Title: 3D Turtle Geometry, turtle programs, symmetry, and miter joints

Abstract:  In this colloquium, we present a 3D variant of the famous
(2D) turtle graphics that Seymour Papert introduced in the 1960s to
motivate children to write programs.  A turtle program describes a
polyline figure, not through absolute cartesian coordinates, but by
'self-relative' operations.  Turtle geometry studies the properties of
turtle programs and figures.  It turns out that 3D is more challenging
(and interesting) than 2D.

Our goal is to determine various properties of figures through their
generating programs.  In particular, we address symmetries of polyline
figures and whether a 3D polygon, i.e. closed polyline, can be constructed
with miter joints that match all the way round.  To that end, we present
an algebraic framework to reason about programs and their equivalence.
There are various equivalences that play a role.  These equivalences
can be viewed as capturing different semantics of turtle programs.