Hinged tilings

Alfinio Flores 

  Arizona State University

Squares and rhombuses       Hexagons, triangles and rhombuses        Squares and parallelograms       Centers of squares   Hexagons with empty triangles     Octagons and squares     Triangles with empty hexagons   Hexagons linked   Squares and empty octagons       Triangles and dodecagons       References

Introduction

In this article, we form tilings by using rigid pieces (colored) hinged together at their vertices and separated by empty spaces (white). Two rigid shapes are considered hinged when they have one vertex in common, and both are free to rotate around that vertex. In a tiling where the shapes are hinged to each other, when one of the rigid pieces moves, all the other rigid shapes move also around hinges at the vertices of the tiling, and the whole tiling adjusts its global shape. The colored tiles remain the same shape all the time, they just change their positions and angles with respect to each other. To adjust for the movements of the rigid pieces, the shapes of the empty spaces change, but remain congruent to each other.

These kinds of tilings are called hinged tilings or hinged tessellations (Wells 1991, p. 101). Wells (1988) and Frederickson (2002) present some examples of hinged tilings. In the last three examples given in this article, rigid polygons are linked to each other with rigid rods that are hinged to vertices of the polygons. Early work in the kinds of transformations illustrated here was developed by Stuart (1963). Stuart went beyond static printed figures. He used first finger movies to convey a dynamical feeling of hinged transformations for polyhedra.

Here we deal with two-dimensional shapes. We use a dynamic geometry program (Jackiw 2001) and its internet component (JavaSketchpad 2003) to represent these tilings. You can drag the shapes to make them rotate around the hinges to get a kinesthetic feeling and to visually follow the changes as tilings morph into other tilings. We encourage you to experiment first with the interactive figures listed on the left to get this dynamical feeling and to explore the transformations. As part of the exploration, we invite you to focus on specific angles of the empty shapes, as some angles will generate regular tilings or other interesting tilings. In the interactive figures we ask you to guess what shapes will be formed for different angles of the empty shapes. You can also see how different kinds of regular tilings are related to each other, and to other tilings that are not regular, via these transformations. The text below and the static figures show some of the interesting tilings that are generated for specific angles. They also provide answers to the questions asked with the interactive figures. Some of these examples can be reached at different stages of the continuos transformations of one tiling into another made possible by the interactive figures. You can reverse the transformations by dragging in the opposite direction.

Hinged squares

The first tiling is formed by rigid squares hinged together. The empty spaces form rhombuses that can be changed in shape. Look at the angle of the empty rhombus at the vertices of the hinges.

A tiling consistent of only squares can be obtained in two ways, when the angle of the rhombus is 90˚ (empty space becomes a square) and when the angle is 180˚ or 0˚ (empty space disappears).
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In figure 1, drag the point to move the squares until the rhombuses have a 60˚ angle. If you divide each rhombus into two equilateral triangles you would have a regular tiling formed by squares and equilateral triangles.


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Hexagons, triangles, and rhombuses

The second tiling consists of rigid regular hexagons and equilateral triangles, leaving rhombuses as empty spaces.

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In figure 2, as one of the vertices of the tiling is moved, the others move too.  A special case of the rhombus is when the angle is 90˚, in which case the rhombus will be a square, thus a regular tiling of the plane with hexagons, squares, and triangles is obtained.



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When the angle of the rhombus are 60˚ and 120˚, the rhombuses can be divided into two equilateral triangles. This will give rise to a regular tiling of the plane formed by hexagons surrounded and separated by triangles. There is another such tiling with hexagons and triangles that is the mirror image of the one shown.


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When the angle is zero, the rhombuses disappear, and a new regular tiling is formed with only hexagons and triangles. Three hexagons surround each triangle.


Squares and parallelograms

Rigid squares of two sizes form the third hinged tiling.

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In figure 3, when the angles of the empty prallelogram are 90˚, we have rectangles separating the squares.

When the angle is 0˚ we have a tiling of squares of two different sizes.


Centers of squares around a parallelogram

In figure 4, we can see that the centers of the four squares on the sides of a parallelogram form a square for any angle of the parallelograms.

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Hinged hexagons

Figure 5 shows a tiling of hexagons and triangles that can be transformed to a tiling with only hexagons. In this case every other vertex of the hexagons is hinged. When non-hinged vertices of two hexagons touch each other, we get equilateral triangles.

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Octagons and squares
Figure 6 shows how a tiling of octagons and squares can be opened to leave some empty spaces and then be closed again into another tiling of the same kind, but where now polygons share different sides. Every other vertex of an octagon is hinged.
         

           

Hinged triangles

Figure 7 transforms a tiling using only triangles into a tiling of empty hexagons surrounded by triangles.
              
  


Hexagons with linkages

In figure 8 a tiling consisting of yellow hexagons can be transformed into a tiling of yellow hexagons surrounded by empty hexagons. An interesting intermediate case is when the empty spaces are equilateral triangles. If we subdivide these into four smaller equilateral triangles we obtain a regular tiling of hexagons and equilateral triangles.



Squares with linkages

In figure 9 squares are linked to other square with a rigid rod. A tiling consisting only of squares can be transormed into a regular tiling of squares and regular octagons.  For an angle of 60˚, we can imagine subdividing the four pointed star into four triangles surrounding a square. An interesting intermediate case is when the empty spaces form bigger squares.



Triangles with linkages

In figure 10 a tiling of equilateral triangles can be transformed into a regular tiling of equilateral triangles and regular empty dodecagons. What are some of the intersting intermediate special cases?
 



References

Frederickson, Greg N. Hinged dissections: Swinging and twisting. Cambridge, England; Cambridge University Press, 2002.

Jackiw, Nicholas. The Geometer's Sketchpad (Version 4) [Computer software]. Emeryville, CA: Key Curriculum Press, 2001

JavaSketchpad. Emeryville, CA: KCP Technologies, 2003.

Stuart, Duncan. Polyhedral and mosaic transformations. Student Publications of the School of Design, North Carolina State University 12 no. 1 (1963): 2-38.

Wells, David. Hidden conections double meanings. Cambridge, England: Cambridge University Press, 1988.

Wells, David. The Penguin dictionary of curious and interesting geometry. London: Penguin Books, 1991.

Squares and rhombuses       Hexagons, triangles and rhombuses        Squares and parallelograms       Centers of squares      Hexagons with empty triangles   Octagons and squares   Triangles with empty hexagons    Hexagons linked     Squares and empty octagons       Triangles and dodecagons   References