A Four-Color Theorem for Periodic Tilings
نویسنده
چکیده
There exist exactly 4044 topological types of 4-colorable tile-4-transitive tilings of the plane. These can be obtained by systematic application of two geometric algorithms, edge-contraction and vertex-truncation, to all tile-3-transitive tilings of the plane.
منابع مشابه
Hexagonal Lattice Systems Based on Rotationally Invariant Constraints
In this paper, periodic and aperiodic patterns on hexagonal lattices are developed and studied using a given set of constraints. On exhaustively enumerating all possible two-color patterns for up to a combination of four constraints, 106 distinct periodic tilings are obtained. Only one set of four constraints out of a total of 24 157 possible sets generated a pattern that is aperiodic in nature.
متن کاملQuasi-periodic configurations and undecidable dynamics for tilings, infinite words and Turing machines
We describe Turing machines, tilings and in#nite words as dynamical systems and analyze some of their dynamical properties. It is known that some of these systems do not always have periodic con#gurations; we prove that they always have quasi-periodic con#gurations and we quantify quasi-periodicity. We then study the decidability of dynamical properties for these systems. In analogy to Rice’s t...
متن کاملComputing (or not) Quasi-periodicity Functions of Tilings
We know that tilesets that can tile the plane always admit a quasiperiodic tiling [4, 8], yet they hold many uncomputable properties [3, 11, 21, 25]. The quasi-periodicity function is one way to measure the regularity of a quasiperiodic tiling. We prove that the tilings by a tileset that admits only quasiperiodic tilings have a recursively (and uniformly) bounded quasi-periodicity function. Thi...
متن کاملOpen Questions in Tiling
In the following survey, we consider connections between several open questions regarding tilings in general settings. Along the way, we support a careful revision of the de nition of aperiodicity, and pose several new conjectures. We also give several new examples. Over the years a number of interesting questions have been asked about the combinatorial complexity of tilings in the plane or oth...
متن کاملFamilies of tessellations of space by elementary polyhedra via retessellations of face-centered-cubic and related tilings.
The problem of tiling or tessellating (i.e., completely filling) three-dimensional Euclidean space R(3) with polyhedra has fascinated people for centuries and continues to intrigue mathematicians and scientists today. Tilings are of fundamental importance to the understanding of the underlying structures for a wide range of systems in the biological, chemical, and physical sciences. In this pap...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1996