In mathematics, a **seashell surface** is a surface made by a circle which spirals up the *z*-axis while decreasing its own radius and distance from the *z*-axis. Not all seashell surfaces describe actual seashells found in nature.

The following is a parameterization of one seashell surface:

x
=
5
4
(
1
−
v
2
π
)
cos
(
2
v
)
(
1
+
cos
u
)
+
cos
2
v
y
=
5
4
(
1
−
v
2
π
)
sin
(
2
v
)
(
1
+
cos
u
)
+
sin
2
v
z
=
10
v
2
π
+
5
4
(
1
−
v
2
π
)
sin
(
u
)
+
15
where
0
≤
u
<
2
π
and
−
2
π
≤
v
<
2
π
\\

Various authors have suggested different models for the shape of shell. David M. Raup proposed a model where there is one magnification for the x-y plane, and another for the x-z plane. Chris Illert proposed a model where the magnification is scalar, and the same for any sense or direction with an equation like

F
→
(
θ
,
φ
)
=
e
α
φ
(
cos
(
φ
)
,
−
sin
(
φ
)
,
0
sin
(
φ
)
,
cos
(
φ
)
,
0
0
,
0
,
1
)
F
→
(
θ
,
0
)
which starts with an initial generating curve
F
→
(
θ
,
0
)
and applies a rotation and exponential magnification.