Tessellation
Given a set of planar figures, a tessellation is a covering of the plane
with copies of figure from the set such that the figures do not overlap.
The figure that is used to tile the plane is a tessellation element and may
consist of one or more pieces.
A regular tessellation is a tiling with just one type of regular
polygon, such that a vertex of a polygon touches a side of another
polygon only at an endpoint of the side. There are only three regular
tessellations of triangles, of squares, and of hexagons.
Tessellation: triangles
Tessellation: squares
A semiregular tessellation allows more than one type of regular
polygon, but again a vertex may touch a side only at an endpoint of that
side, and the same sequence of polygons must appear in order around
each vertex. Including the regular tessellations, there are eleven
semiregullar tessellations.
Octagon and squares
Dodecagon and triangles
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There are also tessellations of the plane in which a vertex of one
polygon is coincident with an interior point of an edge of another polygon.
There tessellations thus allow two adjacent polygons to meet at edges that
do not completely coincide; we call them non-edge-to-edge tessellation.
Non-edge-to-edge squares
Unequal squares
Just tessellation
 {GC} to {4} (4)
Greek Cross to square
If both figures can tile the plane, then the dissection can be formed
by letting their areas equal and superposed them at the right place, then
beautiful dissection were easily constructed.
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Tessellation: {GC}, {4}
This dissection can be derived by superposing a tessellation of Greek
Crosses with a tessellation of square, as shown upon. Infinitely many
four-piece dissections result from shifting one tessellation relative to the
other.
Reference : Frederickson, Greg N (1997). Dissections : plane & fancy, p105~106.
 {12} to {4} (6)
Dodecagon to square
Sometimes a tessellation element can be derived from cutting and
rearrangement, like this dodecagon to square which is discovered by
Lindgren (1951).
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Tessellation: {12}, {4}
element of {12}
First, he cut the dodecagon into four pieces that rearrange to give a
tessellation element. He then superposed the associated tessellation over a
tessellation of squares to give a 6-piece dissection.
Reference : Frederickson, Greg N (1997). Dissections : plane & fancy, p107.
Complete the Tessellation
In the May 1926, Geoffrey Thomas Bennett discovered a marvelous
5-piece dissection of octagon to square. How did he made it ? He uses a
novel technique that Harry Lindgren has called completing the
tessellation.
We know that polygons that are tessellation elements or that we can
transform easily into tessellation elements often have economical
dissections to other polygons. But what happens when a polygon is not a
tessellation element or cannot be transformed easily into a tessellation
element with a desired repetition pattern? It is sometimes possible to
create a tessellation element by the addition of another polygon. If this
additional polygon can also be added to the target polygon to from a
tessellation element with the same repetition pattern, then an economical
dissection may result from superposing the two tessellation. This is
completing the tessellation.
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 {8} to {4} (5)
Octagon to square
An octagon and a square of equal side length form a tessellation
element. Furthermore, any two squares form a tessellation. Taking the
larger square in the second tessellation to be of area equal to the octagon
and taking the remaining square in that tessellation to be identical to the
square in the first tessellation give two tessellations that we can superpose
them.
Tessellation: {8}, {4}
This dissection has 4-fold rotational symmetry. Also, the dissection
can be partially hinged.
Notice that the added small squares of octagon’s is fully inside the
purposed element, namely the big square. And the added small squares of
big square’s also inside the octagon. But what would happened if the
added polygon is not fully inside the target polygon ?
Reference :
Frederickson, Greg N (1997). Dissections : plane & fancy, p150~151.
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 {6/2} to {6} (6)
Hexagram to hexagon
It is possible to from a tessellation from {6/2}s, if we add two
equilateral triangles for each {6/2}. This is conveniently the same number
of triangles that we add to the hexagon to give a tessellation. Thus we can
superpose these tessellation to get a dissection of {6/2} and {6}.
Tessellation: {6/2}, {6}
But unfortunately the added two triangles cannot both fit in hexagon,
the distance between the extremities of the two triangles is greater than
the width of the hexagram at its narrowest point. Thus one equilateral
triangle in each tessellation must overlap some boundary in the other
tessellation. If triangles from each tessellation overlap, the area
common to both need not be used in the dissection. Positioning the
tessellations as shown in this figure. Each of the triangles that overlaps in
the superposition of the tessellations if cut into three pieces, one of which
is not used, and the other two of which must be turned over in the
dissection.
Reference :
Frederickson, Greg N (1997). Dissections : plane & fancy, p153.
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 {8/3} to {4} (8)
Octagram to square
When completing the tessellation with stars, sometimes adding
regular polygons is not enough to till the plane, then we will add some
irregular figure. Like this octagram to square.
Tessellation: {8}, {4}
We can fit an {8/3} together with a 4-armed twisted cross to form a
tessellation element. Serendipitously, this twisted cross is itself a
tessellation element. Thus we dissect the 4-armed twisted cross to a
square and use a square of this size, along with a square of area equal to
that of the {8/3}, to form a third tessellation.
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Tessellation: 4-armed twisted cross to square
Reference : Frederickson, Greg N (1997). Dissections : plane & fancy, p154~155.
Polyhedral tessellation
A polyhedral tessellation is a tiling of the surfaces of a regular
polyhedron by an 2-dimensional element. Sometime the element cannot
tile the plane, but it still works if it can tile the polyhedron.
 {6/2} to {3} (5)
Mott-Smith’s {6/2} to triangle
Geoffrey Mott-Smith (1964) found several 5-piece dissections of
hexagram to a triangle. This dissection can be fully and variously hinged.
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{6/2} element
Tessellation for {6/2}
Mott-Smith’s dissection can be discovered by using tessellations,
with an element for the hexagram shown in Figure 10.20. Dashed edges
indicate a tessellation element for a triangle. We match edges with short
dashes together, and similarly edges with longer dashes, but the latter
must also match an out-arrow label with an in-arrow label. We join
repetition points only if they have the same label. A or B. Figure 10.22
shows a portion of a tessellation. If we add two more hexagram elements,
then we form a larger element with a hexagonal repetition pattern.
Alternatively, we can fold the four hexagram elements on the dashed lines
to give what will cover the top half of an octahedron. Thus eight of our
hexagram elements tile an octahedron, as shown in Figure 10.23.
Figure 10.23: On an octahedron
Reference : Frederickson, Greg N (1997). Dissections : plane & fancy, p110~112.
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 {L’} to {4} (7)
{L’} element
Szips’s {L’} to square
The Cross of Lorraine {L’}, constructed from 13 small squares.
Since 13=22+32, we expect a nice dissection of an {L’} to square.
Discovering a tessellation element for the plane, Bernard Lemaire
gave an 8-piece dissection in (Berloquin, Le Monde, 1974a). But a Mr.
Szeps sent a clever 7-piece dissection, and Lemair then found a different
7-piece dissection; see (Berloquin, Le Monde, 1974c,d), It appears that
Szeps cut the cross into three pieces to from the tessellation element
shown upon. The outline of a square is shown with its corners anchored at
the repetition points. There are three different types of edges for this
square. As in the previous dissection, edges must be matched in a way
that takes into account the length of dashes and the direction of arrows.
Hexahedra! Tessellation for the Cross of Lorraine
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We cannot tile the plane with such an element, but we can tile the
cube! Below figure shows the complete hexahedral tessellation, folded
out flat. Circular arcs indicate how several of the edges fit together to
form a cube, as shown upon. Lemaire extended Szeps’s dissection to a
similar dissection of the Cross of Lorraine to a Greek Cross in just seven
pieces. We can also derive this dissection by using hexahedral
tessellations.
Cross of Lorraine tessellation superposed on the cube
Reference : Frederickson, Greg N (1997). Dissections : plane & fancy, p113~115.
 {10/2} to {5} (7)
Figure 10.25: {10/2} to pentagon
Dissecting a {10/2} to a {5} at first appears difficult, since we
cannot tessellate the plane with pentagons. Lindgren (1964b) used trial
and error to find an 8-piece dissection. However, we can use the notion of
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tessellations on the surfaces of Platonic solids, as suggested by Figure
10.20 and 10.23. Frederickson gave a 7-piece dissection on (Frederickson
1974).
Figure 10.24: {10/2} element
We use a dodecahedral tessellation with the element shown in Figure
10.24 to create the dissection on Figure 10.25. The repetition points are
the vertices of a pentagon, and the sides of the pentagon are indicated
with dashed edges. The full dodecahedral tessellation is shown in Figure
10.26. Folding the twelve corresponding elements along the dashed edges
and matching them up yield the dodecahedron shown in Figure 10.27.
Figure 10.26: Dodecahedral tessellation for {10/2}
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Figure 10.27: On a dodecahedron
Reference : Frederickson, Greg N (1997). Dissections : plane & fancy,
p112~113.
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