M. C. Escher Exhibition Study Guide

M. C. Escher, Drawing Hands, 1948, Lithograph

The Magical World of M. C. Escher Study Guide Museum of Art - DeLand, Florida 1

The Magical World of M. C. Escher Study Guide Museum of Art - DeLand

The Museum of Art - DeLand is proud to present this study guide as an educational resource to the exhibition, The Magical World of M. C. Escher January 26 through March 25, 2018.

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Who Was M.C. Escher?

M. C. Escher preparing for his famous Hand with Reflecting Sphere, or Self-Portrait in Spherical Mirror, 1935

M. C. Escher, Hand with Reflecting Sphere, 1935, Lithograph

Maurits Cornelius Escher is best known as a graphic artist but was also an illustrator, master printmaker, designer and muralist. He was born in 1898 in Leeuwarden, Friesland in the Netherlands and was the youngest of five sons of a prosperous Dutch civil engineer. Escher excelled at drawing from an early age, but struggled academically. In fact, he never officially graduated because he failed his final exams in secondary (High) school. Escher went on to study architecture at the urging of his father, but changed course when an instructor and mentor encouraged him to develop his drawing and printmaking skills.

It was after an extended visit to Italy and Spain in the early 1920s that Escher began perfecting his printmaking techniques, creating images of many realistic landscapes inspired by his travels. During this trip, Escher visited the Alhambra Palace in Granada, Spain which was to have an immense influence on his life. He was overwhelmed by the beauty of the 14th century palace and in particular, by the various styles of Moorish tiles that decorated the building. It was during these travels that Escher began to develop his meticulous observation of nature and his passion for the geometrical regularity he noticed in the world surrounding him.

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The Alhambra Palace in Granada, Spain

It was in 1936 on another visit to Spain and the Alhambra Palace that Escher’s interest in the regular division of the plane developed into his first mathematically-inspired tessellation designs. Escher was not trained as a mathematician, but he took a great interest in geometry and what he called ‘the logic of space.’ As his career progressed, Escher became more and more interested in space, illusion and math.

He studied these subjects and created the work he is best known for, such as images of impossible worlds, repeating patterns and reality-bending shapes that require the viewer to see the images from new perspectives. As a result, Escher’s works are more often associated with mathematics and psychology than art history.

Tiling designs from the Alhambra.

One of the Alhambra tilings sketched by M. C. Escher in 1936.

Source: gruban - http://www.flickr.com/photos/gruban/11341048/

Over the course of his career, Escher had become a research mathematician of the highest order developing his own categorization system which covered all the possible combinations of shape, color and symmetrical properties. As such, he had unknowingly studied areas of crystallography years in advance of any professional mathematician working in this field. Eventually, Escher made numerous woodcuts utilizing each of the 17 symmetry groups.

By the 1950s, Escher had received international recognition and was featured in Time and Life magazine articles. He was in demand both for artistic audiences, and for scientific ones. In a lecture in 1953 Escher said, “I have often felt closer to people who work scientifically (though I certainly do not do so myself) than to my fellow artists.” In 1955 he was knighted by the King of the Netherlands. During the 1960s, Escher’s popularity increased further as he was embraced by the ‘hippie’ generation, with its love of 52

psychedelic art. This success was capped off by the publication of The Graphic Work of M.C. Escher, which included 76 prints and a commentary by the artist.

M. C. Escher died on March 27, 1972, at the age of 73. He was a prolific and passionate artist dedicated to his craft. His imaginative way of thinking and unique graphics have had a continuous influence in mathematics, science and art, as well as in popular culture. He left a legacy of art that continues to fascinate mathematicians, scientists, artists, philosophers, and museum-goers around the world.

Escher Fast Facts 

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This distinguished graphic artist shared a common trait with a few of his legendary predecessors, Leonardo da Vinci, Michelangelo, Holbein, and Durer – all of them were lefthanded.

He is sometimes called “a one-man art movement” and is referred to as the Father of Modern Tessellations.

He made over 2000 drawings and 448 lithographs, woodcuts, and wood engravings.

His work can be divided into two periods. Before the days of geometry, reflections, architecture, and impossible structures, Escher primarily produced drawings and prints of nature. During this time period, he focused on insects, plants, and landscapes, all of which would form a part of his later work.

After Escher made contact with mathematicians, he made much more complicated structures and his work became much deeper mathematically. In particular, Escher was excited by the work of English mathematician Roger Penrose. The Penrose Triangle, which was devised and popularized by Roger Penrose, was partly inspired by depictions of impossible objects by Escher.

In turn, Escher’s famous lithograph prints Waterfall and Ascending and Descending were inspired by the work of Roger Penrose.

Contrary to popular belief, Escher had little background in or talent for math, but that all changed after he read a paper by George Pólya on “plane symmetry groups,” repetitive patterns on two-dimensional surfaces. The paper inspired his work for decades to come, though Escher admits he understood little of the mathematical theory behind it. As his interest in geometries developed, Escher would study topology, work with H.S.M. Coxeter on tessellations, and form a lasting collaboration with Roger Penrose that explored mathematically impossible forms — which inspired his canonical Ascending and Descending.

Some of Escher’s work may have anticipated many deep features of modern cosmology. Today, cosmologists think that the universe may indeed be Escher-shaped. In 1959, Escher created a woodcut titled Circle Limit III using just basic drawing tools. Nearly 40 years later, mathematicians confirmed that it was an astonishingly accurate representation of space as it edges towards infinity, absolutely right to the last millimeter. Modern day mathematicians understand much more clearly the implications of Escher’s works.

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Mathematics and geometry teachers often use his prints to demonstrate to their students how math and science can be a source of poetry and beauty. Psychology textbooks offer them as proof of the claim that our perceptions of reality are, in fact, constructions.

Escher’s mind-bending visions have provided inspiration for the creators of The Simpsons, as well as film-makers including Jim Henson, whose 1986 film Labyrinth starring David Bowie includes a homage to Relativity, and Christopher Nolan, who created a dizzying, Escher-like dream sequence for his 2010 blockbuster Inception, in which the streets of Paris are seen to fold, buckle, and warp.

During the ’60s, Escher’s work found mainstream popularity, as hippies delighted in its supposedly “psychedelic” qualities. (It used to be believed, incorrectly, that the plant at the center of Balcony was cannabis.)

When Mick Jagger wrote to ‘Maurits’ asking for permission to reproduce one of his pictures on the cover of the Rolling Stones’ album Through the Past Darkly, Escher refused, informing the rock star’s assistant: “Please tell Mr. Jagger I am not ‘Maurits’ to him.”

Escher’s stunning representation of three-dimensional objects in two-dimensional drawings, as well as his fantastic world-building techniques in works like Metamorphosis and Relativity, make him an influential figure for video game designers and digital artists today.

Escher Quotes "I have had a fine old time expressing concepts in visual terms, with no other aim than to find out ways of putting them on paper. All I am doing in my prints is to offer a report of my discoveries."

"I don't grow up. In me is the small child of my early days."

"I try in my prints to testify that we live in a beautiful and orderly world, not in a chaos without norms, even though that is how it sometimes appears.”

“Order is repetition of units. Chaos is multiplicity without rhythm.”

“I can't keep from fooling around with our irrefutable certainties. It is, for example, a pleasure knowingly to mix up two and three dimensionalities, flat and spatial, and to make fun of gravity.”

“I never got a pass mark in math ... Just imagine - mathematicians now use my prints to illustrate their books.”

“Science and art sometimes can touch one another, like two pieces of the jigsaw puzzle which is our human life, and that contact may be made across the borderline between the two respective domains.”

“He who wonders discovers that this in itself is wonderful.

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Escher’s Art Techniques M.C. Escher is one of the world's most famous graphic artists. He was skilled in a number of different printing techniques such as woodcuts, lithographs and mezzotints. In 1946, he first tried his hand at mezzotints and made eight by 1951, when he decided that the process was, as he said, “too great a test of his patience.” Despite his proficiency with many art forms, Escher favored woodcuts.

Woodcut & Wood Engraving Woodcut originated in China during the eighth century. This relief technique of stamping from woodblocks was used to print textiles before it was applied to paper. Wood engraving is a much newer process invented by Thomas Bewick in the 18th century. The difference between a woodcut and a wood engraving is that for the woodcut, the artist uses the side grain, while for the wood engraving, the end grain is used.

The same three steps are required for both: 1. The artist draws the image on the piece of wood and carves away the areas that must remain white.

2. With a roller, ink is applied to the raised surface of the wood.

3. A printing press or a hard smooth tool is used to press the paper on the woodblock to transfer the image.

As with all prints, the image cut or carved in the wood is the reverse of the final print.

M. C. Escher, Dream (Mantis Religiosa), 1935, Wood Engraving

M.C. Escher, Sky and Water l, 1938, Woodcut

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Lithography The term lithography comes from the Greek “lithos” (stone) and “graphy” (writing). The lithography technique is based on the physical property that oil and water do not mix. Because of the equipment used and the knowledge and skill required for the printing process, lithography lends itself to collaboration between an artist and a printer. When the printing is done by the artist himself, it is so noted in pencil in the lower margin.

Five steps are required to produce a lithograph: 1. In lithography, a limestone or metal plate is polished so that its surface is uniform and free of grease and oils.

2. The artist draws directly on the plate, using special oil-based lithographic pencils.

3. Water is then applied on the surface, which is repelled by the greased image but absorbed elsewhere.

4. With a roller, oil-based ink is applied on the entire surface of the stone or metal plate. It is repelled in the wet areas, but clings to the grease pencil drawings.

5. A printing press is used to transfer the image to the paper.

M.C. Escher, Reptiles, 1943, Lithograph

Mezzotint The term “mezzotint” comes from the Italian “mezzo” (half) and “tinta” (tone), and means halftones. It is not lines, but shaded and light areas that form the image. Mezzotint achieves tonality by roughening a metal plate with thousands of little dots or burrs, made by a metal tool with small teeth, called a "rocker." When ink is applied to the metal plate, it is trapped between the roughened burrs. The mezzotint technique is rarely used because it is laborious and time consuming.

Four steps are required to produce a mezzotint: 1. A copper or steel plate is prepared by roughing it with a “rocker” (a curved, notched blade that resembles the teeth of on a comb) so that the surface is entirely filled with thousands of crossing lines of rough dots. At this stage, if the plate were inked, it would print a rich, uniform black.

2. A special tool is used to burnish and smooth the areas that must be lighter, reducing the amount of ink that will be absorbed in step 3.

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3. The ink is applied uniformly with a roller over the plate, burnished areas retaining less ink. A completely smooth area holds no ink, and so produces white space.

4. A printing press is used to press the paper on the inked copper plate to produce the image. Usually, no more than 50 quality prints can be produced from one plate.

M. C. Escher, Eye, 1946, Mezzotint

Escher’s Themes/Subjects In the early part of his career, Escher focused on the subject of landscapes that were inspired by his travels to Italy and Spain. He would often incorporate unusual or dramatic perspectives in his prints and drawings from this period. Some of this work included images of insects and plants as well. Around 1935 after another visit to Spain, his attention shifted from the subject of landscape to something he described as "mental imagery." In the later 1930s, Escher introduced the theme of the regular division of the plane and tessellations that were inspired by the decorative designs he had seen in the Alhambra Palace. Escher employed this concept in the creation of his Metamorphosis prints, where one shape or object turns into something completely different. This evolved into one of Escher’s favorite subjects. He became increasingly fascinated with complex architectural mazes involving perspective games and puzzles, the representation of impossible spaces or objects, reflections, visual paradoxes, levels of reality, relativity, and infinity. All of these interests developed into subject matter for his artworks.

Although Escher was not trained in mathematics, a definite mathematical influence emerged in his work. Initially, his approach to the math was more institutive and visual rather than theoretical.

However, over the course of his career that changed, and he began studying mathematical theory and collaborated with noted mathematicians on various projects and papers. Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. His Circle Limit series demonstrates this concept and his ability to create perfectly consistent mathematical designs.

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M. C. Escher, Day and Night, 1938, Woodcut

This print is one of Escher's first to show the influence of the Alhambra’s tile work with its abstract, positivenegative geometric shapes that spurred his interest in the regular division of the plane and tessellations.

Day and night landscapes are shown as mirror images, where the white birds merge with a daylight sky and the black birds blend to create a night sky.

This is Escher’s first print to focus primarily on his idea of relativity, how one object is seen in relation to another. It also addresses the ideas of perspective, paradox, and architectural mazes.

M. C. Escher, Other World, 1947 Wood Engraving and Woodcut

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This iconic image incorporates several themes that Escher frequently employed in his work including a visual game in which he transformed a flat pattern into three-dimensional objects, paradox, perspective, and spatial illusion.

Although Escher often used paradoxes in his works, this is one of the most well-known examples.

M. C. Escher, Drawing Hands, 1948, Lithograph

Ascending and Descending is one of Escher’s most recognizable “impossible objects” images. It features two ranks of human figures trudging forever upwards and eternally downwards respectively on an impossible four-sided eternal staircase. This work was inspired by the British mathematician Roger Penrose and his father, the geneticist Lionel Penrose. Roger Penrose created the Penrose Triangle or Tribar, which is constructed impossibly from three 90degree angles. Escher incorporated three of the Penrose Triangles in his 1961 print, Waterfall.

M. C. Escher, Ascending and Descending, 1960, Lithograph

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M. C. Escher, Metamorphosis II, 1939-1940, Woodcut

The concept of this piece is to morph one image into a tessellated pattern and then slowly alter that pattern eventually to become a new image. This print deals with levels of reality, a theme that appears throughout Escher's work and is a natural offshoot of his preoccupation with time and space. The juxtaposition of fantasy and reality can be seen in this print that plays with reflection not only as a symmetry operation, but also as a psychological or metaphysical experience. In this work, Escher explores an interest in cycles and infinity. In this series, Escher told a story through the use of pictures.

This is one of a series of four woodcuts by Escher depicting ideas from hyperbolic geometry. In the Circle Limit series, Escher experimented with the problem of producing an infinitely repeating pattern in a finite figure. This is one of the two kinds of non-Euclidean space that Escher depicted in this series. This one is based on the French mathematician PoincarĂŠ and the other model was inspired by the work of H.S.M Coxeter.

Can you image what it would be like to be inside this space?

M. C. Escher, Circle Limit III, 1959, Woodcut

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M. C. Escher, Snakes, 1969, Woodcut

Escher had been in poor health since 1964 and Snakes was his final work. It took six months to complete and was finally unveiled in July 1969. Unlike the Circle Limit series, where Escher actually carried through the rendering of smaller and smaller figures to the smallest possible sizes, the infinite decrease of size – and infinite increase in number – is only suggested in Snakes. Even more unusual is the space implied in the image. Here the space heads off to infinity both towards the rim and towards the center of the circle, as suggested by the shrinking, interlocking rings. If you could enter into this space, how would your journey contrast to one in Circle Limit II? 14 11

A Closer Look and Activities Inspired by Escher’s Art Regular Division of the Plane/Tessellations The word “tessellation” comes from the Latin word “tessera” which means “small stone cube.” Ancient Roman mosaics are early examples of tessellations. Tessellations are repeating shapes that cover a surface without overlapping or leaving gaps. Typically, the shapes making up a tessellation are polygons or similar regular shapes. However, Escher was captivated by every kind of tessellation—regular and irregular—and took special delight in what he called “metamorphoses,” in which the shapes changed and interacted with each other, and sometimes even broke free of the plane itself. Examples of common tessellations include checkerboards, soccer balls, tiled floors, brick walls, honeycombs, quilt patterns, and some wallpaper designs.

Symmetry and Tessellations In art and design, symmetry refers to when one side of an object or image balances the other, like drawing a line down the middle of your face and noticing that each side is identical. There are different types of symmetry, including bilateral symmetry, which is when two sides surrounding a dividing line (like the face example) are exactly the same, and radial symmetry, when the symmetry revolves around a central point (like the spokes on a wheel).

Symmetry comes from a Greek word meaning 'to measure together' and is widely used in the study of geometry. Mathematically, symmetry means that one shape becomes exactly like another when you move it in some way: turn, flip or slide. For two objects to be symmetrical, they must be the same size and shape, with one object having a different orientation from the first. There can also be symmetry in one object, such as the example of the face. Not all objects have symmetry; if an object is not symmetrical, it is called asymmetrical.

Three types of mathematical symmetry are commonly found in tessellations. These are translational, rotational and glide reflection symmetry. Escher incorporated four types of symmetry in the regular division of the plane when creating his tessellations, including reflection, translation, rotation, and glide reflection.

Reflection Symmetry If points of a figure are equally positioned about a line, the figure has reflection symmetry, or mirror symmetry. The line is called the reflection line, the mirror line, or the axis of symmetry. The axis of symmetry separates the figure into two parts, one of which is a mirror image of the other part. The simplest case of reflection symmetry is known as bilateral symmetry.

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For example, each of the following figures exhibits bilateral symmetry:

The heart and butterfly each have a vertical axis of symmetry, and the lobster has a horizontal axis of symmetry. The arrow has an axis of symmetry at an angle. If you draw the reflection line though any one of these figures, you will notice that for every point on one side of the line there is a corresponding point on the other side of the line. If you connect any two corresponding points with a segment, that segment will be perpendicular to the axis of symmetry and bisected by it (cut into two equal length segments).

Bilateral symmetry is the most common type of symmetry found in nature, occurring in almost all animals and many plants. Cognitive research has shown that the human mind is specially equipped to detect bilateral symmetry. In fact, humans are especially good at detecting bilateral symmetry when the axis of symmetry is oriented vertically.

Some objects or images can have more than one axis of reflection symmetry. Here are some examples, with the reflection axes shown as dotted red lines. (Pay special attention to the diagonal reflection axes in the cross. These are easy to overlook.)

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Translation Symmetry 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. Every translation has a direction and a distance.

Rotational Symmetry To rotate an object means to turn it around. Every rotation has a center and an angle. If points on a figure are equally positioned about a central point, the object has rotational symmetry. A figure with rotational symmetry appears the same after rotating by some amount around the center point. The angle of rotation of a symmetric figure is the smallest angle of rotation that preserves the figure. For example, the figure on the left can be turned by 180° (the same way you would turn an hourglass) and will look the same. The center (recycle) figure can be turned by 120°, and the star can be turned by 72°.

For the star, where did 72° come from? The star has five points. To rotate it until it looks the same, you need to make 1 / 5 of a complete 360° turn and this is a 72° angle rotation.

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Glide Reflection Symmetry A glide reflection combines a reflection with a translation along the direction of the mirror line. In glide reflection, reflection and translation are used concurrently. Glide reflections are the only type of symmetry that involves more than one step.

A figure, picture, or pattern is said to be symmetric if there is at least one symmetry that leaves the figure unchanged. For example, the letters in ATOYOTA form a symmetric pattern: if you draw a vertical line through the center of the "Y" and then reflect the entire phrase across the line, the left side becomes the right side and vice versa. The picture doesn't change.

ATOYOTA If you draw the figure of a person walking and copy it to make a line of walkers going infinitely in both directions, you have made a symmetric pattern. You can translate the whole group ahead one person, and the procession will look the same. This pattern has an infinite number of symmetries, since you can translate forward by one person, two people, or three people, or backwards by the same numbers, or even by no people. There is one symmetry of this pattern for each integer (positive, negative, and zero whole numbers).

Source: The Four Types of Symmetry in the Plane written by Dr. Susan Addington California Math Show susan@math.csusb.edu http://www.math.csusb.edu/ formatted and edited by Suzanne Alejandre

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Translation Symmetry M. C. Escher, Bird Fish, 1938

Rotation Symmetry M. C. Escher, Lizard, 1942

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Glide Reflection Symmetry M. C. Escher, Horseman, 1946

Combination of reflection, rotation and glide reflection symmetry M. C. Escher, Angels and Devils, 1941

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Symmetry and Tessellation Activities Begin by defining symmetry, tessellation and translation (refer to previous text in this study guide and the glossary).

Explain that squares, triangles and hexagons are tessellations since, when placed side by side, they will cover an area without any gaps or overlaps.

Demonstrate that if you take a piece away from any of these shapes, they are no longer a tessellation because the shapes will not fit perfectly next to each other -- there will be gaps where the pieces were taken away.

Explain that the key to creating tessellations is if you take away a piece of a shape and put the piece back on another part of the shape, you will once again have tessellation, because the piece you give back will fit into the hole where it was taken away.

Show examples of M.C. Escher’s tessellations. You can find examples on the following links: http://www.tessellations.org/eschergallery1thumbs.shtml; http://www.mcescher.com/gallery/symmetry/; Pinterest and Google In the following activities three mathematical rules of repetition are used:  Shifting the position of a shape (something mathematicians call translation.)  Rotating a shape to a new position (mathematicians don’t have a fancy name for this; they just call it rotation.)  Flipping a shape over so it looks like a mirror reflection of itself (mathematicians call this reflection.)

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Objectives for the following Symmetry Tessellation Activities: 1) Identify the line and rotation symmetries of two-dimensional figures to solve problems. 2) Perform translations, reflections, rotations, and dilations of two-dimensional figures using a variety of methods (paper folding, tracing, graph paper). 3) Draw the results of translations, reflections, rotations and dilations of objects in the coordinate plane, and determine properties that remain fixed; e.g., lengths of sides remain the same under translations.

Activity One: Translation Symmetry Tessellation (Grades 6-8) In this activity, students will transform a rectangle into a more interesting shape, then make a tessellation by repeating that shape by applying Translation Symmetry.

Note: this activity can be adapted for grades 4 and 5.

Materials: • Index card 3" x 5" • Ruler • Scissors • Blank paper • Pencil • Transparent tape • Colored markers or pens • 2.5" x 3" grid paper Directions: 1) Cut an index card in half, creating a 2.5" x 3" rectangle.

2) Find the area of the rectangle (length x width).

3) Draw a line between two adjacent corners on one of the long sides of the rectangle. Your line can be squiggly or made up of straight segments.Whatever its shape, your line must connect two corners that share one side of the rectangle.

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4) Cut along the line you drew. Take the piece you cut off and slide it straight across to the opposite long side of the rectangle. Line up the long, straight edges of the two pieces and tape them together.

5) Can you tessellate with this shape? Try tracing this shape several times, creating a row going across a piece of paper. Line up the cut edges of the shape as you trace it.

6) Now draw another line that connects two adjacent corners on one of the short sides of the shape.

7) Cut along this new line. Take the piece you cut off and slide it straight across to the opposite side of the shape. Line up the straight edges and tape them together.

8) You have now created a shape that can be used as a pattern to make a tessellation. What’s the area of this shape? Write the letter A on one side of the shape and turn it over and write the letter B on the other side.

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9) On the grid paper, carefully trace around your pattern shape. Can you figure out where to place the pattern so that your paper will be covered with repetitions of this shape with no overlaps and no gaps?

Try to cover your whole sheet of paper by tracing the pattern, moving it, then tracing it again. If you start with side A facing up do you ever have to turn it over to side B to make your tessellation? If you only have to slide the piece without flipping it over or rotating it, then you are making a translation tessellation. In math, translation means shifting the position of a shape without moving it in any other way.

10) Look for a creative way to color in the resulting design on your sheet of paper. Does your shape look like a fish? A bird? An elephant?

11) You have learned about tessellations and seen some examples, but what exactly goes into making a tessellation? When creating one, what features do you have to think about? Make a list of those features.

Activity Two: Reflection Symmetry Tessellation (Grades 6-8) In this activity, students will transform a rectangle into a more interesting shape, then make a tessellation by repeating that shape by applying Reflection Symmetry.

Note: this activity can be adapted for grades 4 and 5.

Materials: • Index card 3" x 5" • Ruler • Scissors • Blank paper • Pencil • Transparent tape • Colored markers or pens • 2.5" x 3" grid paper 21 24

Directions: 1) Cut an index card in half, creating a 2.5" x 3" rectangle.

2) Draw a line between two adjacent corners on one of the long sides of the rectangle. Your line can be squiggly or made up of straight segments. Whatever its shape, your line must connect two corners that share one side of the rectangle.

3) Cut along the line you drew. Take the piece you cut off, FLIP it over and then slide it across to the opposite long side of the rectangle. Line up the straight edge of the piece with the straight edge of the opposite edge of the rectangle. Tape the piece in place.

4) Now draw another line that connects two adjacent corners on one of the short sides of the shape.

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5) Cut along this new line. Take the piece you cut off, FLIP it over and then slide it straight across to the opposite side of the shape. Line up the straight edge of the piece with the straight edge of the shape.

Tape the piece in place.

6) You have created a shape that you can now use as a pattern to make a tessellation. Write the letter A on one side of the pattern, then turn it over and write the letter B on the other side.

7) On your grid paper, carefully trace around your pattern shape. It may help to position the squared-off corner (formerly the edge of the index card) in one corner of the grid. Can you figure out where to place the pattern piece so that your paper will be covered with repetitions of this shape with no overlapping and with no gaps? Try to cover your whole sheet of paper by tracing the pattern, moving it, then tracing it again. If you start with side A facing up, do you ever have to turn it over to side B to make your tessellation? If you have to flip your piece over, you are making a reflection tessellation. If you also had to move the piece to a new position, you have used translation.

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8) Look for a creative way to color in the resulting design on your sheet of paper.

9) You have learned about tessellations and seen some examples, but what exactly goes into making a tessellation? When creating one, what features do you have to think about? Make a list of those features.

When you cut a shape out of paper, then flip it over, the flipped shape looks like a mirror image of the original shape. So a tessellation made with this technique is called a reflection tessellation. Your hands can help you understand the concept of mirror reflection. Your two hands are the same shape—but your right hand is a mirror reflection of your left hand (and vice versa).

Activity Three: Rotation Symmetry Tessellation (Grades 6-8) In this activity, students will transform a square into a more interesting shape, then make a tessellation by repeating that shape by applying Rotation Symmetry.

Materials: • Index card 3" x 5" • Ruler • Scissors • Blank paper • Pencil • Transparent tape • Colored markers or pens • 2.5" x 2.5 grid paper Directions: 1) Draw a 2.5" x 2.5" square on your index card.

2) Cut out the square from the index card.

3) Draw a line between two adjacent corners on one side of the square. Your line can be squiggly or made up of straight segments. Whatever its shape, your line must connect two corners that share one side of the square.

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4) Cut along the line you drew. Take the piece you cut off (without flipping) and slide it to an adjacent side of the square. Line up the straight edges and tape them together.

5) Now draw another line that connects the two corners on the side adjacent to the cut side of the square.

6) Cut along this new line. Take the piece you cut off (without flipping) and slide it to its adjacent side.

Line up the straight edge of the cut piece with the straight edge of the square, and tape them together.

7) You have now created a shape you can use as a pattern to make a tessellation. Write the letter A on one side of the shape; turn it over and write the letter B on the other side.

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8) On your grid paper, carefully trace around your pattern piece. Try to cover your whole sheet of paper by tracing the pattern, then move it and trace it again. If you start with the side A facing up, do you ever have to turn it over to side B to make your tessellation? If you have to flip your piece over, you have made a reflection tessellation. If you also had to move the piece to a new position you have also used translation. If you have to turn or rotate the shape to make your tessellation, then you have made a rotation tessellation.

9) Look for a creative way to color in the resulting design on your sheet of paper.

10) You have learned about tessellations and seen some examples, but what exactly goes into making a tessellation? When creating one, what features do you have to think about? Make a list of those features.

Š The Exploratorium, www.exploratorium.edu

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Activity Four: Tessellations (Grades K-5) Making tessellations combines the creativity of an art project with the challenge of solving a puzzle.

Part One: Tessellations with One Shape (This is challenging for grades K-1) Materials: • Pattern blocks (multiple sets) You can purchase these online, or download and print out paper patterns here: http://mason.gmu.edu/~mmankus/Handson/manipulatives.htm • Index card 3" x 5" Directions: 1) Choose one pattern block shape. Do you think this shape will cover your card? Using only your shape, cover the index card, leaving no spaces in between. How many shapes did it take to cover the card? How many fit across? How many fit up-and-down?

2) Check out other people’s work—did their shapes cover the paper? It is possible to do this with every pattern block shape. The process is called tessellating.

Part Two: Tessellating with Two Shapes (This is challenging for grade 1) Materials: • Pattern blocks (multiple sets), excluding the orange squares and white rhombuses • Triangle grid paper (See Resources for template) • Markers • Plain paper • Pencils • Tape Directions: 1) Now that you know that you can tessellate with any one of these shapes, try choosing two shapes.

Put the two blocks together to make a unit, and use a small piece of tape to hold them together. How many sides does this new shape have?

Grades 3- 5 If a side of the triangle = 1, what is the perimeter of the new shape?

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2) Describe out loud how you have arranged the two shapes. (For example: “The green triangle is above the yellow hexagon. One side of the triangle matches exactly with one side of the hexagon.”) Now build 10 more of the same unit, taping each unit together.

3) Can you tessellate space (cover the paper leaving no spaces in between) with this unit? If not, try making a new unit with two blocks.

4) Look at other people’s work as well. Were they able to tessellate space with their shapes? Describe out loud how you arranged the units to make a tessellation.

5) If you placed your units side-by-side without turning or flipping them, you made a translation tessellation. If you had to turn your units to fit them together (like a pinwheel), you made a rotation tessellation. If you had to flip your units over to the other side, you made a reflection tessellation. You may have had to do one, two, or all three of these things to make your tessellation.

6) You have made a repeating pattern, or periodic tessellation. How many ways can you arrange your two blocks into a unit to make periodic tessellations? Look at your classmates’ work to see if they have tried anything that you haven’t.

7) Once you have made a tessellation that you like, you can preserve it by either tracing the shapes or drawing them freehand on the triangular grid paper.

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Part Three: Tessellating with Three Shapes (Grades 3-5) Materials: • Pattern blocks (multiple sets), excluding the orange squares and white rhombuses • Triangular grid paper (See Resources for template) • Markers • Plain paper • Pencils • Tape Directions: 1) Repeat the steps from the previous exercise, but this time try three different pattern block shapes.

Some units will not tessellate. Were you able to tessellate space in a repeating pattern? Was this easier or harder than with one or two shapes?

2) If you placed your units side-by-side without turning or flipping them, you made a translation tessellation. If you had to turn your units to fit them together (like a pinwheel), you made a rotation tessellation. If you had to flip your units over to the other side, you made a reflection tessellation.

You may have had to do one, two, or all three of these things to make your tessellation.

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3) After you’ve tried as many combinations of translation, rotation, and reflection as you can, don’t be afraid to make a new unit. Record any units that don’t tessellate so that you won’t repeat them.

4) Once you have made a tessellation that you like, you can preserve it by drawing it onto the triangular grid paper. To do this, you can either trace the shapes onto the paper, or draw them freehand.

Part Four: Tessellating Three-Dimensional (3D) Space Materials: • Cubes (sugar cubes, wooden blocks, or any cubes that are easy to obtain) • Rectangular prisms (shoe boxes, toothpaste boxes, tissue boxes—as long as they are all the same size and shape) • Cylinders (soda cans, paper towel tubes, soup cans, or any cylinders that are all the same size and shape) • Spheres (marbles, tennis balls, or any spheres that are all the same size) • Unsharpened pencils (If you rubber band a bunch of pencils together and look at them from the end, you will see a tessellating honeycomb pattern.)

• Any other groups of identical three-dimensional objects Directions: 1) Explore tessellating with three-dimensional objects and notice the similarities to and differences from working with two-dimensional objects.

2) Start with the cubes. Can you stack the cubes together to fill three-dimensional space without leaving any gaps? To test this out, see if there are any spaces between the cubes that you can stick your pencil into. If not, you have tessellated three-dimensional space.

3) Now try each of the other shapes. Which ones tessellate space? Which ones do not?

© The Exploratorium, www.exploratorium.edu

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Activity Five: Exploring Symmetry with Your Name

The letter P is written in a grid of squares as shown:

Ask students to explain the steps necessary to achieve the results below:

(A combination of rotations about the center of the grid and reflections in the two lines through the center achieve the results.)

When the same combination of rotations and reflections is applied to the “A� example, the result is which of the following: a, b, c, d, or e?

Take the letters in your first name and perform the following symmetrical operations on them: 1) Reflect each letter through a horizontal line, and then rotate each letter 180 degrees.

2) Which letters in your name look different and which look the same? Explain why some letters look the same.

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Symmetry Worksheet

Name _____________________

Do these dotted lines represent lines of symmetry? Write yes or no.

A.

E.

B.

C.

F.

G.

D.

H.

Draw a line of symmetry on these shapes.

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Draw the second half of each symmetrical shape.

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TEACHER ANSWER FORM Do these dotted lines represent lines of symmetry? Write yes or no.

A.

B. YES

E. YES

C.

D.

YES

NO

F.

G. YES

YES

H. NO

NO

Draw a line of symmetry on these shapes?

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Activity Six: Escher and Polyhedra While most famous for his tessellations, Escher also had a strong affinity for polyhedra.

In this activity, students will identify the various polyhedra in a selection of Escher prints. (It may be helpful for the students to view the artwork online in order to see a larger, more detailed image.)

Begin this activity by providing the following definition: In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges. The word derives from the Greek poly (many) plus the Indo-European hedron (seat). Polyhedra are based on polygons, two dimensional plane shapes with straight lines. The plural of polyhedron is "polyhedra" (or sometimes "polyhedrons").

A Polyhedron is defined as having: Straight edges Flat sides (called faces) Corners (called vertices) In addition to being defined by the number of edges, faces and vertices they have, polyhedra are also defined based on whether their faces are all the same shape and size. Like polygons, polyhedra can be regular (based on regular polygons) or irregular (based on irregular polygons). Polyhedra can also be concave or convex. One of the most basic and familiar polyhedra is the cube. A cube is a regular polyhedron, having six square faces, 12 edges, and eight vertices.

1) Share and discuss the following images with students. Ask if they can identify some objects in the classroom that are examples of polyhedra.

Five Regular Polyhedra (Platonic Solids)

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Prisms are simple polyhedrons that have identical polygon ends and flat parallelogram sides.

Pyramids are polyhedrons with a polygon base and sides meeting at a single top point or apex.

2) Next, give students the Escher worksheet and have them identify and list the types of polyhedra Escher used in each image.

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Escher Artwork Activity Sheet ARTWORK

TYPES OF POLYHEDRONS/POLYHEDRA

Escher, Reptiles, Mezzotint, 1943

Escher, Crystal, Mezzotint, 1947

Escher, Star, Wood Engraving, 1948

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Escher, Flatworms, Lithograph, 1959

Escher, Waterfall, Lithograph, 1961

Name _____________________________________

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Escher Artwork Teacher Answer Key 

Reptiles Dodecahedron



Crystal compound of the cube and octahedron



Star stellated rhombic dodecahedron, compound of three octahedral, compound of two cubes with common 3-fold axis, compound of two tetrahdra (also known as a stella octangula), compound of a cube and an octahedron cuboctahedron, rhombicuboctahedron, square trapezohedron, trapezoidal icositetrahedron, triakis octahedron, all five platonic solids



Flatworms tetrahedron and octahedron



Waterfall compound of three cubes and the first stellation of the rhombic dodecahedron

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Activity Seven: Exploring Models of Regular and Advanced 3-D Shapes Students will use simple materials to investigate regular or advanced 3-dimensional shapes.

Objectives: Grade 3: • Analyze and describe properties of two-dimensional shapes and three-dimensional objects using terms such as vertex, edge, angle, side and face.

Grade 4: • Describe, classify, compare and model two-and three-dimensional objects using their attributes.

Grade 5: • Predict what three-dimensional object will result from folding a two-dimensional net, then confirm the prediction by folding the net.

Grade 6: • Classify and describe two-dimensional and three-dimensional geometric figures and objects using their properties; e.g. interior angle measures, perpendicular/parallel sides, congruent angles/sides.

Materials:  Drinking straws (whole and half lengths)  Paper clips (use with older students) or modeling clay/Playdoh (use with younger students)

Part One: Create Models of Regular and Advanced 3-D Shapes Directions: 1) Show students how to construct a rectangular prism made from drinking straws held together with small clumps of modelling clay. OR Bend the paperclips so that the 2 loops form a “V” or “L” shape as needed, widen the narrower loop and insert one loop into the end of one straw half, and the other loop into another straw half.

2) Place the skeleton next to a solid rectangular prism.

Ask students: • How are these two figures the same?

• How are these two figures different?

• Which figure is a skeleton of a rectangular prism? Why is it called a skeleton?

(The drinking straws are like the bones of the figure.)

• What parts of the figure does the skeleton show? (edges and vertices) • What part of the figure do the drinking straws show? (edges) Have students estimate the number of drinking straws that were used to build the skeleton. Dismantle the skeleton and count the drinking straws. Highlight that some drinking straws are long and some are short.

3) Now, show the following graphic example and have students select which shape they will construct.

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4) After students have constructed several skeletons, ask questions such as the following: • What was easy about constructing the skeletons? What was difficult?

• How are the skeletons like the solid three-dimensional figures? How are the skeletons different from the solids?

• Which figure has the fewest edges? How many edges does it have?

• Which figure has the most edges? How many edges does it have?

• Which figure has edges that are all the same length?

• Which figure has edges that are different lengths?

• Which figure has edges that form a square (triangle, rectangle)?

• What did you learn about three-dimensional figures from this task?

Note: This activity can be adapted for multi-grade levels by increasing the degree of complexity of the skeletons that they model. Students may also combine shapes by taping them together.

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Part Two: Working with Nets Students will describe three-dimensional shapes by the number of edges, faces, and/or vertices as well as types of faces. They will identify and build a three-dimensional shape from two-dimensional representations of that object.

Materials:  Copies of net templates (See Resources for additional net templates)  Scissors  Pencils  Markers  Tape Directions: 1) Begin by defining “net” (a pattern that can be folded to make a three-dimensional figure).

2) Ask students to predict the 3-dimensional figure each of the following nets represents. Record your predictions on the chart provided.

3) Give students the following directions: Before you cut the net out, follow the instructions below for each of the nets:  Mark the congruent faces with a red dot. If there are 2 different sets of congruent faces, like 4 congruent rectangles and 2 congruent squares on one net, mark the second set with blue dots.  Trace in purple along the lines between any pairs of faces you think will be perpendicular when you cut out the net and make the figure.  Lightly color in each pair of faces you think will be parallel when you cut out the net and make the figure. Use a different color for each pair.

4) After you’ve made all the predictions listed above, cut out each net along the heavy outline, fold it on the dotted lines, and tape it together to form a 3-dimensional figure.

5) On your chart, write in the actual 3-dimensional figure each net represents.

Note: This activity can be adapted for older students by selecting more complex nets. (See resources for additional net patterns.)

(Source: www.mathlearningcenter.org/media/Bridges.../B5SUP-C3_Geom3DShapes_1211.pdf)

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Net

Prediction

Actual 3-D Figure

a b a c a d a e a Name _________________________________

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Net Template One

b

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Net Template Two

e

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Activity Eight Escher and the Möbius Strip One of the mathematical objects that fascinated Escher was the Möbius strip, which has only one surface. His woodcut Möbius Strip II (1963) depicts a chain of ants marching forever around/over what at any one place are the two opposite faces of the object—which are seen on inspection to be parts of the strip's single surface. The work refers to Escher’s interest in duality.

Escher described the image this way, “An endless ring-shaped band usually has two distinct surfaces, one inside and one outside. Yet, on this strip, nine red ants crawl after each other and travel the front side as well as the reverse side. Therefore, the strip has only one surface.”

The Möbius strip was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858. The Mobius strip is known for its unusual properties. In essence, this strip is an easily constructed topological (which refers to the study of geometric space) paradox: A surface with only one side and one edge.

M. C. Escher, Möbius Strip II (Red Ants), 1963, Woodcut

A bug crawling along the center line of the loop would go around twice before coming back to its starting point. Cutting along the center line of the loop creates one longer band, not two.

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Part One: Create A Möbius Strip For this activity, student will construct their own Möbius Strip and conduct several experiments with their strip to gain a better understanding of its properties.

The objectives for this activity for grades 6 – 8:    

Students will analyze characteristics and properties of two-and three-dimensional geometric shapes and develop math arguments about geometrical relationships.

Students will build new mathematical knowledge through problem solving.

Students will make and investigate mathematical conjectures.

Students will organize and consolidate their mathematical thinking through communication; students will communicate their mathematical thinking coherently and clearly to peers, teachers and others.

Begin the activity by showing the students a Möbius Strip that you have constructed. Explain that the Möbius Strip is an interesting surface. It looks like any other regular surface and upon close inspection it appears to be a 2-dimensional object. However, the surface becomes more interesting when we try to decide how many sides this surface has. If we take a flat piece of paper, then we can demonstrate that it has exactly 2 sides. We could, for instance, color one side of our paper blue and the other side red, and we would never run into any problem, but this is not true for the Möbius Strip. Demonstrate how if you take a marker and start coloring one side of the Möbius Strip you have in fact colored the entire surface proving that the Möbius Strip is a one-sided surface.

Explain that in real world applications, Möbius Strips have been used as conveyor belts (to make them last longer, since "each side" gets the same amount of wear) and as continuous-loop recording tapes (to double the playing time) and as ribbons for typewriters or computer printer cartridges. In the 1960s, Sandia Laboratories used Möbius Strips in the design of versatile electronic resistors. Free-style skiers have christened one of their acrobatic stunts the Möbius Flip.

Materials:  Long strips of paper (about 11” – 14” long)  Scissors  Tape Directions: Warm-up questions: Make an annulus (loop) by taping two ends of a paper strip together.

How many edges does the annulus have?

How many sides does the annulus have?

Draw a loop down the middle, and cut the annulus along that loop. What do you get?

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1) First, cut out a long, thin strip of paper (about 3cm wide). (To simplify the process, you can simply cut down the edge of a plain sheet of paper to make your strip.)

2) Place the paper horizontally, and label the top left corner "A"; the top right corner "B"; the bottom left corner "C"; and the bottom right corner "D.” A

B

C

D

3) Twist the A-C side a half turn and bring it to the B-D side. Hold the two ends in your hands, give the AC side of the strip a half twist and join it to the B-D side.

4) Match the letters, A to D and B to C and tape the edges together. Once the edges are taped, you have completed the Möbius strip.

Part Two: Experimenting with the Möbius Strip 1) Draw a line along the middle of the strip. Using a pen or pencil, start at any point in the middle of the strip and draw a line all the way around without lifting your pen. Eventually, the pen will end up back at the point you started drawing. You have drawn a line on both sides of the loop - but without lifting your pen or crossing any edge. How did this happen?

Start at a different point in the Möbius strip and see if the same thing happens.

2) Take a highlighter and start coloring the edge of the Möbius strip without lifting the highlighter from the strip. Continue with the marker until you reach the point at which you started. You'll find both edges are colored. This indicates that the Möbius strip has only one edge!

3) Cut the Möbius strip along the central line you drew earlier. With a pair of scissors, poke a hole into the middle of the Möbius strip and cut along the line until you reach the beginning cut. It does not, as you'd expect, fall apart into two separate loops; instead you now have a single, larger one-sided loop.

4) Cut the Möbius strip 1/3 of the way away from the edge. Like you did cutting through the center line, take the scissors and this time cut about 1/3 of the way from the edge of the strip. Continue cutting until you reach the original cut. When you have finished cutting, you should have one small ring and one larger ring that are connected together.

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Part Three: “Double Twisted” Band Möbius Strip Construct a “Double Twisted” Band 1) Repeat the previous steps for making the Möbius strip, but this time give the strip a full twist before taping the ends together. a. Draw a loop down the middle. What do you notice? b. How many edges does the band have? c. How many sides does the band have? d. Now cut the band down the middle. What happens? Describe the object you get.

2) Use your strip to complete the following table:

Number of half twists Number of edges Number of sides 0 1 2 3 4 5

Part Four: Construct the Möbius Cross For this activity you will need three crosses. Take two strips of paper, and connect them in the form of a cross.

1) Take one of the crosses and tape together two opposite "arms" into an untwisted loop. Then tape the other two arms in another untwisted loop. The result will look like a twisted figure eight. Cut both loops down their middles. What do you get after cutting one of the loops? What do you get after cutting the second loop?

2) Take the second cross. Again, tape opposite ends into loops. This time make one plain loop and one Mobius band. What do you think you will end up with after cutting? Cut the loops to find out.

3) Using the third cross, tape opposite ends together to make two loops. This time, twist both loops into Möbius bands. What do you think you will end up with after cutting? Cut the loops to find out. © 2006-2018 Anneke Bart and Bryan Clair http://mathstat.slu.edu/escher/index.php/M%C3%B6bius_Strip_ Exploration

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Activity Nine: Escher and the Impossible Object (Grades 9-12)

M. C. Escher, Waterfall, 1961, Lithograph

When viewing a two-dimensional depiction of a three-dimensional object, the human brain automatically attempts to construct a model of the 3-D object in their mind. Escher took advantage of this phenomenon by creating “impossible” objects: objects which can be translated two-dimensionally but are impossible to construct three-dimensionally. Depending on the angle from which the impossible object is viewed and the way in which the viewer interprets it in their mind, there will be different perceptions of it. It is a form of an optical illusion.

One interesting aspect of Escher’s optical illusions involves the mathematical conundrum of the “impossible” or “Penrose” triangle — an object that appears in Escher’s work, like Waterfall, above.

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In this activity, students will study the art of M. C. Escher, specifically as a predecessor to the Op Art Movement. Students will learn how to use geometric shapes to create an artwork that appears to feature an impossible construction.

Materials:  9” X 12” sheets of white drawing paper  Soft lead pencil  Colored pencils  Drawing pencil and eraser  Markers  Ruler Directions: 1) Begin the activity by showing examples of M. C. Escher’s art works including the following: Other World, 1947, Wood Engraving and Woodcut; Relativity, 1953, Lithograph; Waterfall, 1961, Lithograph; Convex and Concave, 1955, Lithograph; Print Gallery, 1956, Lithograph; Belvedere, 1958, Lithograph; Ascending and Descending, 1960, Lithograph; etc.

2) Divide students into groups of three and have them research the work of M. C. Escher, focusing on his Impossible Objects including the Penrose Triangle. Students should explore Escher’s connection to the Op Art Movement and to the work of Op artists Bridget Riley, Richard Anuszkiewicz, and Victor Vasarely.

Each group will write a summary that compares and contrasts the connection between Escher’s work and the Op Art Movement.

3) Have each group present their research findings and conclusions with the class. Conduct a class discussion on the information presented by the groups.

4) Begin by drawing an equilateral triangle in the center of your white drawing paper. Now, extend the ends of the lines past where they join.

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5) Draw lines from these tips, extending them beyond the corners of the inner triangle. Be sure to keep the lines parallel.

6) Draw in the 'corners.'

7) Draw in the final long lines to connect the corners.

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8) Complete your drawings by making sure that the triangles are shaded with tonal values or colored pencil until the illusion is complete. Students will share completed pictures with class.

9) Create an original design featuring optical illusions, based on M.C. Escher’s work, incorporating the Penrose Triangle and tessellations.

Extension: Students can also do the exercise on drawing an impossible square and incorporate this into their final project.

You can find YouTube videos demonstrating how to draw optical illusions/impossible shapes like the following:  Simple Optical Illusion Shape https://www.youtube.com/watch?time_continue=2&v=nt2kRenWFRo  Impossible Triangle https://www.youtube.com/watch?list=PL2aqxaEANPST2KETB2qsajxDALP1W0WaC&v=d4dPnNE Rqzs  3-D Oval Optical Illusion https://www.youtube.com/watch?list=PL2aqxaEANPST2KETB2qsajxDALP1W0WaC&time_contin ue=12&v=udSxy6BGSMY

Curriculum Links: Art – 1) Identify various principles of design e.g. ways to arrange the elements like scale, variety, balance, contrast, rhythm, harmony, dominance, proportion, pattern/repetition; 2) Make informed links between the use of visual qualities and intentions of artists.

Mathematics – 1) Develop thinking, reasoning, communication, application and metacognitive skills through a mathematical approach to problem solving; 2) See and make linkages among mathematical ideas, between mathematics and other subjects, and between mathematics and the real world.

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How to Draw an Impossible Square

Step 1

Step 4

Step 2

Step 5

Step 3

Step 6

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Part Two: Four Intersecting Triangles Puzzle Materials:  Cardstock paper  Triangle template (or students can make their own)  Clear tape  Cutting mat  Scissors (pointed) or X-Acto knife Directions: Print enough sheets using the template for each student or have students create their own patterns by drawing an equilateral triangle with an equilateral triangle hole with sides half as long as the original triangle. They can use the first one as a pattern for the other three.

1) Use sharp-pointed scissors or an X-Acto knife to cut out the template. If using an X-Acto knife, cut on a cutting mat or an old magazine.

2) Make a cut through each of the triangles on one side. Don't put it at the middle of the side or it will be harder to assemble the model.

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3) Link two of the triangles together and move the triangles so that the midpoint of one side is at the vertex of the triangular hole. The opposite sides should look the same. Once the triangles are in place, use a small piece of tape to hold them together.

4) Now continue to link the triangles together. This is not exceptionally difficult; however, it is more challenging than it looks.

5) The triangles must be linked so that each triangle has another triangle side's midpoint in each of the vertexes of its triangular hole. This will also make it so that the midpoint of the side opposite this vertex is in the vertex of the triangular hole of the opposing triangle. It might help to think you are creating regular hexagrams or stars of David where each triangle is rotated with respect to the other.

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6) In order to link the fourth triangle, you will probably have to do some bending and twisting to get it into the proper position.

7) Tape the triangles back together.

Note: You can make triangles larger or smaller.

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Activity Ten: Escher and Insect Symmetry Early in his career, Escher drew inspiration from nature, making studies of insects, landscapes, and plants such as lichens, all of which he used as details in his artworks. Some of the characteristics found in Escher’s work can be attributed to his meticulous observation of nature and his passion for the geometrical regularity he noticed in the world surrounding him.

M. C. Escher, Grasshopper, 1935, Woodcut

M. C. Escher, Dragon Fly, 1936, Wood Engraving

M.C. Escher, Scarabs, 1935, Wood Engraving

M. C. Escher, Butterflies, 1950, Wood Engraving

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Part One: Insect Symmetry (Grade 2) For this lesson, students will investigate the insect world and complete symmetrical drawings of various insects.

Materials:  Insect Symmetry Worksheets  Pencils  Markers and or crayons  9” x 12” white drawing paper  Examples of M. C. Escher’s insect prints and other insect images, including butterflies Directions: 1) Begin by showing examples of works by M. C. Escher depicting various insects like the ones on the previous page. Also show an image of a butterfly or other insect that clearly illustrates mirror symmetry.

2) Ask the class to share some things they notice about the images. Some guiding questions you could ask are: What are the patterns in the butterfly's wings? How are the wings shaped? What things do you notice in the other insect images? How are they similar or different from the butterfly?

Once students touch on the idea that the wings match in some way, introduce the word "symmetry."

Explain that something has symmetry if it can be split into two mirror-image halves. For example, a butterfly is symmetrical because you can fold a picture of it in half and see that both sides match.

3) Have the students use library and electronic resources to do their own research on M. C. Escher and the insect world.

4) Next, have students identify the body parts of insects (head, thorax, abdomen, legs, wings, antennae), and then compare and contrast insect groups (such as butterflies, beetles, dragonflies, and ants).

5) Demonstrate how to take the sheet of paper and fold in half and trace to get the mirror image of the insects.

6) Instruct students to fold the worksheet and then use their pencils to trace the images seen through the paper. Inform them that they may need to hold the folded paper against a sunny window or a light table so the image on the other half is easy to see.

7) Have students compare and contrast their completed drawings.

8) Now, explain that students will design their own symmetrical insects that incorporate pattern and color.

9) First, fold the white drawing paper in half vertically. Use a pencil to draw one half of the insect along the folded edge (line of symmetry). Be sure the center of the insect body meets the fold. Add detailed patterns to the wings (if present) and body of your insect. Make sure all of your pencil lines are dark.

10) Turn over the folded paper. Students trace the image seen through the paper. Inform them that they may need to hold the folded paper against a sunny window or a light table so the image on the other half is easy to see.

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11) Open the paper. Students add color to the design with crayons or markers. As each area is colored, fill in the same area in the same color on the other half to make the insect symmetrical. Color all areas including the background.

Note: Students could also create their insects using shapes cut from colored construction paper.

Science and Art Extension: Drawing Lifecycle of Insect 1) Choose an insect such as a butterfly, beetle, ladybug, cricket, or another similar insect.

2) Draw its lifecycle and color it.

3) Present your lifecycle to the class.

4) Have a class discussion and compare and contrast how the various insects would look like at their different stages (metamorphosis).

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Insect Symmetry Worksheet 1

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Insect Symmetry Worksheet 2

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Insect Symmetry Worksheet 3

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Part Two: Insect Styrofoam Prints (Grades 4-5) Note: This project can be conducted in a regular classroom setting.

Materials:  Water soluble markers  Pieces of styrofoam trays or plates cut into 3” x 3” squares (Grocery stores will often donate the trays. You can also use the thicker styrofoam plates for this project. You can also send out a request for clean trays to be recycled from home and brought in for the project.)  Ball point pen  Pencil  Sturdy plastic spoons  White drawing paper (regular copy paper will work for this part) for test print  Paper cut into 3” x 3” squares (again regular copy paper will work)  9” x 9” white drawing paper  Ruler  Sponges cut into 3” x 3” squares  Spray bottle or bowl with water  Paper towels  Newspaper Directions: 1) Begin by showing examples of M. C. Escher’s prints depicting various insects, including his tessellations with insects.

2) Review the terms: congruent figures (congruent figures are figures with the same shape and same size), symmetry, mirror image or reflection symmetry, and line of reflection with the students.

3) Conduct a class discussion on how Escher incorporated geometric concepts into his artwork.

4) Have students use library and electronic resources to do their own research on M. C. Escher’s insect prints and tessellations. Also have students research and review insect body parts and their symmetrical properties.

5) Give each student a sheet of 9” x 9” white drawing paper and show them how to draw a 3” x 3” grid on the paper. Tell them to keep their pencil lines light.

6) Give each student a sheet of white paper. Tell them to make at least four sketches of insect designs on the paper. The images should include pattern in the designs.

7) Now provide students with one 3” x 3” square of paper. Instruct them to fold it in half to create a line of symmetry.

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8) Select your best sketch and redraw one side of it in the half square. The folded edge of the paper is the line of symmetry and the center of the image should be drawn along this edge of the paper. The image should touch at least two sides of the paper. You may need to make some adjustments to the image to get it to touch on two sides. The image should include some shapes to create pattern.

3” x 3” Square

Square folded in half with fold to the right.

Image drawn on the folded square with the center of image touching the folded edge.

9) Now, fold the paper over and trace the other half of the image. The image should be symmetrical.

10) Place drawing on top of a 3” x 3” piece of styrofoam that will function as the “printing plate” and use a ballpoint pen or dull pencil to trace over the image. (If you can’t hold the image in place, use tape to secure the paper to the styrofoam.) Apply some pressure as you trace in order for the styrofoam printing plate to be indented.

11) Remove the paper and check the styrofoam “printing plate” to make sure that the image has transferred and all of the lines are indented into the plate.

12) Try at least two test prints first on the white drawing paper or copy paper. Begin by covering the work surface with some newspaper.

13) Take the sponge and dampen it lightly by dipping in the bowl of water or by wetting with the spray bottle and squeezing out most of the water. The sponge should not be dripping wet. Place the damp sponge on the sheet of white test print paper and leave in place while you complete the next steps.

14) Color the printing plate with the markers. You may need to go over the colored area a couple of times to get good even coverage.

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15) Remove the sponge and place the styrofoam printing plate colored side down onto the damp area of the paper. Do not let the plate move once you place it on the paper.

16) Using the back of the plastic spoon, rub (burnish) the styrofoam printing plate by applying hard even pressure. Make sure that you have covered the entire surface of the plate. Carefully lift one corner of the plate to see if the entire image has transferred. If it has, then lift your plate off the paper.

17) You can clean the plate with damp paper towels and try new colors or you can go over the plate with the same colors.

18) Once you are satisfied with your printed image, you are ready for the final part of the project.

19) You are going to print your image onto the 3� x 3� grid paper. The objective is to create a new design similar to the M. C. Escher tessellations by flipping, sliding and or rotating your printing plate.

The finished image may look something this:

20) Start in the center section of your grid. Place the damp sponge in the top center square. Color your plate and then repeat the printing process as you did with the test prints.

21) Now print the middle center square, but, this time, turn the printing plate so the bottom is at the top.

How did the design change? What do you have to do next to get a pattern that connects?

22) Continue printing each square until you have completed all nine squares.

23) What do you think would happen to the design if you placed the printing plate in another direction?

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Vocabulary Angle - The amount of rotation between two lines that meet at a common point. See also vertex.

Array - A rectangular arrangement of objects into rows and columns.

Attribute - A quantitative or qualitative characteristic of an object or a shape (e.g., color, size, thickness).

See also property.

Circle - A two-dimensional shape with a curved side. All points on the side of a circle are equidistant from its center.

Cone - A three-dimensional figure with a circular base and a curved surface that tapers to a common point.

Congruence - The property of being congruent. Two-dimensional shapes or three-dimensional figures are congruent if they have the same size and shape.

Cube - A three-dimensional figure whose six faces are squares that are congruent.

Cylinder - A three-dimensional figure with two parallel and congruent circular faces and a curved surface.

Decagon - A ten-sided polygon.

Diagonal symmetry - Symmetry in which the line of symmetry is diagonal. See also horizontal symmetry and vertical symmetry.

Dodecagon - A twelve-sided polygon.

Edge - The line segment at which two faces of a three-dimensional figure meet.

Exterior angles - The angles through which one would turn at the corners of a shape if walking around the boundary of the shape. The sum of the exterior angles of any polygon is 360° or one whole turn.

Face - A flat surface of a three-dimensional figure.

Flip - See reflection.

Geometric solid - A manipulative in the shape of a three-dimensional figure. Common geometric solids include spheres, cubes, prisms, and pyramids.

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Geometry - The study of mathematics that deals with the spatial relationships, properties, movement, and location of two-dimensional shapes and three-dimensional figures. The name comes from two Greek words meaning earth and measure.

Hendecagon - An eleven-sided polygon.

Heptagon - A seven-sided polygon.

Hexagon - A six-sided polygon.

Horizontal symmetry - Symmetry in which the line of symmetry is horizontal. See also diagonal symmetry and vertical symmetry.

Interior angles – The angles inside the boundary of a shape. (See illustration for exterior angles).

Irregular polygon - A polygon whose side or angle measures are not equal. See also regular polygon.

Line of symmetry - A line that divides a shape into two parts that can be matched by folding the shape in half along this line.

Mathematical model or Model- A representation of a mathematical concept using manipulatives, a diagram or picture, symbols, or real-life contexts or situations.

Net - A pattern that can be folded to make a three-dimensional figure.

Nonagon - A nine-sided polygon.

Non-traditional shape or non-standard shape - A two-dimensional shape that might not be identified easily because its form is different from a prototype (mental image) of the shape. See also traditional shape. Non-traditional triangles:

Octagon - An eight-sided polygon.

Orientation / Direction - The orientation of a shape may change following a rotation or reflection. Its orientation does not change following a translation.

Parallel - Extending in the same direction, remaining the same distance apart. Parallel lines or parallel shapes never meet because they are always the same distance apart.

Parallelogram - A quadrilateral that has opposite sides that are parallel.

Pentagon - A five-sided polygon.

Pentomino - A manipulative consisting of arrangements of five squares that are congruent and joined along their sides. The following diagram illustrates five of the twelve different pentominoes.

Polygon - A closed shape formed by three or more straight sides. Examples of polygons are triangles, quadrilaterals, pentagons, and octagons. See also regular polygon and irregular polygon.

Polyhedron - A three-dimensional figure that has polygons as faces.

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Prism - A three-dimensional figure with two bases that are parallel and congruent. A prism is named by the shape of its bases, for example, rectangular prism, triangular prism.

Property - A characteristic that determines (defines) membership in a class. See also attribute.

Pyramid - A three-dimensional figure with a single base that is a polygon and other faces that are triangles. A pyramid is named by the shape of its base, for example, square-based pyramid, trianglebased pyramid.

Quadrant - One of the four regions formed by the intersection of the x-axis and the y-axis in a coordinate plane.

Quadrilateral - A four-sided polygon, rectangle. A parallelogram with four right angles. Opposite sides are equal in length.

Rectangular prism - A three-dimensional figure with two parallel and congruent rectangular faces. The four other faces are also rectangular.

Reflection (Also called “flip”) - A transformation that turns a shape over an axis. The shape does not change size or shape, but it may change position and orientation.

Regular Division of the Plane - A series of drawings by the Dutch artist M. C. Escher which began in 1936. These images are based on the principle of tessellation, irregular shapes or combinations of shapes that interlock completely to cover a surface or plane.

Regular polygon - A closed shape in which all sides and all angles are equal. See also irregular polygon.

Regular shape – A shape in which all the sides are the same length AND all the angles are the same size.

Relationship - In mathematics, a connection between mathematical concepts, or between a mathematical concept and an idea in another subject or in real life. As students connect ideas they already understand with new experiences and ideas, their understanding of mathematical relationships develops.

Rhombus - A quadrilateral with all sides equal in length.

Right angle - An angle that measures exactly 90 degrees.

Rotation (Also called “turn”) - A transformation that turns a shape around a fixed point. The shape does not change size or shape, but it may change position and orientation.

Shape – (See two-dimensional shape.)

Side - An outer boundary (a straight or curved line) of a two-dimensional shape.

Skeleton - A three-dimensional figure showing only the edges and vertices of the figure.

Slide - See translation.

Spatial sense - An intuitive awareness of one’s surroundings and the objects in them.

Sphere - A three-dimensional figure with a curved surface. All points on the surface of a sphere are equidistant from its center. A sphere looks like a ball.

Square - A quadrilateral that has four right angles and four equal sides.

Square-based pyramid - A three-dimensional figure with a base that is square and four triangular faces.

Standard shape – (See traditional shape.)

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Symmetry - The quality of a two-dimensional shape having two parts that match exactly, either when one half is a mirror-image of the other half (line symmetry), or when one part can take the place of another if the shape is rotated.

Tessellation (or Tiling) is when a surface is covered with a pattern of flat shapes so that there are no overlaps or gaps.  A regular tessellation is a pattern made by repeating a regular polygon. There are only 3 regular tessellations: triangle, square and hexagon.

Triangles 3.3.3.3.3.3

Squares 4.4.4.4

Hexagons 6.6.6

Vertex – A vertex is just a “corner point.” For a regular tessellation, the pattern is identical at each vertex.

6 6

6



6

6

Three hexagons meet at this vertex, and a hexagon has 6 sides.

So this is called a "6.6.6" tessellation.

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A Semi-regular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same.

There are only 8 semi-regular tessellations:

3.3.3.3.6

3.6.3.6

3.3.3.4.4

3.12.12

3.3.4.3.4

4.6.12

3.4.6.

4.8.8

To name a tessellation, go around a vertex and write down how many sides each polygon has, in order ... like "3.12.12"; Always start at the polygon with the least number of sides, so "3.12.12" not "12.3.12."

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Three-dimensional figure - (Also called “figure”) - An object having length, width, and depth. Threedimensional figures include cones, cubes, prisms, cylinders, and so forth. See also two-dimensional shape.

Tiling - The process of using repeated congruent shapes to cover a region completely.

Traditional shape - (Also called “standard shape”) - A two-dimensional shape that can be easily identified as a square, rectangle, triangle, and so on, because its form matches a prototype (mental image) of the shape. Also called a standard shape. See also non-traditional shape.

Transformation - A change in a shape that may result in a different position, orientation, or size.

Transformations include translations, reflections, and rotations.

Translation - (Also called “slide”) - A transformation that moves a shape along a straight line to a new position in the same plane. The shape does not change size, shape, or orientation; it changes only its position.

Trapezoid - A quadrilateral with at least one pair of parallel sides.

Triangle - A three-sided polygon.

Triangle-based pyramid - A three-dimensional figure with a triangular base and three triangular faces.

Triangular prism - A three-dimensional figure with two parallel and congruent triangular faces and three other rectangular faces.

Two-dimensional shape - (Also called “shape”) - A shape having length and width but not depth. Twodimensional shapes include circles, triangles, quadrilaterals, and so forth. See also three-dimensional figure.

Turn - See rotation.

Vertex - The common point of the segments or lines of an angle or of edges of a three-dimensional figure.

Vertical symmetry - Symmetry in which the line of symmetry is vertical. See also horizontal symmetry and diagonal symmetry.

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More Escher Resources Adventures in Perception - https://laughingsquid.com/m-c-escher-documentary/ Dutch filmmaker Han Van Gelder captured the artistic incredible process of M.C. Escher in the 20minute documentary entitled “Adventures in Perception“. The film, part-biography, part artist demonstration was created for The Ministry of Foreign Affairs, Netherlands program “Living Art The Netherlands”. The first half of the film shows a lot of his work accompanied by a dissonant score by Felix Visser. About halfway through, there are shots of Escher at work, with narration offering biographical information sprinkled with quotes taken from Escher’s own statements over the years.

Inspirations http://www.openculture.com/2012/02/inspirations_film_celebrating_the_mathematical_art _of_mc_escher.html Inspirations is a short film by Cristóbal Vila about M.C. Escher (1898-1972), the Dutch artist who explored a wide range of mathematical ideas with his woodcuts, lithographs, and mezzotints.

Math and the Art of M. C. Escher http://mathstat.slu.edu/escher/index.php/Math_and_the_Art_of_M._C._Escher This textbook is intended to support a mathematics course at the level of college algebra, with topics taken from the mathematics implied by Escher's artwork. Materials included for grades K-12.

M. C. Escher - http://www.mcescher.com/ The official website published by the M.C. Escher Foundation and The M.C. Escher Company, B.V.

MC Escher, Images of Mathematics... by Shawn Taylor https://www.youtube.com/watch?v=t-Gcz9FIB4w MC Escher, Images of Mathematics... is a comprehensive overview of Escher and his art. The film runs about eleven minutes.

Tessellation Lesson Plans - https://www.lessonplanet.com/article/elementaryart/tessellation-lesson-plans Various lesson plans on tessellations.

Tessellations.org - http://www.tessellations.org/ This site has information about all aspects of tessellations including: their history, do-it-yourself tessellation lessons, and galleries of examples by school students, guest artists, the webmasters and M. C. Escher.

Tessellation Tutorials - http://mathforum.org/sum95/suzanne/tess.intro.html Tutorials and templates for making tessellations.

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The Mathematical Art of M.C. Escher - http://platonicrealms.com/minitexts/MathematicalArt-Of-M-C-Escher/ Topics include: Tessellations, Polyhedra, The Shape of Space, The Logic of Space, and Self-Reference

The Mathematical Art Of M.C. Escher - https://www.youtube.com/watch?v=Kcc56fRtrKU Overview of M. C. Escher and his tessellations by the BBC about 4 minutes in length.

#25: Relativity by M. C. Escher - http://www.scottmcd.net/artanalysis/?p=548 Article with a detailed analysis of Escher’s Relativity.

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The Magical World of M. C. Escher

Resources s

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Activity Nine Four Intersecting Triangles Puzzle Template

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How to Make Tetrahedron from a Small Envelope Materials: Small triangular flap envelope (3 5/8� x 6 1/2�)

Pencil Ruler Scissors Markers (optional)

Directions: 1. Take the envelope and open it as shown in the illustration.

2. Cut off the flap following the fold on the envelope.

3. Use the pencil and ruler to draw a diagonal line from corner to corner.

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4. Cut out the triangle along the top of the envelope. Be sure to cut along the diagonal lines.

5. Now fold the two triangles along the line. Be sure to crease well. Then back fold and crease again. It is very important to have sharp clean creases.

6. Unfold the envelope and push the two corners into the center. You may have to adjust a little to get the two sides to fold into each other.

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7. You can also draw patterns or color the sides of the tetrahedron.

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Net for a Cube Template

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Net for a Triangular Prism Template

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Net for a Rectangular Prism Template

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Net for a Triangle-Based Pyramid Template

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Net for a Square-Based Pyramid Template

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Net A—Cube Template

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Net B—Rectangular Prism Template

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Net C—Hexagonal Prism Template

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Net D—Triangular Pyramid Template

Net E—Square Pyramid Template

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Net F—Triangular Prism Template

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Pattern Block Shapes – Hexagons

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Pattern Block Shapes – Triangles

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Pattern Block Shapes – Trapezoids

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Pattern Block Shapes – Squares

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Pattern Block Shapes – Large Rhombuses

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Pattern Block Shapes – Small Rhombuses

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Shape and Form Examples

Cube

Cone

Cylinder

Sphere

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Triangular Grid Paper for Activity Four Tessellations

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