Tessellations by Recognizable Figures

From EscherMath
Jump to navigationJump to search
Horseman. M.C. Escher, 1946.

Relevant examples from Escher's work:


A Quick Introduction to Creating Escher Like Tessellations

Have you ever wondered how M.C. Escher made some of his wallpaper patterns? Here are some examples:

Regular-division-45.jpgRegular-division-67.jpg Regular-division-96.jpg

All M.C. Escher works © Cordon Art BV - Baarn - the Netherlands. All M.C. Escher works (c) 2007 The M.C. Escher Company - the Netherlands. All rights reserved. Used by permission.


It is not as difficult as it looks. If you want to make drawings like this, all you need is some geometry. These regular divisions of the plane go by several different names. They are called tessellations, tilings and even wallpaper patterns. You of course all know examples of these tilings. Think about the tile in your bathroom. You most often have an entire pattern made up out of squares. Some older bathrooms may actually have hexagonal (six sided) tiles. If you do, the pattern will resemble that of chicken wire. If you look at the side of a brick building, you will see a tessellation by rectangles. We will see that all of Escher’s patterns are derived from just a handful of geometric patterns. There are several different techniques that Escher used, and sometimes he combined techniques as well. The simplest example of an Escher like tessellation is based on a square. We will start out with a simple geometric pattern, and then change that ever so slightly.

Escher-trial-1.png

In this example all the vertical pieces were made to look like a lightning bolt, and the horizontal pieces like another jagged piece. Note that all vertical pieces are changed in the same way, and that all horizontal pieces look the same as well. Now it becomes an inkblot test. What do you see? Some will think it looks like inter locked men, others may see birds. By decorating each of the newly formed tiles you will create an Escher like tessellation. This method is the simplest way to create Escher –like tessellations, and also works well if you start with a tessellation by rectangles or parallelograms. You should give this a try! It’s not hard, and it’s a lot of fun.

Rotation about a vertex This technique only works with the square, rhombus and the hexagon. If you rotate through a vertex, then you will map one side onto another, and for this to work we need all sides to have the same length. Below you see an example worked out for a square tessellation.


Escher-trial-2.png

Rotation about the midpoint of a side.

This technique is the most general. This method works for all possible geometric tessellations, not just the six used by Escher. This technique is fun to experiment with on the computer. If you take care not to overload the memory capacity of your machine, you can start with any 3- or 4-gon, and create an Escher-like tessellation.

  1. Draw a 3- or 4-gon
  2. Create a midpoint on each of the sides.
  3. Modify one half of each of the sides.
  4. Rotate each side 180° about the midpoint. Repeat.

Triangle Example

Escher-trial-3.png

Kite example

Escher-trial-4.png

Escher's Polygon Systems

Escher created his tessellations by using fairly simple polygonal tessellations, which he then modified using isometries. This chapter gives a brief overview of Escher's own categorization system for tessellations and contains instructions for creating tessellations by recognizable figures using some of Escher's simpler techniques.

Escher organizes his tessellations into two classes, systems based on quadrilaterals, and triangle systems built on the regular tessellation by equilateral triangles. The bulk of Escher's tessellations are based on quadrilaterals, which the novice will find much easier to work with. The less common triangle systems are easily identified because three or six motifs will meet at a point, and the entire tessellation will have order 3 or order 6 rotation symmetry.

For a complete discussion of Escher's systems, read Visions of Symmetry (Chapter 2), which also reprints each page of Escher's notebook "Regular Division of the plane into asymmetric congruent polygons". Here, we give only a brief summary of his quadrilateral systems.

Within the quadrilateral systems, Escher's categorization has two factors: the type of polygon and the symmetries present in the tessellation. Specifically, he assigns a capital letter to each type of polygon and a Roman numeral to each type of symmetry. The polygons are:

  • A - Parallelogram
  • B - Rhombus
  • C - Rectangle
  • D - Square
  • E - Isoceles Right Triangle

Note that each is a special type of quadrilateral except for E, the isoceles right triangle. These 45°-45°-90° triangles fit together to form a square grid with order 4 rotation symmetry.

Escher's ten different systems of symmetry do not correspond exactly to the wallpaper groups. The wallpaper group for a figure describes the isometries of the figure. Escher's systems also describe the isometries, but are additionally concerned with the relative positions of the different motifs. There are five wallpaper groups that have no reflections and no order 3 rotations, and all of Escher's symmetry systems correspond to one of these five wallpaper groups.

Symmetry types for quadrilateral systems
System Translations Rotations Glide-Reflections
I
Translation in both transversal and diagonal directions none none
II
Translations in one transversal direction 2-fold rotations on the vertices; 2-fold rotations on the centers of the parallel sides none
III
Translations in both diagonal directions 2-fold rotations on the centers of all sides none
IV
Translations in both diagonal directions none Glide-reflections in both transversal directions
V
Translations in one transversal direction none Glide-reflections in one transversal direction. Glide-reflections in both diagonal directions
VI
Translations in one diagonal direction 2-fold rotations on the centers of two adjacent sides Glide-reflections in one diagonal direction. Glide-reflections in both transversal directions, but only in the direction of the sides without rotation point
VII
none 2-fold rotations on the centers of the parallel sides Glide-reflections in one transversal direction. Glide-reflections in both diagonal directions.
VIII
none 2-fold rotations on the four vertices Glide-reflections in both transversal directions
IX
none 4-fold rotations on diagonal vertices; 2-fold rotations on diagonal vertices none
X
none 4-fold rotations on three vertices; 2-fold rotations on the center of the hypothenuse none

Escher created at least one tessellation with each of the possible systems in his categorization. His sketches were organized into five folio notebooks, the Regular Division of the Plane Drawings. Each of these drawings is carefully numbered and marked with Escher's categorization.

For instance, in Sketch #96 (Swans), notice the system IV-D denoted below the sketch. System IV-D means that the underlying geometric tessellation is based on a square and that there must be translations in both diagonal directions, no rotations, and glide-reflections in both transversal directions.

It is hard to see what Escher means by 'transversal directions'. In this sketch #96, you need to turn the sketch at an angle so as to see rows and columns of touching swans, alternating black and white colors. These strips alternate a swan with it's mirror image, so swans along these strips (the 'transversal directions') are alternately reflected. It's a different usage of the term 'glide reflection' than we're used to seeing. In fact, using the mathematical definition of glide reflection this sketch has two different types of glide reflection symmetry, both in the vertical direction.

Tessellating With Translations

The simplest and most flexible tessellations are Escher's Type I systems, which can be based on a paralellogram, rhombus, rectangle or square.

Schematic for Escher's Type I Tessellation Systems
Creating a tile for a system I-A tessellation. (See Sketch #105 (Pegasus)).

To create one of these tessellations, follow these steps:

  1. Draw a parallelogram. This is easy on graph paper, as you can count squares to ensure the opposite sides are parallel and the same length.
  2. Alter the top edge of one parallelogram by replacing it with a curved or crooked line.
  3. Translate that edge to the bottom of the paralleogram.
  4. Alter the left edge of the parallelogram
  5. Translate the left edge to the right side.

This gives a figure which tessellates, and with luck its outline will suggest a recognizable motif that you can develop with further alterations to the edges. Finish by creating more copies of the motif by translation.

The resulting tessellation has symmetry group p1.

Escher's Regelmatige vlakverdeling, Plate I is an illustrated description of this process. A grid of parallelograms appears in panels 1-4, then develops bent or curved edges in panels 5-7. Finally, with the addition of detail, the tile becomes a bird or a fish.

Escher made many sketches using system I. Some good examples to look at include Sketch #38 (Moths), Sketch #73 (Flying Fish), Sketch #74 (Birds), Sketch #105 (Pegasus), Sketch #106 (Birds), Sketch #127 (Birds), and best of all Sketch #128 (Birds) where it is very easy to see how the bird motif developed from a square tile.

Tessellating With Glide Reflections

This simple arrangement of parallelograms is a good starting point for creating tessellations with glide reflection symmetry:

Tess-parallelograms.svg

The pattern has horizontal translation symmetry, and vertical glide reflection. To create an interesting tessellation from it:

Creating a tile for a system V tessellation. (See Sketch #97 (Bulldogs)).
  1. Draw a parallelogram (or rectangle).
  2. Alter the top edge of the parallelogram by replacing it with a curved or crooked line.
  3. Glide-reflect that edge to the bottom of the paralleogram.
  4. Alter the left edge of the parallelogram
  5. Translate the left edge to the right side.

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. Along with Sketch #97 (Bulldogs), Escher used this technique in Sketch #108 (Birds) and Sketch #109 (Frogs). 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:

Tess-kites.svg

Alternately, draw the long diagonals and erase horizontal edges to form a tessellation by "dart" shapes:

Tess-darts.svg

Creating a tile for a system V tessellation based on kites. (See Sketch #66 (Winged lions)).

To create a tessellation using the kite or dart pattern above, follow these steps:

  1. Draw the kite or dart pattern by starting with parallelograms.
  2. Alter one top edge of the kite or dart by replacing it with a curved or crooked line.
  3. Glide-reflect that edge to the bottom of the kite or dart.
  4. Alter the other top edge of the kite or dart.
  5. Glide-reflect that edge to the bottom of the kite or dart.

Escher classified this sort of tessellation as Type IV. Good examples of tessellations based on the kite shape are Sketch #62 (Sniffers), Sketch #66 (Winged lions), Sketch #67 (Horsemen), and Sketch #96 (Swans). 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.

Type IV tessellations based on the rhombus (left) and on the square (right)

Tessellating With Rotations

Escher's Type II: these tessellations have 2-fold rotations at all the vertices and 2-fold rotations in the centers of opposite sides.


Escher Type II tessellations,

Other Interesting Methods

Cutting a tile into two pieces is a simple way to get added flexibility. The dividing line rarely needs to obey any symmetries, and so can be drawn freely. A good example of this is Escher's Sketch #76 (Birds and horses), which could be started as a Type IV or Type V grid of parallelograms each divided into two. Other examples based on translation only are Sketch #52 (Birds and frogs) and Sketches #47-50 which are the basis for Verbum.


Pentagon tessellation. Escher's favorite Alhambra pattern.

Heesch Types

Related Sites