benice

Hello everyone! Thank you for your appreciation. Hi michel123456, I made them just for fun. The epicycloid curves have some applications in gear design. (http://en.wikipedia....ki/Cycloid_gear) If k (defined in this page) is a rational number, then the curve is closed and periodic. If k is an irrational number, then the curve never closes. Hi Daedalus, we may superpose different curves with different colors on the same picture to make more beautiful patterns. I use an open software called GeoGebra(v4.0) to make animations.

(1)Polygonal Epicycloid (2)Polygonal Epitrochoid

Answers: A funny 3D image: A funny curve: The above graphics were created completely using math formulas and math softwares.

They are isosurfaces of the same function Fun(x,y,z) at different levels.

Two living things. Three animal-like faces. Thanks for your reply.

Hint: Consider the intersection of a horizontal plane with the surface. The intersection consists of a family of funny curves. (An example curve is shown below. It has nothing to do with the answer.) Cut off the upper surface above the horizontal plane to get a funny 3D image. (This curve is obtained from another surface.)

The surface in above image is a portion of the graph of some mathematical equation. Let's do some brainstorming: Can you guess what's inside the surface? .