Today, the Facebook company rebranded itself as Meta. PR aside, what mathematically describes its new Lemniscate-like loop logo (try saying that quickly three times)? The logo in question

Answer: A Lissajous curve (also called a Bowditch curve), part of a family of parametric equations where the $$x$$ and $$y$$ coordinates are sinusoids.

$$x = A\sin(at + \delta), \quad y = B\sin(bt), \quad 0 \leq t \leq 2\pi.$$

Here is the equation for the logo ($$A=3, a=1, \delta=5\pi/9, B=2, b=2$$). Try tweaking the parameters and see how it changes!

After tweaking around or refreshing your high school algebra knowledge, you can find that $$A$$ and $$B$$ control the width and height of the curve. The ratio of $$\frac{a}{b}$$ determines the number of lobes in the figure. Lastly, $$\delta$$ is called the ‘phase difference.’ It is the difference in starting point between the $$x$$ and $$y$$ sinusoids.

From high school trigonometry, you might remember that the value of a sine function corresponds to the height of a unit circle (“SOH” in “SOH-CAH-TOA”). We can use that trick to plot the Bowditch curve. $$A$$ and $$B$$ are now radii and $$\delta$$ is now an angle offset for their starting positions. $$a$$ and $$b$$ control the speeds of rotation of the circles.

If we consider the Lissajous curve to be an image of a three-dimensional wireframe, then $$\delta$$ controls the angle at which we view it.

Interpretation of $$\delta$$ being the angle at which we view a 3D wireframe. Ignoring that closer parts of the wireframe appear larger, this is precisely our original curve in 2D. For an interactive view of the wireframe, see here.

The wireframe has the parameters

$$x = A\cos(t), \quad y = A\sin(t), \quad z=B\sin\left(\frac{b}{a}t\right), \quad 0 \leq t \leq 2\pi.$$

A good mathematical exercise would be to show that the projection of this onto a plane through the $$z$$-axis indeed results in the original Bowditch curve! Given that the company has the augmented reality ‘metaverse’ in mind, they certainly chose Lissajous curves as a figure that lives simultaneously in two and three dimensions.

If you’re interested in how I made the animations for this post, here is the source code for generating them. I used Manim!

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2021-10-28 18:33 -0500