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Amazing Epicycles

A computer animation by Jason Schattman of twirling rods that fly back and forth across a circular room in eye-catching patterns, often creating optical illusions and "whooshy" wave-like motion. I coded these animations in the Processing programming language. The mathematical ingredients are simple sine and cosine. Although the animations look very different, they're all generated from the same template: at the start of each animation, a "flock" of rods of equal length are placed along the edge of a circular room at equal intervals. Each rod flies back and forth across the center of the room at a certain frequency, while also spinning about its own center at a constant rate. Varying the number of rods, their lengths, or their rate or direction of spin produces a different animation. To create the knot-like patterns in the 2nd half of the video, another ingredient is added: the rods are distributed along the edge of the room as before, but are released from the room's edge at slightly different times instead of all at once. Each rod leaves the edge a tiny bit later than the rod behind it, creating the wave-like interference patterns you see. More math animations from my channel ********************************************************* All 6 trig functions on the unit circle    • All 6 Trig Functions on the Unit Circle   On the beautiful geometry of imaginary numbers & complex functions (can be enjoyed even without knowing what that means)    • Twisting the Plane with Complex Numbers   Optical illusions made using trig functions    • Optical Illusions   Fancy "card tricks" animated using mathematical pretzels (called Lissajous curves)    • Видео   Sound waves in an oval room:    • Sound Waves in an Ellipse   Fly through the 3D Sierpinski pyramid:    • Fly Thru a 3D Sierpinski Fractal   Drawing on a spinning white board:    • Amazing Spirograph   More detail for the mathematically inclined ******************************************** At the start of each animation, there are N rods of length L placed at equal intervals along the edge of a circular room with radius R. Each rod flies back and forth across the center of the room in a sinusoidal pattern with frequency f, while spinning about its own center at angular speed w. The distance r(t) of a rod's center from the center of the room at time t is computed as r(t) = R cos(ft). Each variation in N, L, R, w and f produces a different animation. To create the knot-like patterns in the 2nd half of the video, phase shifts are added to the timing of the rods' flights across the center. This is done by generalizing r(t) to the formula r(t) = R cos(f(t-cn)) where c is the difference in phase shift between adjacent batons, so that cn is the total phase shift of rod number n. By adjusting c, you can get a 3- or 4- or 5-headed knot pattern, or anything in between. In each frame, the endpoints (x1(t), y1(t)) and (x2(t), y2(t)) of the nth rod are calculated using the formulas x1(t) = xC + r(t) cos(kn) + (L/2) cos(wt) y1(t) = yC + r(t) cos(kn) + (L/2) sin(wt) x2(t) = xC + r(t) cos(kn + pi) + (L/2) cos(wt) y2(t) = yC + r(t) cos(kn + pi) + (L/2) sin(wt) where (xC, yC) is the center of the room, and k = 2pi/N is the angular gap between the rods' centers with respect to the center of the room, so that kn is the angle the rod's center makes with the horizontal.