# "Turning a Snowball Inside Out"

In the 1960s, Bernard Morin, a blind mathematician showed a way to turn a sphere inside out,
if the surface is allowed to pass through itself, but must not be punctured, ripped, or creased.
The convoluted path to execute this complex task leads through "Morin's surface"
at the half-way point, where equal amounts of the inside and of the outside surface
of the original sphere are shown to the world.
We plan to celebrate this fascinating surface and exhibit it to a large audience
by building a large model of it -- in snow!
Since the surface self-intersects, and since it is of some interest to see what is going on
"on the inside", we have chosen to represent the smooth surface by a gridded mesh of struts.
This should also make for a very dramatic sculpture in its own right!

Computer rendering of Morin's Surface by John Sullivan.

Simplest polyhedral Approximation of the Morin Surface by Richard Denner.
More images from the Cuboctahedral Sphere Eversion.

A partial cardboard model of the polyhedral Morin Surface by Tien Tan.

A 3D-print model of the above shape, using a subdivision surface.

A first large scale model to study how to approximate the surface with only a few struts.

Shape of the surface adjusted to fit the blocks of the snow-sculpting championships.

The first task in the snow-sculpting competition will be to carve this shape.

Low-order polyhedral approximation of this surface, showing the approximate complexity that we aim for.

But we want the various curves to flow smoothly over the original Morin Surface, as indicated by the tape ...

... or in this drawing.

And all four lobes should end up in a reasonably large footprint at the base on top of the pedestal.

Another model (made from pipe cleaners) to study the topology and interconnectivity of the struts.

One challenge is to model this on a CAD tool, so I can make a maquette.
The few CAD tools I have knowledge of do not have the right primitives.
Alex (my student) and I are currently thinking about modeling each smooth strut sequence
as a sweep of a trapezoidal cross section along a separate interpolating spline.

We start with a model of all the self-intersection lines in the Morin surface.

Modeling the basic shape with carefully chosen sweeps along the self-intersection lines

The we use a sweep of a cross section, modeled as a Bézier curve, to create the arms of the Morin surface.
This gives more control over the surface shape.

We can also readily choose the number of struts and have better control where they fall.
If only these struts were nicely curved ... !
Also, we need to work on the way that these struts disappear into the sculpture base.

--Here is a nice model of the Morin Surface made on the 3D printer, but ...

Complexity is still a mayor issue when considering this for a snow sculpture!

Some of the possible variations (simplifications) include:  not opening up all the windows,
perhaps, keeping the surface contiguous and connected through the four inversion mouths,
and only carving out the windows where the struts start to diverge dramatically.

Here is an even more promising option:

Suppose, we just carve the struts from two of the lobes, leaving the other two lobes solid ?
This would also nicely demonstrate that we see two different sides of this surface.
We can claim that it acts like a "one-way mirror" : from one side it looks solid, from the other transparent!
This seems quite doable !

As an intermediate checkpoint, we built a physical model of this "Half-Morin" surface on the FDM machine.
We can contemplate to stand the surface on its cross junction, so it opens like a flower in the up-direction.
This upside-down orientation makes this a more "flamboyant" sculpture!
I would not have dared doing this with the full skeleton, but with the half-skeleton, this should work.

-This is also how the model came out of the machine; all the supports are still in place.

-After partial removal of the supports and some clean-up.

-All supports removed, and turned upside-down into the original orientation.
(Note. There was a problem with the fabrication file due to some unresolved contour intersections.
This accounts for the band of additional gray support material in the wrong place.)

-A closer look at the intersections between the full and the skeletal branch.
I arranged it so that right at that intersection line there is a "rib" of the skeletal branch; this makes for a nice termination.

Refinement of the strut layout

-Using 6 longitudinal struts and 9 ribs -- this looks quite elegant !

However, there are some serious structural engineering concerns:
While this shape looks elegant, I now have some doubts whether this heavy solid "worm" will hold up by itself.
The skeletal branches attached to it will only provide little support for the "loose" ends of this worm.
If anything, the skeleton would like to have something to lean to and get some support for itself.

What should we do ?
Make the struts much thicker ?
or turn the whole thing upside-down again?
In a "bridge" configuration, this crucial worm has a better chance of holding up.

--  Best rendering of current design: two side views and a top view.

The latest model under construction in the FDM machine after 2 days.

Page Editor: Carlo H. Séquin