Pen marks X Y position for Pluto and Neptune respectively. This knotwork pattern results because Pluto makes two orbits for every three orbits of Neptune around the Sun. The ratio of their frequencies is 3 to 2, a kind of "orbital resonance." That traces out a "Lissajous knot" pattern, I discovered. To bring out the analogy between the orbital frequencies and musical frequencies (once called the music of the spheres), when Pluto is directly below Sun we hear note E. When Neptune is directly below Sun we hear note B, a "perfect fifth" about E. B has 3/2 the frequency of E, just as the orbital ratio is 3/2. You can adjust "speed" but the ratios of the angular frequencies remains 3 to 2. Spacebar to hide speed variable and white position lines x and y. Where those lines intersect a dot is placed. The line intersections trace out the Lissajous pattern.
Neptune orbits 3/2 times as fast as Pluto. Whenever Y moves 3/2 times as fast as X in a wave or orbit, the Lissajous pattern results. In earlier project, I drew line between Pluto and Neptune at each point. After doing lots of Lissajous patterns for music (perfect 5ths), I realized the 3/2 resonance of Pluto-Neptune would be brought out with a dot placed on the X for one planet and the Y of the other. The angular frequencies are not affected by the radius of the orbits. Nonetheless, we set Pluto's radius as 1.33 times that of Neptune, because Pluto is 3.7 billion miles from Sun. Neptune is 2.78 billion miles from Sun.
