Now general spoiler warning. On 2015-12-13 20:31, rwg wrote:
Eavesdroppers: Spoiler warning!
Wow, Fred, I didn't expect anybody to get that without seeing an adjoining stream in the opposite phase. --rwg
The cross section of one of those rotors is a quarter-circle, mirrored. An elementary and tedious-looking problem is to find the area of the "fat square" formed by the intersection of two perpendicular rotors. I once bletched through the calculus, took one look at the answer, and kicked myself.
Here is Mathematica's rather impressive computation of the area: In[3]:= Expand[ FullSimplify[ Area[ImplicitRegion[ And @@ (EuclideanDistance[{x, y}, ReIm[#]/Sqrt[2]] < 1 & /@ (I^ Range@4)), {x, y}]]]] Out[3]= 1 - Sqrt[3] + \[Pi]/3 I.e., rational - algebraic + transcendental. A diagram if you need it: http://gosper.org/areaansa.png . Notice quadrilateral ACBO. A still shorter command for the area might use |z-i/sqrt 2| < 1 and ... etc, but Mathematica pleads ignorance. Should it? What is the area of the region |z|<1? i pi? --rwg
On 2015-12-13 19:44, Fred Lunnon wrote:
Cross-sectional area of rotor = 2(pi - 2) ; rotors meet and surround squares in alternating checkerboard pattern. Areas of pair adjacent boli sum to 2 - 4(pi-2) = 10 - 4pi , heights = 1 ; so volume of each = 5 - 2pi .
It would improve the demo to equip rotors and -- er -- boli with distinct colour-schemes! I wonder if the hiccup at wrap-around is connected with the fact that Firefox cannot pause the simulation.
WFL
On 12/13/15, rwg <rwg@sdf.org> wrote:
Neat problem: The traveling bicuspidoids intersect the horizontal plane in circular arcs of radius -1. They are vertically 1 unit apart. What is one's volume? Hint: No calculus. --rwg
On 2012-03-25 12:01, Bill Gosper wrote:
For reasonable clarity and looping, w/o downloading: http://toobnix.org/?p=741 --rwg The pause between periods may be my fault, but I avoided the canonical mistake of rendering the same frame at both ends of the interval.
On Sun, Mar 18, 2012 at 7:01 PM, Bill Gosper <billgosper@gmail.com> wrote:
To run it <http://gosper.org/pumper_2.mp4> continuously, download with Miro, QuickTime-View-Loop. The pumped shapes split lengthwise resemble boats, so a grand entrance to a math museum would be a chain of these boats in an annular canal, passing through four rotors acting as an (approximate) air|water lock, with another four for the exit. Oh, the insurance. --rwg No manual entry for overboard