The dodecaflower (right) is formed by making each face from a flower with 10 petals made from ellipses and then replicating each face to form a dodecahedron (regular 12-sided shape). This cannot be printed on a desktop 3D printer, but can be printed on a laser sintering machine. Shapeways will print these in plastics or metals and mail the results.
The Square Cup
"The Square Cup" is not at all square. All of the profiles are formed from z=mx^2 with the handle cutout formed by slicing out cylinders across a profile of y=mx^2. This was printed in ceramic by Shapeways. They no longer print this ceramic, but will do it in porcelain for around £3500!!!
The Christmas Star (right) is a popular object on Thingiverse. This can be printed on a home 3D printer in translucent PLA and then inserted with LEDs. A quick and effective christmas decoration.
The ring to the Left is a render of a ring for printing in Metal. This was requested by a friend whose partner is a keen climber. The figure-of-eight knot is a common climbing knot.
The pendant to the right (sinewave pendant) is formed from a algorithmically generated pattern.
The buckyball was my first attempt to print something much bigger than the 3D printer. This is a truncated icosahedron (a 32-sided polyhedra) consisting of 60 3D printed vertices and 90 pencils sharpened at both ends. So this can be made at home using a desktop 3D printer. The result was then suspended from a modified ceiling rose to form an unusual lampshade (although it doesn't provide much shade!)