The cube, the epicycles and the human face

立方体、本轮与人脸

Written by Andrei N. Ciobanu on 15 Jun, 2026 作者：Andrei N. Ciobanu，2026年6月15日

This article should be read with the opening scene of Westworld in mind. What if Plato and Leibniz were right? Maybe some mystics were right as well. As above, so below. Everything would fit nicely if the oscillatory behaviour of nature were more than a mathematical construct. Maybe mathematics is ontological. What if sinusoids are not here merely to follow and approximate a shape, but to interfere and give birth to one? 阅读本文时，请联想《西部世界》（Westworld）的开场场景。如果柏拉图和莱布尼茨是对的呢？也许一些神秘主义者也是对的。“如其在上，如其在下”（As above, so below）。如果自然的振荡行为不仅仅是一种数学建构，那么一切都将完美契合。也许数学本身就是本体论的。如果正弦曲线不仅仅是为了追踪和近似某种形状，而是通过干涉来孕育出形状呢？

In any case, I found the idea cool and decided to put it into practice. Here is a cube where three of its faces contain epicycle constructions that work together to describe a human face. On each face of the cube, you can also see the component sinusoids without them interfering with one another. Pause [ ⏸ ] Reset view 无论如何，我觉得这个想法很酷，并决定将其付诸实践。这里有一个立方体，其三个面上包含着协同工作的本轮（epicycle）结构，共同描绘出一张人脸。在立方体的每个面上，你还可以看到组成这些形状的正弦曲线，它们互不干扰。暂停 [ ⏸ ] 重置视图

The concept

概念

If you are already familiar with the topic, the idea shouldn’t seem too complicated. First, we sample a series of points from the surface of a face. Each point has three coordinates: $x$, $y$, and $z$. Finding a good way to sample the surface was probably the most difficult part. The order of the points is important. The Fourier reconstruction follows them one after another. If we randomly change their order, the result will no longer look like a face. 如果你已经熟悉这个主题，这个想法看起来应该并不复杂。首先，我们从人脸表面采样一系列点。每个点都有三个坐标：$x$、$y$ 和 $z$。找到一种好的表面采样方法可能是最困难的部分。点的顺序至关重要，傅里叶重构是按照点的顺序依次进行的。如果我们随机改变它们的顺序，结果将不再像一张人脸。

Epicycles work in two dimensions, but our points live in three dimensions. To solve this, we create three projections: $(x,y), (y,z), (z,x)$. We calculate the Fourier coefficients separately for each projection. This gives us three groups of epicycles, one for each face of the cube. We then combine their endpoints to obtain a point in three-dimensional space. We only use a limited number of circles, so the generated point does not always lie exactly on the face. Before drawing the final line, we move it back onto the surface. As a bonus, each face of the cube also shows the original sinusoids without allowing them to interfere with one another. 本轮在二维空间中运作，但我们的点存在于三维空间中。为了解决这个问题，我们创建了三个投影：$(x,y), (y,z), (z,x)$。我们分别为每个投影计算傅里叶系数。这为我们提供了三组本轮，立方体的每个面对应一组。然后，我们将它们的端点组合起来，以获得三维空间中的一个点。由于我们只使用了有限数量的圆，生成的点并不总是精确地落在表面上。在绘制最终线条之前，我们会将其移回表面。作为额外福利，立方体的每个面还展示了原始的正弦曲线，且互不干扰。

Related Articles

相关文章

I’m so glad you made it to the end. The following articles are loosely related to this one: 很高兴你能读到最后。以下文章与本文有一定关联：

• Apr 09, 2026 — The Scroll Manifesto 2026年4月9日 — 滚动宣言
• Apr 08, 2026 — A mathematical problem 2026年4月8日 — 一个数学问题
• Mar 27, 2026 — What Is Your Name in Another Universe? 2026年3月27日 — 你在另一个宇宙叫什么名字？
• Mar 19, 2026 — The associations people make 2026年3月19日 — 人们的联想
• Mar 16, 2026 — The Shape of Inequalities 2026年3月16日 — 不平等的形状

Source Code & Contributions

源代码与贡献

Spot an error or have an improvement? Open a PR directly for this article. 发现错误或有改进建议？请直接为本文提交 PR。