The recently discovered amazing aperiodic monotile (or "einstein") of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss tiles the plane only if reflections of the monotile are allowed. Is there hope of modifying their construction to produce an aperiodic monotile that tiles the plane without the need for reflected tiles?
Geometry – Aperiodic Monotile Without Reflections?
aperiodic-tilingeuclidean-geometryplane-geometrytiling
Related Solutions
No, you cannot three-color that tiling.
Here's a finite part of the tiling from page 10 top left of the article, the tile numbered 2 here is the one darkened on that figure. This part cannot be three-colored.
To prove this, start from the three adjacent tiles marked 0, 1, 2, and call their colors yellow, blue and red respectively. Tile 0 is surrounded by a sequence of adjacent tiles 1, 2, 3, 4, 5, 6, so these must be colored blue and red alternatingly, 5 being blue, 6 being red. Tile 7 is adjacent to tile 5 and 6 and so must be yellow; while tile 8 is adjacent to tile 1 and 6 and so also must be yellow. Now tile 9 is adjacent to tiles 6 and 8 and so must be blue, but tile 10 is adjacent to 6 and 7 and so also must be blue, but 9 and 10 are adjacent so this coloring is invalid.
(a second answer because this one is an answer)
So, I misled myself staring at the H8 in Smith et al. The way to solve this is to look at the F-supertile. That tile has 5 edges, and 4 of them are F-tiles that make up the fractal (at the next expansion level). So if we just ignore other parts of the F-supertile, it will expand to form at least part of the fractal.
The F-tiles surrounding the F-supertile each belong to a fylfot (3 F-tiles). By carefully including those F-tiles as well during the expansion, you get a curve which never draws the same edge twice.
- $A \to +A[+F]-BA[+F]-C+$
- $B \to -D+[--F]BA[-F]+B-$
- $C \to -D+[--F]BA$
- $D \to BA[+F]-C+$
- $F \to -D+[--F]BA[-F]+BA[-F]+F$
Where at level 0 symbols $C$ and $D$ are lines 1 unit long, $A$, $B$, and $F$ are lines $\dfrac{\phi}{\sqrt{2}}$ long, $-$ and $+$ are left and right turns by $\dfrac{\pi}{3}$, $[$ is stack push, and $]$ is stack pop.
Running code: https://trinket.io/python/df6f9fa4db
import turtle
import math
# use the simpler 'Golden Key' f-tile from
# Socolar, 'Quasicrystalline structure of the Smith monotile tilings'
# https://arxiv.org/pdf/2305.01174.pdf
phi = (1 + math.sqrt(5))/2
root2 = math.sqrt(2)
def expand(order, a, stack, s0, s1):
for op in s0 if order <= 0 else s1:
mono_op_map[op](order - 1, a, stack)
def op_push(stack):
stack.append([turtle.pos(), turtle.heading()])
def op_pop(stack):
pos, hd = stack.pop()
turtle.up()
turtle.setpos(pos)
turtle.setheading(hd)
turtle.down()
# The F-supertile in Smith et al has 5 sides, 4 of which are F-tiles.
# to get an L-system for the F-tile fractal, we just expand those.
# Choosing rules carefully avoids repeating any edge.
mono_op_map = {
"a": lambda o, a, s: turtle.forward(a*phi/root2),
"b": lambda o, a, s: turtle.forward(a),
"A": lambda o, a, s: expand(o, a, s, "a", "+A[+F]-BA[+F]-C+"),
"B": lambda o, a, s: expand(o, a, s, "a", "-D+[--F]BA[-F]+B-"),
"C": lambda o, a, s: expand(o, a, s, "b", "-D+[--F]BA"),
"D": lambda o, a, s: expand(o, a, s, "b", "BA[+F]-C+"),
# A free edge from a fylfot, allowed to branch everywhere.
# Use 'B' instead of 'F' above to get a sponge.
"F": lambda o, a, s: expand(o, a, s, "a", "-D+[--F]BA[-F]+BA[-F]+F"),
"+": lambda o, a, s: turtle.right(60),
"-": lambda o, a, s: turtle.left(60),
"[": lambda o, a, s: op_push(s),
"]": lambda o, a, s: op_pop(s),
}
stack = []
start = "A"
order = 6
size = 70/(order*order)
turtle.speed("fastest")
# this system is super slow, disable animation entirely
#turtle.hideturtle()
turtle.tracer(0, 0)
expand(order, size, stack, start, start)
turtle.update()
Best Answer
The same authors have just released a preprint claiming a positive answer to this question.
EDIT: Here is a picture of the reflection-free aperiodic monotile:
More visualizations and other data are available at this web page of one of the authors.