Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in $\mathbb{C}$. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.

中文翻译：

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黎曼不可微函数的一些几何性质

黎曼的不可微函数是一个连续但几乎无处可微的函数的著名例子。有强有力的数字证据表明，它的一个复杂版本代表了与副法向流或涡流丝方程相关的实验中的几何轨迹。在此设置中，我们在 $\mathbb{C}$ 中分析其图像的某些几何属性。这篇笔记的目的是断言它的图像的 Hausdorff 维不大于 4/3 并且它没有切线。