  A Jordan net (resp. web) is an embedding of a unital Jordan algebra of
dimension $3$ (resp. $4$) into the space $\mathbb{S}^n$ of symmetric $n\times
n$ matrices. We study the geometries of Jordan nets and webs: we classify the
congruence-orbits of Jordan nets (resp. webs) in $\mathbb{S}^n$ for $n\leq 7$
(resp. $n\leq 5$), we find degenerations between these orbits and list
obstructions to the existence of such degenerations. For Jordan nets in
$\mathbb{S}^n$ for $n\leq5$, these obstructions show that our list of
degenerations is complete. For $n=6$, the existence of one degeneration is
still undetermined.
  To explore further, we used an algorithm that indicates numerically whether a
degeneration between two orbits exists. We verified this algorithm using all
known degenerations and obstructions, and then used it to compute the
degenerations between Jordan nets in $\mathbb{S}^7$ and Jordan webs in
$\mathbb{S}^n$ for $n=4,5$.
