Is there a generalization of the helix from $\mathbb{R}^3$ to $\mathbb{R}^4$? The helix is a curve $x(t) \in \mathbb{R}^3$ defined by:
$$
x(t) = \begin{bmatrix}
\sin(t) \\
\cos(t) \\
t
\end{bmatrix}
$$
and it takes the classic shape:

Does this have a natural extension from $\mathbb{R}^3$ to $\mathbb{R}^4$?  (Or even $\mathbb{R}^n$?)


What I've tried so far:
The classic $\mathbb{R}^3$ helix curve above has two nice properties:

*

*$x(t)$ has constant distance from the axis of propagation $\hat{e}_3$, where $\hat{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$

*$x(t)$ has constant angular velocity when projected onto the plane normal to $\hat{e}_3$.  i.e. the vector $(x_1(t), x_2(t))$ has polar coordinates $(r, \theta) = (1, t)$, so $\dot{\theta} \equiv 1$.

The classic helix can be viewed as a parametric walk of a circle in $\mathbb{R}^2$, with the parameter $t$ added as the third dimension.  A natural extension to a helix in $\mathbb{R}^n$ would be a parametric walk of a curve on a hypersphere in $\mathbb{R}^{n-1}$, with parameter $t$ added as the nth dimension.  So for $\mathbb{R}^4$, one could choose a spherical spiral to walk the sphere in $\mathbb{R}^3$, and use parameter t as the 4th dimension:
$$
x(t) = \begin{bmatrix}
\sin(t) \cos(ct) \\
\sin(t) \sin(ct) \\
\cos(t) \\
t
\end{bmatrix}
$$
The first three components are rendered on wikipedia as:

This construction matches the two properties I listed:

*

*$x(t)$ has constant distance from the axis of propagation $\hat{e}_4$, where $\hat{e}_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$

*When $c=1$, $x(t)$ has constant angular velocity when projected onto the 3-plane normal to $\hat{e}_4$.  i.e. the vector $(x_1(t), x_2(t), x_3(t))$ has spherical coordinates $(r, \theta, \phi) = (1, t, t)$, so $\dot{\theta} = \dot{\phi} \equiv 1$.

It's technically a direct extension of the $\mathbb{R}^3$ helix, since $c=0$ induces an identical curve (up to a projection.)  But it still feels a little arbitrary, and the closed form will be quite ugly in higher dimensions.
Is there a generally accepted extension of the classical circular helix in $\mathbb{R}^3$ to $\mathbb{R}^4$?  (Or even $\mathbb{R}^n$?)  And do its properties or construction at all resemble the above?

After some research, I've learned that there are interesting generalizations of helices in $\mathbb{R}^n$, defined in terms of derivative constraints, Frenet frames, etc. such that even polynomial curves can behave as helices. [Altunkaya and Kula 2018].  However, that's much more general than I'm seeking, since those are aperiodic, and may have unbounded distance from the axis of propagation.  But the existence of such work is promising - I just don't know how to search this space well.
 A: After a few hours of digging around and thinking, I've found a way to more naturally express the spherical spiral idea in my question.
I'm still not sure if my construction or properties make sense though, so I won't mark my own answer as correct here.  Someone else with broader geometry knowledge should weigh in instead of me.

One can write the classic $\mathbb{R}^3$ helix in cylindrical coordinates $(\rho, \phi, z)$:
$$
\begin{bmatrix}
x_1(t) \\
x_2(t) \\
z(t)
\end{bmatrix}
= 
\begin{bmatrix}
\sin t \\
\cos t \\
t
\end{bmatrix}
\implies 
\begin{bmatrix}
\rho(t) \\
\phi(t) \\
z(t)
\end{bmatrix}
= 
\begin{bmatrix}
1 \\
t \\
t
\end{bmatrix}
$$
Cylindrical coordinates are a hybrid of $\mathbb{R}^2$ polar coordinates $(r, \theta)$, plus an additional cartesian coordinate $(z)$.  In the diagram below, the helix would propagate vertically, winding around the $L$ axis.

So we can apply the same kind of hybrid using $\mathbb{R}^3$ spherical coordinates $(r, \theta, \phi)$ with $(z)$ to get the "hypercylindrical" coordinates $(\rho, \phi_1, \phi_2, z)$ and write the $\mathbb{R}^4$ helix from the question just as easily.
$$
\begin{bmatrix}
x_1(t) \\
x_2(t) \\
x_3(t) \\
z(t)
\end{bmatrix}
= 
\begin{bmatrix}
\sin t \cos t \\
\sin t \sin t \\
\cos t \\
t
\end{bmatrix}
\implies
\begin{bmatrix}
\rho(t) \\
\phi_1(t) \\
\phi_2(t) \\
z(t)
\end{bmatrix}
= 
\begin{bmatrix}
1 \\
t \\
t \\
t
\end{bmatrix}
$$
and the pattern naturally extends for the general $\mathbb{R}^n$ helix.  We use $\mathbb{R}^{n-1}$ hyperspherical coordinates to write the helix in $\mathbb{R}^n$ hypercylindrical coordinates
$$
\begin{bmatrix}
\rho \\
\phi_1 \\
\phi_2 \\
... \\
\phi_{n-3} \\
\phi_{n-2} \\
z
\end{bmatrix}
= 
\begin{bmatrix}
1 \\
t \\
t \\
... \\
t \\
t \\
t
\end{bmatrix}
$$
This trivially meets my listed properties, because

*

*$\rho=1$ means constant (unit) distance from the axis of propagation $\hat{e}_n$.

*$\phi_k = t \implies \dot{\phi_k} = 1$, so angular velocity is also constant in all angular coordinate dimensions.

Like I've said, though, I'm not sure those properties actually make sense for $\mathbb{R}^n$ helices.
A: Any answer to this question is necessarily going to be a bit arbitrary, but here are a few thoughts:

*

*We have an interesting map $\theta \mapsto (\cos \theta, \sin \theta) : \mathbb{R} \to S^1$. The helix is the graph of this map.

*In this spirit, we might consider that the graph of a parametrization of a manifold is a generalized helix. For example, we have the spherical coordinates parametrization of the 2-sphere $(\theta, \phi) \mapsto (\cos\theta \sin \phi,\sin \theta \sin \phi, \cos \phi) : \mathbb{R}^2 \to S^2$. We could consider its graph, a subset of $\mathbb{R}^2 \times \mathbb{R}^3 = \mathbb{R}^5$ to be a generalized helix.

*We could also concentrate on the fact that $\mathbb{R}$ is the universal cover of $S^1$. So maybe, given a submanifold $M \subset \mathbb{R}^n$, we should consider the graph of the projection $\widetilde M \to M$ to be a generalized helix. Since $S^2$ is its own universal cover, we just get another copy of $S^2$ back in this case.

A: I think another idea which might be a generalization of helices to $\mathbb{R}^n$ is that we take the parametrization of a $\mathbb{R}^{n-1}$ sphere and then we translate it by a function of the parameters which are used at generating the sphere mentioned before in a direction which is not in the (affine) linear subspace, which the sphere lies in.
For example in 3D it could be a "skew Helix"(for example given by the equation $\textbf{r}=\begin{bmatrix} \cos(t) & \sin(t)+\frac{1}{4}t & \frac{1}{2}t \end{bmatrix}$, though this may not always be in a constant distance from the axis of rotation. I didn't check whether the distance is constant, even though it still remains I think a good generalization in some sense.) or a Helix of which the translation vector is dependent on an exponential function(such is the "generalized Helix" given by the equation $\textbf{r}(t)=\begin{bmatrix} \cos(t) & \sin(t) & e^{\frac{1}{2}t} \end{bmatrix}$).
Or in 4D an object given by the parametric equation $\textbf{s}(t;u)=\begin{bmatrix} \cos(t)\sin(u) & \cos(t)\cos(u) & \sin(t) & t^2\end{bmatrix}$
It's also only an idea, I didn't look up this anywhere.
I study Math, but I don't have a degree yet.
