Every object is mathematical but some are more mathematical than others. This morning I noticed my one year old daughter playing with one of her toys - an open cube with small spheres at each vertex which held many distractions. On one side of this cube was an Archimedes screw, an astonishing mathematical object. She was running a small wheel that ran over this little plastic Archimedes screw. What joy a simple toy holds.
Archimedes Screw is one my favorite mathematical objects. What is your favorite mathematical object (utilitarian, like the Archimedes Screw, or abstract, like E8)?
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breasts! (actually i admire their tendency to achieve structural resonance, so maybe that's engineering....)
As a former math major, I feel embarrassed that I knew that object by its more common name: an auger.
I don't know if I really have a favorite mathematical object, but without giving it too much thought, I've always liked Möbius strips and Klein Bottles.
I really like geodesic domes, I also like compound pendulums though they may not count as a mathematical objects.
ah, Mobius strip! Somewhat like Archimedes screw in its application (think conveyor(?) belts are twisted sometimes as Mobius strips so that they wear out on both sides, saw this first at a industrial size flour grinder when I was a kid)
Aesthetically I like the 120-cell, which is a 4-dimensional object made of 120 regular dodecahedra pasted together on their faces. Nice pictures at
http://en.wikipedia.org/wiki/120-cell
I've always been fond of fractals. It amazes me that they look identical no matter how much you zoom in or out. I guess I'm fascinated by the concept of infinity.
symplectodiffeomorphisms!
Spherical harmonics! Ylm!
My favorite mathematical shape is a torus. With powdered sugar.
I've always been patrial to Mobius strips, various polyhedra and cellular automata. If you've never played around on this site for mathematical simulations/demonstrations, you should definitely check it out:
http://demonstrations.wolfram.com/index.html
I too have a thing for geodesic domes, especially as expressed by R. Buckminster Fuller.
Broadening your categories, however, I would add a simple toilet plunger. The first day of my first attempt at Calc II, our professor strode in and confidently smacked a common, wood handled plunger onto the blackboard. Then we plunged (pun intended) into the world of vectors radiating from planes. Later, we would actually rotate cardboard planes around the plunger handle. Sadly, once I got to graduate level physical oceanography, all the references to plungers ceased.