What is a Sphere Inversion To invert a sphere is to turn it inside out by means of continuous deform

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Turning a Snowball Inside OutModeling the
Intermediate stages of a Sphere
InversionUndergraduate Researcher Alex
Kozlowski Advisor Prof. Carlo Séquin
In the 1960s, mathematicians showed how to turn a
sphere inside out. To do this, the surface must
pass through itself, but no tears or creases are
allowed to form at any point.  This project
attempts to make this intriguing concept more
intuitive to non-mathematicians through a blend
of art, mathematics, and computer science. The
end result is a series of thirteen models
depicting several intermediate stages of this
process.
In addition to virtual models, some physical
models were created using a rapid prototyping
machine. Above A preliminary model of the Morin
surface with all four lobes gridded. Right A
model of the quarter-stage of the inversion.
Construction To create these models we used
SLIDE, a CAD tool designed at UC Berkeley. We
elected to use three dimensional sweeps of Bezier
curves as our primary medium because Bezier
curves gave us the right control over the model
geometry that we desired. Additionally, to gain
an under-standing of what happens beneath the
outer-most surface, we made each model
"semi-transparent" by using a grid structure to
represent certain parts of the surface. The
modeling capabilities of SLIDE were enhanced with
a utility for creating this perforated offset
grid that allows for a very smooth tessellation
as well as for some extra control as to where
individual grid-lines should be placed.
The impetus for this project was the 2004
International Snow Sculpting Competition held in
Breckenridge, Colorado. We spent much of 2003
designing the half-way point of the inversion,
known as the Morin Surface. This model served as
the blueprint for our entry into the competition.
In Breckenridge, we carved the surface out of a
twelve foot block of packed avalanche snow
(picture above). Laminated shots of all thirteen
models were displayed around our sculpture for
the general public to see. Out of fourteen
entries from around the world, our entry won
honorable mention for "Most Ambitious Piece!"
Snow-Sculpting
What is a Sphere Inversion?To invert a sphere
is to turn it inside out by means of continuous
deformation, allowing surfaces to pass through
one another but disallowing puncturing, ripping,
creasing, or pinching of the surface. A Morin
Surface is the halfway-model of sphere inversion.
The surface has four lobes, two showing the
inside and two the outside. A ninety-degree
rotation inter-changes the sides, so the two
halves of the inversion differ by this four-fold
twist. By following the inversion once can easily
tell that the process is symmetric.