Squaring the Hexagon
An investigation into
similarities and equivalence between different types of
polyform. Polyforms have two properties which distinguish
them from each other: the shape of the pieces and the grid
or lattice in which they are constructed. Is it possible to
transform one set of polyforms into a set with a different
appearance, but which can solve the same problems? The
answer is "yes". Possibly the simplest example is to take a
set of pentominoes, and construct a square with missing
corners (Fig. 1). If the puzzle is observed from an
oblique angle and the effects of perspective are ignored,
then the result is shown in Fig. 2. A 90°
rotation results in Fig.3. We have just created the
polyrects from the
polyominoes. The pentominoes with
diagonal symmetry, L, W and X, are the
same in both images, whilst all the other pieces have two
different polyrect equivalents: a long and a short version.
The difference between the polyominoes and polyrects is
therefore one of perspective. Polyrects are equivalent to
polyominoes if every
long and short pair of polyrects is regarded as different rotations of
the same piece. Conversely, polyominoes allowing
reflections but regarding rotations as different pieces, are
equivalent to polyrects.



Figure 1 
Figure 2 
Figure 3 


Polyomino Rects 
Polyrect
Ominoes 
All pieces different.
Reflections allowed, no
rotations. 
Pieces of the same colour are
treated as rotations of a single piece.
Reflections allowed, no
rotations. 


We can repeat this simple
trick with the pentominoes rotated by 45°, resulting with
the
polyrhombs (Figs.46). This
time we have wide and narrow versions of those pentominoes
without mirror symmetry parallel to the grid, and a single
copy of I, T, U and X. If we
regard each pair as different rotations of a single piece,
then the polyrhombs are equivalent to the polyomines. To get
the polyomino equivalent of the polyrhombs, we regard
reflections as different pieces, but allow rotations.



Figure 4 
Figure 5 
Figure 6 


Polyomino Rhombs 
Polyrhomb Ominoes 
All pieces different.
Rotations allowed, no
reflections. 
Pieces of the same colour
are treated as reflections of a single piece.
Rotations allowed, no
reflections. 


Triangle to
Hexagon
Another fairly easy
equivalence to find is that between
polyhexes and
polyiamonds. If the sides of the hexagon are
regarded as being elastic, alternate midpoints can be
stretched outward until a triangle is formed. The inscribed
circle touches the perimeter of each shape where adjoining
pieces can be placed.
The circles also make it
much easier to see the shape of the polyhex equivalents. It
should also be clear from the images below, that there are
no polyiamond equivalents for polyhexes which do not fit
into the grid.



Figure tiled by the 12
hexiamonds. 
The equivalent hexahex
pieces. 
The hex grid for
polyiamond equivalents. 
Square
to
Hexagon
A slightly different
procedure can be used to convert polyhex pieces into
polyomino equivalents. Begin with a square grid as shown in
Fig. 7. This is the grid used by the polyominoes.
Offset alternate rows by half the width of the square, to
get the grid used by the polyhops in Fig. 8. Each of
the single squares in the polyhop grid can be replaced by a
square tetromino to conform with the polyomino grid. Note
that the circles must now be distorted outside the top and
bottom of the squares, in order to make contact. Squashing
this new grid by half in the vertical dimension results in
the polybricks grid, shown in Fig. 9, with each brick
represented by a domino in the polyomino grid. We now take
the midpoints of the long sides of each brick, and pull them
out vertically until the ellipse is once again contained.
The result, shown in Fig. 10, is a grid of
nonregular hexagons. It only remains to stretch this new
grid so that the hexagons are regular, and the ellipses are
restored to circles, Fig. 11.
The propeller tetrahex piece is shown below in
all of these equivalent forms, along with the polyiamond equivalent.
Now we can construct a set of
polyominoes which can emulate all of these pieces.
Polyomino Hexes 

Pieces of the same colour are
treated as rotations of a single piece.
Reflections allowed, no
rotations. 
The equivalent polyhex pieces. 


Hexagon Back
to Square
Polyhex equivalents of
polyomino pieces can also be made. First, we need to
"remove" one of the axes from the hexagonal grid. Fig. 12
shows such a grid where adjacent cells in the vertical axis
are not regarded as connected. Polyhex versions of the five
tetrominoes can be constructed on this grid, as shown in
Fig. 13. These pieces may be reflected horizontally and
vertically, but not rotated, since the same tetrahex in
different rotations represents two different tetrominoes.
This makes it a little more difficult to see the equivalence
between the two sets, especially when the red connection
lines are absent.


Figure 12 
Figure 13 


Polyhex Ominoes 

Pieces of the same colour are
treated as rotations of a single piece.
No reflections or
rotations allowed. 
The equivalent polyomino
pieces. 


Square
to
Pentagon
A piece as small as a domino can be used
as a generating shape to make polyomino equivalents of
polycairos, provided that the puzzle does not
contain any monocairo pieces, in which case an octomino will
work. Figs. 14 and 15 show the polycairo
grid and the equivalent domino grid. Four octomino versions
of monocairos are shown in Fig 16.
The reverse
procedure uses a tetracairo as a generator. These pieces
cannot be reflected in the grid, so we need to add mirrored
versions of the F, L, N, P, Y
and Z pentominoes.



Figure 14 
Figure 15 
Figure 16 



Polyomino Cairos 
Polycairo Ominoes 
The five tricairo pieces shown
with polyomino equivalents.
Rotations and reflections
allowed.

Pieces of the same colour are
treated as reflections of a single piece.
Reflections impossible,
rotations allowed. 


Adding Colour
We can even use the shape
of the pieces to simulate different coloured segments. Two
colours can be represented using 9unit squares to generate
the pieces, with the central square in each block used to
represent the colour. The pieces can be constrained to
follow a chequerboard pattern by blocking corresponding unit
squares on the grid. More than two colours can be simulated
using a larger square as the generating piece, with
different symmetric patterns of holes representing the
colours.
Chequered Polyminoes 


Each omino is formed from 3 x 3
squares, with the central square indicating the colour. 
The equivalent chequered
pieces. 
Chequered grid. 



Squares to
SquareRhombi
Holes can also be used to
differentiate between different shapes. The
polySquareRhombi use both
squares and rhombi as generators. In this case, each piece
is represented by a 4 x 4 block of polyominoes. The square
pieces are left complete, whilst the ones representing
rhombi have two ominoes removed. In the central image below,
the diagonal lines of holes follow the direction of tilt of
the corresponding rhombi. For a minimal set of equivalent
pieces, we can do better: we only actually need the central
four squares of each block. However, using this form makes
it more difficult to see the relationship between the
different sets.
PolySquareRhombi 
Polyomino SquareRhombi 
Grid for polyomino SquareRhombi. 



Kites to Hexes
A grid of
polykites is shown
below, each with an inscribed circle. As was demonstrated
earlier, the circles touch where there is a joint between
adjacent pieces. Note that there are triangular arrangements
of three circles, square arrangements of four and hexagonal
arrangements of six. We can construct equivalent grids of
hexagons in two ways. The first keeps the triangles
together, and the second, the hexagons. The red lines show
the possible joints between adjacent polykite pieces. Using
the first grid, extra hexes may be inserted to prevent
individual polykites becoming disjointed, the drawback being
that polykites of the same order would consist of unequal
numbers of hexes. This process is not possible using the
second, more compact grid.
Polykites 
Polyhex Grid 
Alternative Polyhex Grid




Pons to Hexes
We now perform the same
trick with a grid of
polypons. Here there are triangular
arrangements of three circles and dodecagonal arrangements
of twelve. Again, equivalent grids of hexagons can be made
in two ways. Extra hexes can be inserted into either grid to
keep the polypon pieces contiguous.
Polypons 
Polyhex Grid 
Alternative Polyhex Grid




When we look at the
extended polypons,
things get more complicated. The grid for these pieces
consists of three copies of the one above, which overlap as
shown below. If we imagine these as three separate grids,
then adjacent grids are connected by each parallelogram of
two polypons, although the circles within these pieces do
not touch. Such a pair joining the red and green grids is
highlighted. In the next picture, three copies of the
alternative polyhex grid above are overlaid in equivalent
positions.
Extended Polypons 
Polyhex Grid Without CrossLinks 


Next, the crosslinks
between the three grids have been added. The purple lines
connect cells which are considered to be adjacent in the
blue and red grids, the yellow lines join red and green, and
the cyan lines join blue and green cells. The points where
these lines cross are not relevant, just those at the centre
of a hex cell.
CrossLinked Polyhex Grid


There are positions on the
first grid where pieces overlap which coincide with hexes
containing a star in the hex grid. There can be one to three
cells of a single colour adjoining these cells, or two
opposing cells of different colours. As shown below,
adding two polykites to the end of each adjoining hex will
emulate these conditions. Fig. 21 shows one of these
cells replaced by six polykites, meaning that each single
hex in the crosslinked grid above must be replaced by a
ring of six, or patch of seven hexes.
