From Lissajous to Pas de Deux to Tattoo: The Graphic Life of a Beautiful Loop
From Lissajous to Pas de Deux to Tattoo: The Graphic Life of a Beautiful Loop
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从利萨如到双人舞再到纹身：美丽循环的图形生活
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发表时间：
2011
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通讯作者：
D. McKenna
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作者：
D. McKenna
作者重新实施了一种自定义绘图技术-近30年前为大型钢笔画开发的艺术品-为了重新绘制一个美学上调整的，书法上扩大的，复合的Lissajous图形，却发现它第三次呈现在一个不寻常的数学艺术媒介中：作为纹身。导论.我们如何通过机器精确地绘制数学思想的历史可以追溯到我们当前无处不在的数字计算时代之前很久。在20世纪60年代之前，当计算机程序员迈出创造数学艺术的第一步时，也许最能捕捉灵感的方法是玩弄振荡系统，无论是机械的还是后来的电子的。傅立叶毕竟在1807年展示了一个相当惊人的事实，即频率和振幅变化的简单三角函数sin（）和cos（）的叠加（尽管是无限的）可以用来表示具有几乎任意形状的任意周期函数。后来的诺贝尔奖获得者迈克尔逊将使用一种仪器来机械地求和和绘制傅立叶级数。1法国数学家利萨如使用镜子、光源和音叉建造了一种仪器来帮助他探索简谐系统x（t）= Ax sin（λxt+ φ），y（t）= Ay sin（λyt）的输出（1）（其中时间/角度t以弧度测量，As是振幅，λs是正弦频率，φ是相移）。一种主要的激光显示，这些被称为二维李萨如曲线。它们在视觉上是优雅的，典型的封闭曲线，在平面上由一个以原点为中心的矩形限定，边长为2Ax × 2Ay，其中Ax和Ay是等式（1）中的振幅。图1：各种李萨如曲线，单位振幅（因此它们由正方形而不是矩形包围），并使用各种频率比。最右边的图也使用相位φ 6= π/2。在世纪的后半叶，和声学[1]作为探索某些复合李萨如图形的一种手段而具有巨大的魅力，由于摩擦力的影响，这种图形在视觉上更加有趣，迈克尔逊也认为这台机器的行为不正常，因为后来被误认为是吉布现象。虽然吉布斯在迈克尔逊的困惑之后不久就向物理学界解释了这种奇怪的不连续性，但这种现象最早是由一位24岁的Trinity数学学生亨利·威尔布拉姆（Henry Wilbraham）描述的，他比吉布斯早了50年。参见[7]。桥梁2011：数学，音乐，艺术，建筑，文化
The author re-implemented of a custom drawing technique—developed nearly 30 years ago for large pen plotter artwork—in order to redraw an aesthetically tuned, calligraphically widened, compound Lissajous figure, only to find it rendered a third time in an unusual medium for mathematical art: as a tattoo. Introduction. The history of how we draw mathematical ideas accurately by machine goes back long before our current era of ubiquitous digital computation. Prior to the 1960s, when computer programmers took their first baby steps towards creating mathematical art, perhaps the most imagination-capturing method was to play with oscillatory systems, either mechanical or later electronic. Fourier had, after all, shown in 1807 the rather startling fact that a superposition (albeit infinite) of simple trigonometric sin() and cos() functions of varying frequencies and amplitudes could be used to represent arbitrary periodic functions having nearly arbitrary shapes. The future Nobel laureate Michelson would later use an apparatus to mechanically sum and graph Fourier series.1 Using mirrors, light sources, and tuning forks, the French mathematician Lissajous built an apparatus to help him explore the output of the simple harmonic system x(t) = Ax sin(λxt+ φ) , y(t) = Ay sin(λyt) (1) (where time/angle t is measured in radians, the As are the amplitudes, the λs are the sinusoidal frequencies, and φ is a phase shift). A staple of laser light shows, these are known as two-dimensional Lissajous curves. They are visually elegant, typically closed, curves bounded in the plane by a rectangle centered at the origin, with sides 2Ax × 2Ay, where Ax and Ay are the amplitudes from equation (1). Figure 1 : Various Lissajous curves, with unit amplitudes (so they are bounded by squares, not rectangles), and using various frequency ratios. The rightmost figure also uses a phase φ 6= π/2. In the second half of the 19th century, harmonographs [1] held great fascination as a means of exploring certain compounded Lissajous figures, made more visually interesting by the effects of friction-caused Michelson also thought the machine was misbehaving due to what would later be misattributed as Gibb’s Phenomenon. Although Gibbs explained the curious discontinuity to the physics community soon after Michelson’s confusion, the phenomenon was first described by a 24-year-old Trinity math student, Henry Wilbraham, 50 years earlier than Gibbs. See [7]. Bridges 2011: Mathematics, Music, Art, Architecture, Culture