TITLE: Geodesics on the 3-Sphere vs Hopf Circles
QUESTION [3 upvotes]: Regarding $\mathbb S^3$, the $3$-sphere:
I'm trying to reconcile claims that seem contradictory. On page 2 of this article: "Sculptures in $\mathbb S^3$" by Schleimer and Segerman one finds the quote:
"Any plane, meeting the origin in $\mathbb R^4$, cuts $\mathbb S^3$ in a great circle. The great circles are the geodesics, or locally shortest paths, in the geometry on $\mathbb S^3$. Just as for the usual sphere, $\mathbb S^2$, two distinct great circles meet at two points, [$x$ and the antipode of $x$]"
That makes sense to me.
However on page 103 of "Three-Dimensional Geometry and Topology" by William Thurston, I find the quote: "Each complex line (one-dimensional subspace) in $\mathbb C^2$ intersects $\mathbb S^3$ in a great circle, called a Hopf Circle. Since exactly one Hopf circle passes through each point of $\mathbb S^3$,..."
A one-dimensional line in $\mathbb C^2$ is analagous to a  plane in $\mathbb R^4$.
The first quote says more than one great circle can pass through a single point of $\mathbb S^3$, while the second quote says the opposite.
What am I confusing here?
Secondly, it sounds like Thurston is saying that the plane does not need to intersect the origin for its intersection with $\mathbb S^3$ to form a geodesic. Is that correct? If so, how does on make a small circle on $\mathbb S^3$?

REPLY [2 votes]: "Complex"

In the second quote, they are referring to complex line. So if $v\in \mathbb S^3$ the plane spanned by $\{v, \sqrt{-1} v\}$ is the unique complex line passing through $v$. Thus there is a unique Hopf circle passing through $v$. 
However there are lots of real two dimensional planes (which are not complex) passing through $v$, and thus lots of great circles passing through $v$. 
Also, let me point out (as anon did) that two great circles in $\mathbb S^3$ might not intersect: just take two 2-dimensional plane $L_1$, $L_2$ in $\mathbb R^4$ which interesct only at the origin. Then the great circles cut out by $L_1, L_2$ will not intersect in $\mathbb S^3$.
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