Escher made many sketches using system I. Figures with bilateral symmetry are naturally easier to make into recognizable figures, because many natural forms have bilateral symmetry. To create a tessellation by bilaterally symmetric tiles, we need to start with a geometric pattern that has mirror symmetries.
However, these mirror symmetries should not lie on the straight sides of the polygon tiles. If they do, the straight sides must remain straight and there is no longer flexibility to make a recognizable figure.
This is a very simple method for generating a tessellation by two different tiles. Each of the two tiles has bilateral symmetry. Begin with a tessellation by rectangles.
The vertical mirror symmetries down the centers of the rectangles will remain in the final tessellation. Repeat the resulting figure in a checkerboard pattern, leaving spaces which form the other tile of the tessellation. Notice that the horizontal strips of tiles form frieze patterns with pm11 symmetry, which explains why the horizontal translation is by two tiles - the vertical mirror lines must be spaced at half the translation length. The resulting figure tessellates in a pattern similar to wood shingles, and gives a tessellation with symmetry group cm.
This simple arrangement of parallelograms is a good starting point for creating tessellations with glide reflection symmetry:. The pattern has horizontal translation symmetry, and vertical glide reflection. To create an interesting tessellation from it:. This gives a figure which tessellates. Repeat identical copies of it to the left and right, and repeat mirror image copies above and below. The resulting tessellation has symmetry group pg.
Escher would describe this as a Type V system, although it doesn't fit exactly into his categorization. Another good example is Sketch 17 Parrots , though it is a slight variant. For a shape that lends itself even more towards recognizable figures, divide each parallelogram into two halves by drawing its short diagonal.
Then, erase the horizontal edges to form a tessellation by "kite" shapes:. Alternately, draw the long diagonals and erase horizontal edges to form a tessellation by "dart" shapes:.
Escher classified this sort of tessellation as Type IV. The birds in Sketch 19 Birds are only slightly altered from the dart scaffolding, although Escher's visible grid of rhombuses suggests he went about the construction in a completely different manner.
Escher wrote in his summary chart that Type IV tessellations have translations in both diagonal directions and glide-reflections in both transversal directions. This means that motifs that share a side are reflected images, and motifs that touch at corners diagonally are translated images. There are many ways to use rotation symmetry as the basis for a tessellation, and only some simpler ones will be described in this section. Escher's Type II tessellations begin with a grid of parallelograms squares, rectangles, and rhombuses also work.
The tile will be distorted into a shape that can tessellate using 2-fold rotations at all four vertices and 2-fold rotations in the centers of one pair of opposite sides, as shown by the red dots in the figure below.
Repeat identical copies of it by translating up and down, and repeat rotated copies of it to the left and right. This results in a tessellation with symmetry group p2.
All triangles tessellate, and all quadrilaterals tessellate. Altering half of each side and filling in the other half by rotation will also give a tessellating shape. In fact, this technique quite general and works for many geometric tessellations. Fill in the tessellation by rotations. The best example is Sketch 88 Seahorse , because Escher's geometric scaffolding for the sketch is also in his notebook.
In the scaffolding, the underlying shape appears to be a triangle but should really be viewed as a quadrilateral with two sides in a straight line, giving four vertices and four midpoint rotation centers. Another good example is Sketch 9 Birds. In Sketch 9 Birds , each bird is derived from a quadrilateral which you can find by connecting the points with four birds coming together.
In Escher's scaffolding for the sketch, there is a visible grid of paralleograms which he obviously used to lay out the picture. His scaffolding is an easier way to build this type of tessellation by hand, and relies on a theorem of Euclidean geometry:.
Escher would have drawn the grid of parallelograms, constructed the midpoints of each side of the parallelogram, and then altered the parallelogram to the bird form allowing the sides to bend and corners to move. Other examples of Type III tessellations are Sketch 90 Fish and Sketch 93 Fish , where in the latter the eyes and mouths of the fish destroy the rotation symmetry of the silhouette.
Starting with a pattern of squares can produce a resulting tessellation with an order 4 rotation and symmetry group p4. Escher made no tessellations using this technique, but did do something similar with his Type X tessellations.
Rotation about a vertex can be applied to a regular hexagon as well, and Escher used this as the basis for one of his most successful tessellations, Sketch 25 Reptiles. There are 17 possible ways that a pattern can be used to tile a flat surface or 'wallpaper'. Between and Escher produced 43 colored drawings with a wide variety of symmetry types while working on possible periodic tilings.
He adopted a highly mathematical approach with a systematic study using a notation which he invented himself. There are 4 ways of moving a motif to another position in the pattern. These were described by Escher. A translation is a shape that is simply translated, or slid, across the paper and drawn again in another place.
The translation shows the geometric shape in the same alignment as the original; it does not turn or flip. A reflection is a shape that has been flipped. Most commonly flipped directly to the left or right over a "y" axis or flipped to the top or bottom over an "x" axis , reflections can also be done at an angle. If a reflection has been done correctly, you can draw an imaginary line right through the middle, and the two parts will be symmetrical "mirror" images.
To reflect a shape across an axis is to plot a special corresponding point for every point in the original shape. Rotation is spinning the pattern around a point, rotating it. How to Solve a Hexagon. How to Calculate Volumes of Pentagonal Prisms. How to Make a 3D Hexagon. How to Make a Hexagon on Grid Paper. What Are the Degrees of a Quadrilateral? What Are Vertices in Math?
What Are Congruent Shapes? Six Properties of a Parallelogram. How to Find an Angle of a Hexagon. How to Calculate the Missing Angle of a Rectangle. How to Figure Degrees in an Octagon. What Is a Congruence Statement? What Are the Characteristics of a Pentagon, Hexagon
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