25 research outputs found
Cubic Polyhedra
A cubic polyhedron is a polyhedral surface whose edges are exactly all the
edges of the cubic lattice. Every such polyhedron is a discrete minimal
surface, and it appears that many (but not all) of them can be relaxed to
smooth minimal surfaces (under an appropriate smoothing flow, keeping their
symmetries). Here we give a complete classification of the cubic polyhedra.
Among these are five new infinite uniform polyhedra and an uncountable
collection of new infinite semi-regular polyhedra. We also consider the
somewhat larger class of all discrete minimal surfaces in the cubic lattice.Comment: 18 pages, many figure
Method of Efficiently Constructing Negatively Curved Surfaces from Flat Material
An object having a plurality of negative curvatures comprising a plurality of planar sections adjoined together by locking segments
Method of Efficiently Constructing Negatively Curved Surfaces from Flat Material
An object having a plurality of negative curvatures comprising a plurality of planar sections adjoined together by locking segments
An aperiodic monotile
A longstanding open problem asks for an aperiodic monotile, also known as an
"einstein": a shape that admits tilings of the plane, but never periodic
tilings. We answer this problem for topological disk tiles by exhibiting a
continuum of combinatorially equivalent aperiodic polygons. We first show that
a representative example, the "hat" polykite, can form clusters called
"metatiles", for which substitution rules can be defined. Because the metatiles
admit tilings of the plane, so too does the hat. We then prove that generic
members of our continuum of polygons are aperiodic, through a new kind of
geometric incommensurability argument. Separately, we give a combinatorial,
computer-assisted proof that the hat must form hierarchical -- and hence
aperiodic -- tilings.Comment: 89 pages, 57 figures; Minor corrections, renamed "fylfot" to
"triskelion", added the name "turtle", added references, new H7/H8 rules (Fig
2.11), talk about reflection
A chiral aperiodic monotile
The recently discovered "hat" aperiodic monotile mixes unreflected and
reflected tiles in every tiling it admits, leaving open the question of whether
a single shape can tile aperiodically using translations and rotations alone.
We show that a close relative of the hat -- the equilateral member of the
continuum to which it belongs -- is a weakly chiral aperiodic monotile: it
admits only non-periodic tilings if we forbid reflections by fiat. Furthermore,
by modifying this polygon's edges we obtain a family of shapes called Spectres
that are strictly chiral aperiodic monotiles: they admit only chiral
non-periodic tilings based on a hierarchical substitution system.Comment: 23 pages, 12 figure
Aperiodic Hierarchical Tilings
Abstract. A substitution tiling is a certain globally de ned hierarchical structure in a geometric space. In [6] we show that for any substitution tiling in E n, n> 1, subject to relatively mild conditions, one can construct local rules that force the desired global structure to emerge. As an immediate corollary, in nite collections of forced aperiodic tilings are constructed. Here we give an expository account of the construction. In particular, we discuss the use of hierarchical, algorithmic, geometrically sensitive coordinates{ \addresses", developed further in [9]. 1