Mathematics · the experiment
Cut the lemon, send the ball everywhere
You hold the cue. First a round disc, where the ball always leaves a hole. Then Fan and Hasselblatt's lemon, where it ends up going everywhere: you send it into the centre, and you measure how long it stays there.
1 · Shoot in the round disc
Beginner
What do Fan and Hasselblatt prove for their very lopsided lemon?
No cut: the table is a disc. Set the launch angle from the radius until the zone the ball never visits has exactly the radius of the dashed disc, half the radius of the table.
2 · Cut the lemon
Intermediate
The article requires R ≥ r · 30,000 / sin²ζ. For the widest cut it allows, tan ζ = 1/3, hence sin²ζ = 1/10, the large radius must exceed the small one by at least a factor of…
Choose the cut: ζ is the half-angle under which the two corners are seen from the centre, and the large circle is set at the article's bound, R = 30,000 r / sin²ζ. Shoot the ball at 58.3° from the radius, a shot that would leave the centre forbidden in the round disc, and get it into the central disc.
3 · Measure the time spent at the centre
Intermediate
If the table is ergodic, what share of its time does the ball spend in the central disc of radius r/2?
The cut is at its maximum, ζ = 18.4°, and the ball is off again at full speed. Set the radius of the measuring disc until the ball spends half its time inside it.
4 · Measure the bulge of the arc
Expert
At the maximum cut, R ≈ 300,000 r. By how much does the arc of the large circle stand off, at its middle, from the chord joining the two corners, for a table of radius r = 1 m?
Ball sent everywhere
—