4D lift of the tilings by the Smith et al. aperiodic monotile

Hexagons

A discussion between Dale Walton, Craig Kaplan and Yohiaki Araki allowed them to single out a nice way of decorating Tile(1,1), that leads to an interesting tesselation of the plane into regular hexagons, squares, and rhombs of angles, and so that every edge has the same length.

I am not aware that it has been given a name so let me call it here a Causeway, by reference to a famous field of basaltic hexagonal columns. To my knowledge, Yoshiaki Araki is the first to have decorated the tile and released this type of image, based on a suggestion that Craig Kaplan made after seeing another decoration and tessellation made by Dale Walton.

Now the black segments adapt to all tiles in the family Tile(a,b), though there is a twist (pun intended) that I describe below.

The little imperfection is that for a < b/√3, the rhomb becomes inverted and one vertex of the hex (the only vertex not in in the initial set of vertices of the tile) goes outside the interior of the tile. Actually, this will not be a problem for the 4D-lift: it is still possible to define unmabiguously which 4D point will project to this vertex for all (a,b): for the initial example with the tip at (0,0,0,0) and a specific orientation one finds (0,√3,√3,1). So the black family of tilings (that are genuine tilings when a ≥ b/√3) can be lifted to 4D. Better: all the lifted pieces are flat polygons (unlike the lift of the tile outline, which is a skew polygon in 4D). So the lift of the black tiling has a natural filling in 4D.

Note: this may help to define a natural filling of the 4D lift of the tiles themselves.

Pieter Mostert has a very nice 2D animation, that he calls Tetris Move, where the hexagons of a hats tiling are shown, and then stacked to form a full hexagonal tiling of the plane.

He did not program it using a 4D lift, but it can be reproduced by such means: first note that

• The complex vectors ℂ² to which the black segmens lift come into two families: they either lie in the complex vector subspace generated by h₁ = (1,-i) ∈ ℂ² or in the one generated by h₂ = (1,i).

This gives the following for the May tiling:

And with the initial tile outlines removed:

March tiling, with Turtles, gives

and

• Each hex has a uniform colour, each rhomb (or square) has two colors.
• Odd tiles have their hexes yellow, even tiles have it blue (or the other way round).
• All the lifts of blue hexes are parallel, all the lifts of yellow hexes are parallel, and lifts of different colors are mutually orthogonal.
• For the hats tiling, the rhomb is flattened out.

The idea is then to shorten the yellow segments, which can be performed in the basis (h₁,h₂) simply by multiplying the h₂-coordinate by a real number that tends to 0 as times goes, before performing the usual projection L_60° (the one of complex matrix (cos(60°),sin(60°))). You can play with this projection in the projection applet.

At the end of Mostert's animation all the yellow lines have been contracted to 0 and there only remains what is shown above as blue hexagons, all packed. In this case the projection matrix from ℂ² to ℂ is a complex multiple of the matrix \[\begin{pmatrix}1 & i\end{pmatrix}\] and for this matrix, a striking thing happens: all the green and red segments are also projected to the boundary of the hexagons where the blue segments project.

To better see how things map, I will show projections that are slightly off. For this I will use matrices of the form \[\begin{pmatrix}e^{i\alpha_1} & e^{i\alpha_2}\end{pmatrix}\] a family suggested by Pieter.

Let me make one remark: the complex-linear projection will map the hexagons to regular hexagons (or points) because they are regular hexagons contained in complex subspaces of ℂ². Actually the converse is true:

Proposition: The blue and yellow hexagons project to regular hexagons (or points) if and only if the projection matrix is complex-linear.

The animation below illustrates an instance of the March tiling. The animation comes in two parts: the first shows deformations to hexagons of a tiling by Hats, the second deformations to hexagons of its Turtles equivalent. Each animation is repeated with the yellow/blue hexagons (shown here as yellow/white) shown in place of the tile outlines.

Technically this is done as follows: in 4D we first apply matrix diag(c',s') in basis (1,-i), (1,i) of ℂ², where c' = cos(α'), s' = sin(α') and then project with matrix (c,s) with c = cos(α), s = sin(α). The value of α is fixed to 60° for the Hats, and to 30° for the Turtles. The value of α' varies with time from 45° (showing a March tiling) to 0° then to 90° and back to 45°. When α'=0° or 90°, the projection becomes a pure tiling by regular hexagons (red, green, white and yellow are all mapped to the boundaries of the hexagons).

Let us show the projection of an even tile and an odd tile for the hexagonal projection that leaves only white hexagons (projection to the complex vector space generated by (1,-i), parallel to the complex vector space generated by (1,i)).

An unexpected symmetry for the March tiling

Let us look at how hexes and rhombs vary when we perform the usual projection L_α (of complex matrix (cos(60°),sin(60°)))) and vary the angle α between 0° and 90°, for the March tiling. Here we do not perform the contraction of yellow or blue segments.

The video starts at α=60° (Hats) and gets down to 45° (Tile(1,1)), then 30° (Turtles), 0° (Comets), then α increases to 90° (Chevrons) and comes back to 60°.

One sees that the blue and yellow hexagons keep a constant, and equal, side length. Actually there are two kind of rhombs in the full yellow/blue tesselation and their angles are 120°−2α (and its complement 60°+2α) for one rhomb and 2α (and its complement 180°−2α) for the other. Negative angles or >180° correspond to inverted rhombs. Moreover the arrangement look very similar between the image for α and 60°−α (the symmetric of α w.r.t. 30°): the hexagon arrangement for α=60° is like the one for α=0°, the arrangement for α=45° is like the one for α =15°. the arrangement for α=30° is quite symmetric. To get the symmetric of values of α>60° one must consider negative values of α.

α = −30°	α = −30°
α = −15°	α = −15°
α = 0°	α = 0°
α = 15°	α = 15°
α = 30°	α = 30°
α = 45°	α = 45°
α = 60°	α = 60°
α = 75°	α = 75°
α = 90°	α = 90°

Surprisingly, this is different from another correspondence that one would naturally define: the one between tilings with Tile(a,b) and tilings with Tile(b,a). It exchanges α with 90°-α (the symmetric of α w.r.t. 45°).

Another remark concerns the picture already shown at the beginning of this page, showing the hexes and rhombs for α=30°.

Suppose you that someone tiled the plane with copies of the Turtle. Among the two mirror images of the Turtle, there is one whose rotated copies appear more: they form the even tiles, and one whose rotated copies appear less: they form the odd tiles.
Suppose now that they gave you the image of a finite patch, but only showing the hexagons. From this you can deduce which are odd and which are even, and you can deduce the rhombs too and the colors blue/yellow. But from the image we can ne sees that it is hard to figure out which mirror image of the Turtle is the most common one (even tiles) and which is the rare one (odd). A priori it is not clear how to recover the Turtles from such a picture, without supplementary marking (for instance an arrow in the hexagons would be enough).

Playing with the above image I came to make the following:

Conjectural assertion: Consider a Turtles tiling (March flavor) of the whole plane. Consider the associated set of hexagons. Let the most common mirror image have blue hexagons and the less common have yellow hexagons. Then there are exactly two Turtles tilings which has this set of associated hexagons: the original one and one other whose even and odd tiles are mirror images of the even and odd tiles of the original tiling.

Actually it cannot hold in every case: there exists some March tilings (discovered by Socolar, see this article, in particular Figure 18 page 12) of the whole plane for which there exists a path of tiles, infinte on both ends, wriggling but non self-crossing, and on which all tiles can be rotated around their hexagon, see the animated image below. This contradicts the uniqueness above. The assertion may yet hold in for a "generic" March tiling.