So, for those that thought the hat was two shapes, then I have a surprise for you. The Spectre doesn’t need reflections to tile non-periodically, making it a true or chiral einstein (actually two, as you also have a mirrored version). But remarkably nobody noticed it!
For everyone else, we need to deform the sides in a consistent way. A basic form of this is alternating notches (protrusions/intrusions). This Spectre has 55 sides.
But let’s backtrack a little first. It all started with the discovery of the hat then soon after, the turtle. Then we fast forward onto Joseph’s discovery of the evolutionary timeline (JET) from chevron to comet. Slap bang in the middle of it was the Spectre but seemingly boring, so got overlooked (it was periodic after all).
Inspired by Yoshiaki Araki’s fun aperiodic tile maker app https://www.t3puzzle.com/a/, it occurred to me that this polygon had more freedom than the others. So what would happen if I were to restrict it to unreflected tiles only? A pile of machine cut shapes from card followed.
After some study, Craig and I noticed that the angle placement of the middle tile (the “virtual” reflected tile) is offset by 30° relative to those that surround it, which in turn produces a small arc. I named it the Spectre (a spirit or presence perhaps?).
One of Craig’s early observations using Joseph’s hex cluster arrangements.
This example started out as a brute force computational drawing by Craig. He noticed certain groupings of Spectres that were consistent throughout, containing eight and nine Spectres respectively which he highlighted. I added the white line marking to show a wide periodic band from left to right. The green line marking was added later by Craig which heads off at a right angle.
Moving on, Joseph found out that you could simulate the way in which the Spectre tessellates by mixing hats with turtles (hat dominant) and turtles with hats (turtle dominant). It was all coming together nicely.
Then another important breakthrough. Craig produced a brilliant substitution tiling system unearthed a Buddha (a Mystic in the paper), which was a combination of two tiles that produced a symmetrical outline. And also, where every other iteration produced a reflected version. I remember being so thrilled when I saw it.
There are actually two Buddhas/Mystics in a cluster and forming a gentle arc.
A small patch of Spectres incorporating ‘chains’. Those in blue are the ‘oddball’ Spectres that take the place of reflected tiles. Long chains or worms that appear in the hat and turtle tilings do not appear in a Spectre tiling. They have more of a curved nature.
Colours indicate the ‘oddball’ Spectres. Those in red are one of the six orientations to give an idea of the complexity.
Mystics only and with a little image filtering, to help display the organised chaos.
By dilating the Mystics enough times, those ones nearest to one another, converge and create larger areas I call butterflies.
Here’s one way to draw the Spectre. Remember all sides are of unit length 1 except the longer side which is of unit length 2.
The only thing now was to alter the edges of the Spectre, so that unreflected tiles and reflected tiles could not be used in the same tessellation. That was the easy bit, as there are lots of ways to accomplish this.
Below a fantastic example by Yoshiaki Araki of how to manipulate the sides of a straight-edged Spectre to produce bizarre mythical animals.
There was a frantic push at the end for Craig, Joseph and Chaim to get it out there (I just cracked the whip), but we did it! I cannot thank them enough for their efforts.
The journey is finally complete!
More here… https://cs.uwaterloo.ca/~csk/spectre/