The first known instance of the elusive "einstein," a single form that can
be tiled endlessly without repeating a pattern, is a novel 13-sided
design.
Examine carefully! A unique 13-sided design that can be tiled endlessly
without ever duplicating a pattern has been created by mathematicians. It is
known as "the Einstein."
Mathematicians have pondered the possibility of finding a single unique
form that could properly tile a surface without creating any gaps or
overlaps and with a pattern that would never repeat for many years. Of
course, repeating patterns are basic to work with; simply take a glance at a
bathroom or kitchen floor, which is presumably covered in plain rectangular
tiles. You may locate a location where your floor appears precisely the same
as it did in the previous position by picking it up and moving it (referred
to as a "translation" in mathematics), demonstrating that the pattern
repeats itself.
Mathematician Hao Wang postulated in 1961 that it was impossible to create
aperiodic tilings, or tilings that never recur. However, Robert Berger, one
of his own students, outsmarted him by identifying a collection of 20,426
forms that, when meticulously assembled, never repeated. Later, he reduced
that to a set of 104 tiles. It follows that there would never be a repeated
pattern if you were to purchase a set of those tiles and lay them on your
kitchen floor.
The Penrose tiling, which was discovered by Nobel Prize–winning physicist
Roger Penrose in the 1970s, is a set of just two tiles that may be placed in
a nonrepeating pattern.
Since then, mathematicians from all over the world have been on the prowl
for "the einstein," the mythical object of aperiodic tiling. The term comes
from the German translation of the renowned Albert's last name, one stone,
rather than from him. Is it possible for a single tile, or a single "stone,"
to completely cover a two-dimensional space without ever repeating the
pattern it makes?
David Smith, a retired printing technician from East Yorkshire, England,
has recently learned the solution. How did he discover this brilliant
solution? Smith said to
The New York Times, "I'm always playing around and experimenting with shapes." "Practical
experience is usually pleasant. It has a contemplative quality.
The new design was given the moniker "the hat" by Smith and his co-authors
mostly due to its fuzzily fedora-like appearance. The design, which has 13
sides, has long been known to mathematicians, but they had never thought of
it as a potential candidate for aperiodic tiling.
Marjorie Senechal, a mathematician at Smith College who was not involved in
the study, told The Times, "In a way, it has been sitting there all this
time, waiting for somebody to uncover it.
Smith developed two proofs demonstrating that "the hat" is an aperiodic
monotile – an einstein — in close collaboration with two computer
scientists, a mathematician, and another. One method of demonstrating how
the pattern never repeats as the surface area increases included creating
larger and larger hierarchical groupings of the tiles. The team's discovery
that there were an endless number of similar forms instead of just one of
these tiles might serve as the basis for the second piece of evidence. The
team's publication is accessible on the preprint service
arXiv, but
neither the proofs nor it has been peer-reviewed.
These aperiodic tilings are more than just interesting mathematical
phenomena. For starters, they show that several medieval Islamic mosaics
used comparable nonrepeating patterns, as seen in the Penrose tiling at the
Salesforce Transit Center in San Francisco.
Aperiodic tilings are particularly useful for explaining the formation and
behavior of quasicrystals, which are ordered but non-repeating crystal
structures.
