Paper's Title:

Linearly Transformable Minimal Surfaces

Author(s):

Harold R. Parks and Walter B. Woods

Abstract:

We give a complete description of a nonplanar minimal surface in R³ with the surprising property that the surface remains minimal after mapping by a linear transformation that dilates by three distinct factors in three orthogonal directions. The surface is defined in closed form using Jacobi elliptic functions.

Paper's Title:

Purely Unrectifiable Sets with Large Projections

Author(s):

Harold R. Parks

Abstract:

For n≥2, we give a construction of a compact subset of that is dispersed enough that it is purely unrectifiable, but that nonetheless has an orthogonal projection that hits every point of an (n-1)-dimensional unit cube. Moreover, this subset has the additional surprising property that the orthogonal projection onto any straight line in is a set of positive 1-dimensional Hausdorff measure.

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