Very cool for nerds and geeks - Tilted Forum Project Discussion Community

pan6467 · 01-15-2008, 11:13 AM

Got this and had to share it.

I'm not sure I'll get the pic here, but if you click the link you'll see the model.

Quote:

New mathematics-based sculpture at Penn State looks beyond three dimensions

The sculpture, designed by Adrian Ocneanu, professor of mathematics at Penn State, presents a three-dimensional 'shadow' of a four-dimensional solid object.

University Park, Pa. -- Artistic works traditionally carry significance beyond their physical beauty, but a new sculpture in the McAllister Building headquarters of the Penn State Department of Mathematics may carry that tradition to its limits.

The stainless-steel work, a striking object of visual art, also is a mental portal to the fourth dimension, a teaching tool, a memorial to a graduate of the math department, and a reminder of the terrorist attacks of Sept. 11, 2001. The sculpture itself measures about six feet in every direction and is mounted on a granite base about three feet high in order to bring its center approximately to eye level.

The sculpture, designed by Adrian Ocneanu, professor of mathematics at Penn State, presents a three-dimensional "shadow" of a four-dimensional solid object. Ocneanu's research involves mathematical models for quantum field theory based on symmetry. One aspect of his work is modeling regular solids, both mathematically and physically.

In the three-dimensional world, there are five regular solids -- tetrahedron, cube, octahedron, dodecahedron, and icosahedron -- whose faces are composed of triangles, squares or pentagons. In four dimensions, there are six regular solids, which can be built based on the symmetries of the three-dimensional solids. Unfortunately, humans cannot process information in four dimensions directly because we don't see the universe that way. Although mathematicians can work with a fourth dimension abstractly by adding a fourth coordinate to the three that we use to describe a point in space, a fourth spatial dimension is difficult to visualize. For that, models are needed.

"Four-dimensional models are useful for thinking about and finding new relationships and phenomena," said Ocneanu. "The process is actually quite simple -- think in one dimension less." To explain this concept, he points to a map. While the Earth is a three-dimensional object, its surface can be represented on a flat two-dimensional map.

Ocneanu's sculpture similarly maps the four-dimensional solid into a space perceptible to the human observer. His process, radial stereography, presents a new way of making this projection. He explained the process by analogy to mapping a globe of the Earth onto a flat surface.

"We place a light bulb at the north pole of the Earth and we project onto a sheet of paper placed underneath it," he said. "The southern hemisphere, away from the north pole, will remain quite small, while the northern hemisphere, near the projection pole, will become very big and north pole itself will be sent toward infinity."

The technique can be used to make a two-dimensional projection of a cube by first mapping the cube radially onto the surface of a globe. Ocneanu explained, "The edges of our cube become circles on the map, just like straight highways are slightly curved on maps of the Earth. Its angles, however, remain true in this projection, so the map retains the key aspects of the symmetry of the original cube, unlike a photograph of a cube."

When the same technique is applied to project a four-dimensional solid into three dimensions, the inner part of the projection -- equivalent to the south pole on the map -- has smaller, undistorted faces, while the outer part extends toward infinity. Linear edges of the solid become circles in the projection.

However, the projection is conformal, which means that the angles between faces and the way that the faces meet at corners are uniform throughout the projection. The retention of these key characteristics makes the sculpture a powerful teaching tool in addition to a powerful esthetic object.

"When I saw the actual sculpture, I had quite a shock," said Ocneanu. "I never imagined the play of light on the surfaces. There are subtle optical effects that you can feel but can't quite put your finger on." The sculpture has significance in several areas of mathematics related to the study of symmetry, and it can represent structures that are fundamental to many branches of mathematics and physics.

"The sculpture is a new way to represent a classical mathematical object," said Nigel Higson, head of the Penn State Department of Mathematics. "For professionals the sculpture is very rich in meaning, but it also has an aesthetic appeal that anyone can appreciate. In addition, it helps to start conversations about abstract mathematical concepts -- something that is generally hard to do with anyone other than another expert."

The subject of the projection is a regular 4-dimensional solid of intermediate complexity, which Ocneanu calls an "octacube." It has 24 vertices, 96 edges and 96 triangular faces, which enclose 24 three-dimensional "rooms." Windows cut in faces allow the viewer to see within the structure, the same way that a window in a cubic room opens to the inside of the cube. Physically, the sculpture is a giant puzzle of 96 triangular pieces cut from stainless steel and bent into spherical shape.

Ocneanu attributes the success of the project to the machinists and welders of Penn State's Engineering Services Shop, managed by Jerry Anderson. "It turned out way better than I could have imagined," Ocneanu said. "It's very hard to make 12 steel sheets meet perfectly -- and conformally -- at each of the 23 vertices, with no trace of welding left. The people who built it are really world-class experts and perfectionists -- artists in steel."

The sculpture was sponsored by Jill Grashof Anderson, a 1965 graduate of the mathematics department, who provided funds for its development and construction. It is dedicated to the memory of her husband, Kermit Anderson -- also a 1965 mathematics graduate -- who was killed in the World Trade Center terrorist attack on Sept. 11, 2001. She also has sponsored a scholarship in his memory. "I hope that the sculpture will encourage students, faculty, administrators, alumnae and friends to ponder and appreciate the wonderful world of mathematics," said Anderson. "I also hope that all who view the sculpture will begin to grasp the sobering fact that everyone is vulnerable to something terrible happening to them and that we all must learn to live one day at a time, making the very best of what has been given to us. It would be great if everyone who views the Octacube walks away with the feeling that being kind to others is a good way to live."

To view an animation of the sculpture, go to http://www.science.psu.edu/alert/vid...ube%20anim.swf

Daniel_ · 01-15-2008, 12:35 PM

The_Jazz · 01-15-2008, 12:54 PM

The link won't work for me (that may be a "I'm at work" issue, though), but this sounds very cool.

PonyPotato · 01-15-2008, 02:18 PM

Quote:

Originally Posted by The_Jazz

The link won't work for me (that may be a "I'm at work" issue, though), but this sounds very cool.

The link doesn't work for me, either, at school or at home.

robot_parade · 01-15-2008, 02:54 PM

That's dead sexy.

Bear Cub · 01-15-2008, 03:05 PM

I wish I could see the animation, and coming from a welding background, I wish they would give some info on the actual fabricators!

Hain · 01-15-2008, 03:05 PM

Looks like I'll have to get one of those for the living room now.

Arc101 · 01-15-2008, 03:52 PM

Nope the links seem to be down, tried to find another link but nothing seems to work. Can anyone help ?

albania · 01-15-2008, 04:09 PM

I found a few links for animations, the first from penn state is for the "octacube" refered to in pan's post http://www.science.psu.edu/alert/vid...ube%20anim.swf . The second is for a rotating hypercube again from penn state http://www.science.psu.edu/alert/vid...gHypercube.swf . The third is from wiki

PonyPotato · 01-15-2008, 04:12 PM

Here's a link to a few articles about it:

http://www.science.psu.edu/alert/Math10-2005.htm

A link to the animation is on the left side of the page.

blahblah454 · 01-16-2008, 09:42 PM

Wow that is unbelievably cool

01-15-2008, 12:35 PM	#2 (permalink)
Daniel_ Evil Priest: The Devil Made Me Do It! Location: Southern England	Wow! __________________ ╔═════════════════════════════════════════╗ Overhead, the Albatross hangs motionless upon the air, And deep beneath the rolling waves, In labyrinths of Coral Caves, The Echo of a distant time Comes willowing across the sand; And everthing is Green and Submarine ╚═════════════════════════════════════════╝

01-15-2008, 12:54 PM	#3 (permalink)
The_Jazz Asshole Administrator Location: Chicago	The link won't work for me (that may be a "I'm at work" issue, though), but this sounds very cool. __________________ "They that can give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety." - B. Franklin "There ought to be limits to freedom." - George W. Bush "We have met the enemy and he is us." - Pogo

01-15-2008, 03:05 PM	#7 (permalink)
Hain has a plan Location: middle of Whywouldanyonebethere	Looks like I'll have to get one of those for the living room now. __________________ ////\\// IRONSKY.NET \\//\\\\

01-15-2008, 04:12 PM	#10 (permalink)
PonyPotato Soaring Location: Ohio!	Here's a link to a few articles about it: http://www.science.psu.edu/alert/Math10-2005.htm A link to the animation is on the left side of the page. __________________ "Without passion man is a mere latent force and possibility, like the flint which awaits the shock of the iron before it can give forth its spark." — Henri-Frédéric Amiel

01-15-2008, 02:54 PM	#5 (permalink)
robot_parade Junkie Location: San Antonio, TX	That's dead sexy.

01-15-2008, 03:05 PM	#6 (permalink)
Bear Cub The Reverend Side Boob Location: Nofe Curolina	I wish I could see the animation, and coming from a welding background, I wish they would give some info on the actual fabricators!

01-15-2008, 03:52 PM	#8 (permalink)
Arc101 Addict Location: Nottingham, England	Nope the links seem to be down, tried to find another link but nothing seems to work. Can anyone help ?

01-15-2008, 04:09 PM	#9 (permalink)
albania Psycho	I found a few links for animations, the first from penn state is for the "octacube" refered to in pan's post http://www.science.psu.edu/alert/vid...ube%20anim.swf . The second is for a rotating hypercube again from penn state http://www.science.psu.edu/alert/vid...gHypercube.swf . The third is from wiki

01-16-2008, 09:42 PM	#11 (permalink)
blahblah454 Let's put a smile on that face Location: On the road...	Wow that is unbelievably cool