![]() ![]() In contrast, any non-zero δ produces a figure that appears to be rotated, either as a left/right or an up/down rotation (depending on the ratio a/b). For example, δ=0 produces x and y components that are exactly in phase, so the resulting figure appears as an apparent three-dimensional figure viewed from straight on (0°). Finally, the value of δ determines the apparent “rotation” angle of the figure, viewed as if it were actually a three-dimensional curve. For example, a ratio of 2/1 produces a figure that is twice as wide as it is high. The ratio A/B determines the relative width-to-height ratio of the curve. For example, a ratio of 3/1 or 1/3 produces a figure with three major lobes ( see image). Visually, the ratio a/b determines the number of “lobes” of the figure. The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures. Other ratios produce more complicated curves, which are closed only if a/b is rational. Another simple Lissajous figure is the parabola (a/b = 2, δ = π/4). For a ratio of 1, the figure is an ellipse, with special cases including circles (A = B, δ = π/2 radians) and lines (δ = 0). The appearance of the figure is highly sensitive to the ratio a/b - Image 3 (3/2, ¾ and 5/4). Lissajous curve, also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equations: x = A.sin(a.t + δ) and y = B.cos(bt) The Beauty of Math in Science - Lissajous Curve The mathematical explanation is a bit confusing if you haven’t taken a first course in calculus, but if you’re interested, you can check it out here. Even though this half gallon is enough to entirely fill the horn, it’s not enough to even coat a fraction of the inner wall! If the horn’s bell had, for example, a 6-inch radius, we’d only need about a half gallon of paint to fill the horn all the way up. This fact results in the Painter’s Paradox - A painter could fill the horn with a finite quantity of paint, “and yet that paint would not be sufficient to coat inner surface”. Gabriel’s Horn is a three-dimensional horn shape with the counterintuitive property of having a finite volume but an infinite surface area. Gabriel’s Horn and t he Painter’s Paradox ![]()
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