A plane polygon
P
\mathcal {P}
inscribed in a conic
C
C
and circumscribed to a conic
D
D
can be continuously ‘rotated’, as it were. One of the many proofs consists in viewing each side of
P
\mathcal {P}
as translation by a torsion point of an elliptic curve. In the
n
n
-space version, involving torsion points of hyperelliptic Jacobians, there is a
g
=
(
n
−
1
)
g=(n-1)
-dimensional family of rotations, where
g
=
genus
g=\text {genus}
of the hyperelliptic curve; the polygon is now inscribed in one and circumscribed to
n
−
1
n-1
quadrics.