# FAQ overflow

#### QUESTION

The surface show below is very beautiful; however, I don't know its function either as an implicit function or in parametric form.

Anyone have an idea about it and how to draw it with Mathematica? Consider this:

ParametricPlot3D[
RotationTransform[a, {0, 1, 0}][{0, 0, Sin[3 a] + 5/4}],
{a, 0, 2 Pi}, Evaluated -> True] Now rotate this around a circle, while rotating it at the same time around its' origin:

ParametricPlot3D[
RotationTransform[b, {0, 0, 1}][{6, 0, 0} +
RotationTransform[a + 3 b, {0, 1, 0}][{0, 0, Sin[3 a] + 5/4}]],
{a, 0, 2 Pi}, {b, 0, 2 Pi}, PlotPoints -> 40, Evaluated -> True] EDIT:

A color function, omitting surface mesh, fixing direction of rotation and adding a hint of transparency, like the original:

ParametricPlot3D[
RotationTransform[b, {0, 0, 1}][{6, 0, 0} +
RotationTransform[a - 3 b + Pi, {0, 1, 0}][{0, 0, Sin[3 a] + 5/4}]],
{a, 0, 2 Pi}, {b, 0, 2 Pi}, PlotPoints -> 40,
ColorFunction -> (RGBColor[#, 0, 1 - #, 4/5] &[1/2 + {1, -1}.{#1, #2}/2] &),
Mesh -> False, Evaluated -> True] This might be slightly more intuitive way to write ColorFunction using Blend and Opacity in PlotStyle:

ParametricPlot3D[
RotationTransform[b, {0, 0, 1}][{6, 0, 0} +
RotationTransform[a - 3 b + Pi, {0, 1, 0}][{0, 0, Sin[3 a] + 5/4}]],
{a, 0, 2 Pi}, {b, 0, 2 Pi},
PlotPoints -> 40,
PlotStyle -> Opacity[4/5],
ColorFunction -> (Blend[{Red, Blue}, 1/2 + {1, -1}.{#1, #2}/2] &),
Mesh -> False, Evaluated -> True]