. "Along with Leonardo, other great artists and architects of the Italian Renaissance such as Piero della Francesca, Leone Battista Alberti, and Albrecht Durer explored geometry. The compass, straightedge and ruler -- tools of the navigator, astronomer and mathematician -- were also the tools of the artist, architect and philosopher.

"Hmmm, I always thought math and art and science were so different from each other!" says Angelina. "But now I see so many overlaps." They didn't teach these things about Leonardo da Vinci and Durer in my art classes."

"Yes," says Sarah, mathematician Jay Kappraff is not alone in saying that geometry is the bridge between the arts and the sciences. Here is the beautiful polyhedra of goldsmith Wentzel Jamnitzer. He wrote a treatise on perspective for artists in the 16th century."

"Gee, if I knew geometry was really so beautiful and easy to understand I think I would have liked it more. I don't see any points or planes here." says Angelina.

Sarah shakes her head and clicks the kiosk in front of a wall sized sketch. "No, these artists and scientists began with three dimensions rather than one dimension. They made models they could hold to explore angles, symmetry and proportion rather than starting with points, lines and planes. Here is Johannes Kepler's studies of stellated polyhedra. Although we know him for his discoveries in astronomy, he was actually the first scientist to explore space-filling and to diagram the full family of archimedean solids. he even discovered the rhombic dodecahedron!. I bet they didn't teach you that in your astronomy class!"