This question originally appeared on Quora. 
Answer by Frank Heile, Physicist, Software Engineer, Consciousness Theory
The most symmetrical, spherical man-made objects are the four fused quartz spheres made for the Gravity Probe B satellite (GP-B for short). This satellite experiment verified some predictions of General Relativity to new levels of accuracy and required four rotating very accurately spherical fused quartz spheres to act as gyroscopes. The almost perfect fused quartz spheres are coated with superconducting niobium:
For example, if these spheres were scaled up to the size of the Earth, the radial distance from the highest hill to the lowest valley would only be 2.3 meters!
GYROSCOPE FACTS
The gyroscopes are electrically suspended with only 0.001 inch clearance from the housing walls.Sphericity is less than 40 atomic layers from perfect
Size- 3.81 centimeter (1.5-inch) diameter
Composition- Homogeneous fused quartz
Coating- Niobium (uniform layer 1,270 nanometers thick)
Spin Rate- Between 5,000 Ð 10,000 RPM
And from GP-B - Challenges & Solutions :
Polishing the Perfect Sphere
Challenge: Polishing a fused quartz sphere with standard methods creates "hills and valleys," destroying sphericity.
Solution: Create a tetrahedral lapping and polishing machine that brushes the sphere with micro-inch abrasive slurry in random variations.
Result: Each fused quartz sphere deviates less than one micro-inch from peak to valley (25 nm), making them the roundest objects ever created on Earth.
Bonus Fact: In September 2004, GP-B received a certificate from Guinness World Records Limited, acknowledging that the GP-B gyroscope rotors had been entered into the Guinness Database of World Records. The certificate reads as follows:
"The most spherical man-made objects are the fused quartz gyroscopic rotors onboard the Gravity Probe B Spacecraft operated by NASA and Stanford University. Their average departure from mathematically perfect sphericity is only of their diameter."
These gyroscopes had to be extremely accurate, much more accurate than the normal gyros used in inertial navigation. The GP-B satellite was trying to measure an effect predicted by General Relativity that would show up as a drift of only 6 milliarcseconds per year - so the gyroscope needed to have extreme pointing stability over a long period of time. The angle between the axis of rotation of the sphere was compared to the direction to fixed stars to measure this very small effect.
Another interesting comparison would be to compare these gyros to a common billiard ball. A billiard or snooker ball has a specifications for smoothness; if scaled up to the size of the Earth it would result in a 28 kilometers difference between the heights of mountains and deep trenches rather than the 2.3 meters difference for these gyros (see Proof that the Earth is smoother than a billiard ball ). For the Earth itself, the difference between the highest mountain and deepest trench is only about 20 kilometers - however the rotation of the Earth makes the average diameter at the equator differ from the diameter at the poles by 42 kilometers. So, both the Earth and billiard balls are very rough by a factor of approximately 10,000 when compared to the GP-B spherical gyros!