6 research outputs found
Domes over curves
A closed piecewise linear curve is called integral if it is comprised of unit
intervals. Kenyon's problem asks whether for every integral curve in
, there is a dome over , i.e. whether is a
boundary of a polyhedral surface whose faces are equilateral triangles with
unit edge lengths. First, we give an algebraic necessary condition when
is a quadrilateral, thus giving a negative solution to Kenyon's
problem in full generality. We then prove that domes exist over a dense set of
integral curves. Finally, we give an explicit construction of domes over all
regular -gons.Comment: 16 figure
Domes over Curves
A closed piecewise linear curve is called integral if it is comprised of unit intervals. Kenyon\u27s problem asks whether for every integral curve γ in ℝ3, there is a dome over γ, i.e. whether γ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when γ is a quadrilateral, thus giving a negative solution to Kenyon\u27s problem in full generality. We then prove that domes exist over a dense set of integral curves. Finally, we give an explicit construction of domes over all regular n-gons
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Geometrie
The topics discussed at the meeting ranged from geometric evolution equations to minimal surfaces, Riemannian foliations and hyperbolic geometry. Because of a flexible schedule, the 53 participants had ample time for discussion