The Lissajous-type curve $\left(\sin(\omega_1 t), \sin(\omega_2 t)\right)$, with $t \in \mathrm{R}$, is dense in a certain region of the plane.
This can be seen for instance from excellent answers to a previous question of mine: Curve dense inside the unit circle.
Consider now the curve $\left(\sin(\omega_1 t), \sin(\omega_2 t), \sin(\omega_3 t)\right)$ with $t \in \mathrm{R}$, and assume that the three frequencies are incommensurable.
Is this curve dense in some 3-dimensional region of finite volume? Or is it also dense just inside a 2-dimensional surface?
$\textbf{ADDED QUESTION}$:
And how about a similar curve in four dimensions (four sines with incommensurate frequencies); would it be dense in a 4-dimensional region of space? If so, how to prove it?
"fill"... Since the curve is continuously differentiable its image must have enmpty interior. Regarding whether the image can be dense in a non-empty open set, (pairwise) incommensurable is not quite the right condition. In fact
Meaning that $r_1\omega_1+r_2\omega_2+r_3\omega_3=0$, $r_j$ rational implies $r_1=r_2=r_3=0$.
If the $\omega_j$ are not rationally independent there exist integers $n_j$ with $\sum n_j\omega_j=0$ and not all $n_j$ vanish. I think it's clear that this implies the curve is not dense, although I haven't written a formal proof.
Define $$\gamma(t)=(e^{i\omega_1t},e^{i\omega_2t},e^{i\omega_3t}).$$ Say $\Bbb T=\{z\in\Bbb C:|z|=1\}$. It is clear that if the $\omega_j$ are rationally dependent then $\gamma$ is not dense in $\Bbb T^3$.
On the other hand, if the $\omega_j$ are rationally independent then a standard elegant argument using Fourier series shows that $\gamma$ is dense in $\Bbb T^3$ (hence your curve is dense in $[-1,1]^3$). Sketch:
Suppose the $\omega_j$ are rationaly independent. Consider the equation $$\left(\frac1{2\pi}\int_0^{2\pi}\right)^3f(e^{it_1},e^{it_2},e^{it_3})\,dt_1dt_2dt_3=\lim_{T\to\infty}\int_0^Tf(\gamma(t))\,dt,\tag{*}$$which may or may not hold for a given $f\in C(\Bbb T^3)$. Show that $(*)$ holds if $f$ is a character, that is $$f(e^{it_1},e^{it_2},e^{it_3})=e^{i(n_1t_1+n_2t_2+n_3t_3)}$$for some integers $n_j$. (Hint: If $n_1=n_2=n_3=0$ both sides equal $1$; if not, the fact that $\sum n_j\omega_j\ne0$ shows that both sides equal $0$.)
Hence $(*)$ holds if $f$ is a trigonometric polynomial (a linear combination of characters). Since the trigonometric polynomials are dense in $C(\Bbb T^3)$ it follows that $(*)$ holds for every $f\in C(\Bbb T^3)$, hence $\gamma$ must be dense in $\Bbb T^3$.