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Linear Transformations in Works of Art

 Robert Keeney

Academic Setting

School Environment

Rio Rancho High School in New Mexico has varied demographics.  Like its region, there are students from multi-ethnicities and heritages.   A percentage breakdown yields approximately 55 percent Caucasian, 25 percent Hispanic, 7 percent African American, 5 percent Native American, and the remaining from nearly every other section of the globe. The economic categorization follows a fairly normal distribution with a broad middle to upper middle class base.  Unfortunately the lower end of the income scale is more heavily represented than the upper, but the extremes are generally not severe.

            Rio Rancho, as compared to other parts of the state, is distinct however, in that it has a high ratio of transfer students from all around the nation.  A major employer is Intel, which recruits its talent from numerous regions.  As a consequence, there is a wonderful base of varied life experiences in the student population, adding a unique perspective and system of dynamics to the school environment.  Intel’s presence has also influenced an emphasis on the use of technology.  All staff is highly encouraged to become proficient in its uses as applied to their particular curricula.

            The school district is unique in another way.  It is a charter district, and one of its points of emphasis is its cross-curricular integration approach.  Language arts, literature, history and government are all taught together as one course called humanities, and math and science fall together under the roof of “Scimatics.”    It is this central idea that this unit attempts to exploit to even a further degree.

            To graduate from Rio Rancho High School, each student needs to complete four credits of math, four credits of science, and eight of humanities, in addition to vocational education, physical education, communication, and nine electives.  With the noted exception of humanities, each class and credit is a semester long.  The school runs on a four-by-four block system with ninety-minute classes.  At the start of the sophomore year, students join one of the four academies.  They are Business and Technology, Science, Humanities, and Fine Arts.  The core classes are offered in each academy, though the specialty classes are concentrated in the appropriate academies.  Students can take classes from any academy, whether he or she is a member or not.

            Though the possibilities are numerous, this unit will focus on how linear transformations, specifically translations, reflections, rotations, and dilations play a part in art, both contemporary and past, and from Western and non-Western cultures.  Its intent is to be a viable substitution to a standard geometry text’s unit on graphic transformations, though it may certainly be used as a supplement, or in any context deemed suitable by the user.  It is assumed that the audience is a typical high school geometry class, though modifications are easily applied to make it appropriate for special needs, remedial, and enriched students.

Goals and Objectives

As may be the case with any subject, maturation and independence of the learning process is the ultimate goal of this unit.   The approach to achieve this is to allow students to investigate the properties of linear transformations by associating them within an artistic and cultural context, and then using the knowledge gained to express their understandings in a creative forum.  Though the integrated environment is emphasized, the unit still strives to meet the essential concepts of linear transformations, and their distinctions from non-linear transformations.  Both mechanical and technological mediums are employed to vary the investigations.

            The New Mexico state standards met in this unit are compiled in Table 1, included in the Documentation section towards the end of this unit.  They are referred to individually by number in each of the included lesson plans in the Implementation section.

Context and Background

Rationale

If one were to look at the evolution of learning throughout the history of mankind, he would see a stark transformation from its early vestiges to its current Western state.  Envision a Paleolithic boy carefully watching his uncle chip stones to fashion a scraping tool, or a young daughter mimicking the weaving patterns her mother used to produce a basket.  Contrast that with a modern-day class of 30-plus students who are likely only acquainted because they happen to be in the same classroom.  The curriculum they are studying has been dissected and refined to cover very specific objectives determined by organizations and institutions sometimes thousands of miles away.

            Dimension and depth is lost when something is viewed exclusively from a single vantage point.  This phenomenon is not limited to visual experiences, but applies equally well to cognitive processes.  Unfortunately, this is all too often the case in a high school classroom.  With the demands from state curricula and school syllabi, classroom instruction often evolves into an anesthetized package of subject matter devoid of variation, and ultimately of interest and worldly relevance.  But life does not come in those same individual packages.

            Mathematics is an integral part of everyday life and thought, whether one is conscious of the process or not.  An accepted approach in teaching the subject is to help students become more aware of its relevance and depth in their lives.  The intent is to put together a unit where mathematics is tied to the art indicative of cultures around the world.  The thought process concerning patterns and their interrelations are given a visual domain in this way.   Additionally, it is very easy to incorporate historical and social components that bring the topics together as a whole, which is how any subject truly exists in our world. 

            This is not intended to be an attack on Western education’s general approach, because one only needs to briefly consider the accomplishments of its society to realize its successes.  However, there are definite educational benefits to the personal, one-on-one, direct, and relevant instruction that occurred millennia ago.  Though to propose that education attempt to return to that state is unrealistic, some attributes could be incorporated to increase the effectiveness in the modern classroom.  One that is no secret is to make the material more relevant to the experiences and perspectives of the students.  The purpose of this unit is to attempt such a feat, though not so much by using examples and scenarios common to the students’ daily lives, but by expanding the setting in which mathematical instruction is presented. 

            At first glance, the right brain realm of the artist may seem contradictory to the rigors and regimen of the mathematician’s world, but there are crossovers.   The recent discoveries and proliferations of fractal geometry have produced highly aesthetic pictures from apparent chaos.  Geometric designs have been pervasive in art since ancient times.   The more contemporary Dutch artist, Maurits Cornelis Escher focused his work on the study of many mathematical concepts presentable in a visual and artistic context. 

Background 

The word geometry stems from the Greek root words meaning “earth” and “measure.”  It is generally accepted that geometry has its origins in ancient Egypt sometime before 450 BC.  The yearly flooding and redistributing of land by the Nile created the need for the ability to accurately measure parcels of land for taxation and revenue purposes.  This became the domain of  “rope stretchers” who performed the majority of their measurements with ropes knotted at regular intervals. 

            Numerous examples of calculations, formulas, and theorems have survived from those dates, although many were not accompanied by derivations or rigorous proofs.  They were simply recorded as fact.  It is possible that many of the relationships were developed by trial and error (Burton 53).  However, the results were accurate enough to effectively serve their design.  It was possibly not until the Greeks tackled more elusive problems when the more formal system of proofs was developed. 

            As the knowledge of the subject increased, more opportunities arose to investigate questions of a less than utilitarian nature.  One such subject was that of symmetry and repetition, which often produces a pleasing visual effect.  This was certainly known to ancient artisans long before its study was formalized, but it was geometrical inquiry that unlocked its mathematical nature. 

            Both symmetry and repetition of an object can be accomplished by transformations.  A transformation is simply a process where each point on an object or shape is moved in a formulaic way.   The original figure is called the pre-image, and the resultant figure is called the image.  Some transformations will cause a distortion of the pre-image.  Although the image can take on fascinating shapes, the mathematics describing the change can often be complicated and beyond the reach of the average high school geometry student.  Such a change is called a non-linear transformation, and will only be briefly discussed in the unit.  Linear transformations are much simpler to describe, and will be the focus of the unit. 

            The image of a linear transformation “looks” like the pre-image.   Specifically, all of the vertex and arc angles in the image have the same measure as the corresponding angles in the pre-image.   An isomorphism is an even stricter linear transformation where additionally the measures of segment and arc lengths are identical in both figures.  Given this definition, then the image and pre-image of an isomorphism are congruent.  This can be thought of visually as taking an overhead transparency and laying it on a table.  If the transparency were moved by sliding, flipping, or spinning, as long as it ultimately ended up back on the table, it would have undergone an isomorphic transformation.  If the figure were to be enlarged or reduced on a photocopier, then it would have undergone a non-isomorphic, linear transformation. 

            There are basically four linear transformations that are studied in most high school geometry texts.  They are translations, reflections, rotations, and dilations.  Any time a figure is moved on a plane, as long as all of the corresponding angles are equal, then it can be described by one, or a combination of more than one of the above-mentioned transformations.  If more than one transformation is employed, it is called a composite transformation. 

            Probably the simplest transformation, and usually the first type introduced in the subject, is a translation.  This transformation is also referred to as a slide or glide.  In everyday terms, this can be thought of as taking the shape and simply sliding it in a straight-line direction, taking care not to allow any spinning to occur.  When finished, any segments in the image will be parallel to their corresponding segments in the pre-image. 

            In more mathematical terms, a translation is defined similarly.  The image is formed by moving each point in the pre-image by a fixed amount given as a distance, and in a fixed direction usually given as an angle where right or east are zero and moving counterclockwise is the positive direction.   With a little experimentation, it would not be too difficult to see how the result would be what is described in the paragraph above. 

            Often the reflection is the next transformation introduced.  Just as there are countless examples of translations in everyday life, so too do reflections occur everywhere.  This translation is often very intuitive because its mathematical process is the same as its everyday notion.  A reflection is commonly referred to as a flip over a line.  If a shape were printed on a transparency and then flipped over a line so that the line were half way between the image and pre-image, then it would have undergone a reflection. 

            More explicitly, a reflection occurs when each point on an object is placed on the opposite side of the reflection line such that a line drawn through the image and pre-image points is perpendicular to the reflection line, and each point is equidistant to the reflection line.  Carrying out this process on graph paper would produce a “mirror” figure where corresponding segments are not necessarily parallel.  However, non-parallel segments could be extended and would always intersect on the reflection line with the reflection line being the angle bisector. 

            Next on the list of transformations is the rotation.  Often referred to as a spin, it is necessarily done around a given point.  A practical way to view a rotation is to think of it as drawing the figure on a piece of paper and then taking a pin and sticking it somewhere into the paper.  Any spin on the paper would create a rotation.  The pin could be removed and stuck in another place, including inside or on the border of the original figure, to produce a different group of rotations. 

            More precisely, a rotation is generated when a given point of the pre-image is moved by a set angle, yet remains the same distance from the point of rotation.   Since all of the pre-image and image points are moved the same number of degrees, and each remain their respective distances from the center, no distortion of the figure takes place. 

            That is not the case for the remaining transformation however.  When a shape undergoes dilation, it does not result in an exact copy simply moved in a certain manner.  However, it is still very recognizable, because it has just been shrunk or enlarged.   The common analogy of a photocopier is very intuitive.  If the figure is copied, but reduced or enlarged by a given percentage, then a dilation has taken place. 

            A dilation is often the most difficult of linear transformations to explore mathematically, because its precise definition adds an element of complexity.  A dilation cannot occur unless a point is specified.  Strictly speaking; a figure is dilated when each of its points is moved closer to or farther from the point of dilation by a given factor or percentage.   Another way to view this is to draw a ray from the point of dilation through a point on the pre-image.  Multiply the distance between the two points by the dilation factor, and move that point along the ray to the new distance from the dilation point.  When this is accomplished for all of the points on the pre-image, the image will be larger or smaller than the pre-image by the dilation factor.  If the factor is less than one, it will be smaller and nearer to the dilation point, and conversely, if it is larger than one, it will be larger and farther away.  Because a dilation seems to have two elements of change, motion to or from the point of dilation and a change in size, it often is less intuitive than the previous three transformations. 

            In the stricter definitions of the above transformations, the change is described for individual points.  However, since the original relationships between the points are preserved in the process, it is often only necessary to map a few points.  For example, if a triangle were translated, only the three vertices would need to be moved.  Then the points could be connected with segments and the complete translation would be formed. 

            In common sense terms, linear transformations are just an exact way to describe how a shape might be moved around on a surface and resized.  However, it is not possible to move and resize the figure every place on a plane, with only a single transformation.  But if several or all are employed, then every possible size, location, and orientation can be achieved.  Additionally, it is often the case that different combinations or composites can achieve the same result. 

            Each of the three isomorphisms results in a symmetry named after the transformation.  Translation symmetry might be seen in an assembly line of cars.  Reflective or line symmetry appears during a sunset over still water.  This could be described further as horizontal symmetry.  And a stunning example of rotational or point symmetry is found on the petals of many flowers. 

            Examples of symmetry and thus transformations are ubiquitous, including in human endeavors of aesthetics.  A true transformation only occurs in a plane, which might limit this review to such mediums as canvases, tapestries, and walls.  However, it seems to be a natural cognitive process to recognize what occurs as flat surfaces are curved into three-dimensional objects.  Therefore this unit will also look at many of the intricate designs found on pottery. 

            Central to human existence has been shelter.  Often the buildings themselves exhibit reflective and even rotational symmetry, but this unit will focus on wall tilings.  Any regular or irregular shape, design, or picture can be reflected or rotated to make a regular pattern.  If such a pattern is imprinted on a square or rectangular tile and then placed in a band across a wall, it is called a frieze. 

            Frieze groups are linear patterns that are found in architecture all around the globe.  They are common in Roman and Muslim buildings, particularly as borders along walls near the ceiling.   (See Figure 1.)  They can be thought of as a repeating design or picture drawn out in a line, often as far as the structure allows.  To be a mathematical frieze, it must contain a repeating translation.  It can, and often does, also contain rotations and reflections.  Given the large number of combinations conceivable, it would seem that there would be many different patterning possibilities.  However, there are actually only seven different types. 

            As is often the case with composite transformations, different combinations end up with the same result.  Outlined in Table 2 are the seven examples possible, with a brief description of their symmetry.   The names they have been given were coined by the mathematician John Conway, and are easily associated  with footprints. 

            Given enough surface area, a frieze pattern could theoretically go on forever in one dimension, and is thus called an infinite group.  Another in this category, but one which can extend forever in two dimensions, is called a wallpaper group.   The number of possible transformations creates a total of 17 distinct orientations.  Their individual discussions are beyond the scope of this unit, but their examples abound in ancient and modern architecture.  One spectacular source where all 17 can be found is the Alhambra in Spain.  Often considered the pinnacle of Moorish architecture, it has inspired the development of similar ornamentation in numerous cultures, including English wallpaper, carpets, and furniture design (Sheridan). 

            Maybe one of the best-known artists in relatively modern times whose work reflected such ornamental concepts is M.C. Escher.  During 1922, and again in 1936, he traveled to Spain and was inspired by what he saw in the Alhambra.  In his lifetime he created many works on the subject of “space-filling.”  These works came to be known as examples of tessellations, a word derived from the Greek word tesseres, which means “four,” and probably stemmed from the fact that the first tessellations were square tiles (Alejandre). 

            A tessellation occurs when a shape is repeated (translated or reflected) in a plane in both dimensions such that no spaces are left in between.  There are only three regular (equal length sides and equal measure angles) shapes that will tessellate by themselves.  They are a square, an equilateral triangle, and a regular hexagon.  However, modifications of these three, and combinations of other shapes allow for infinite possibilities.  This will be further explored in the lesson plans. 

            Tessellations found in tilings and other traditional works of art are normally strictly geometric in design.  Escher pioneered work where the tessellations were figures of real-life objects.  One of his more intricate and well-known is titled, Horsemen.   It features a horse and rider in such a way that ranks of the mounts drawn nose to tail create “negative” spaces which are the identical outlines to the original characters.  Though it contains many features and details, it is actually just a modification of a tessellating quadrilateral. 

            Escher goes further to combine more than one shape and still fill the plane.   In his works, Sky and Water I and Sky and Water II the white sky between the birds metamorphoses into fish swimming in a dark sea.  Though Escher has taken the artistic liberty of diverging from true tessellations in his final works, his study drawings are mathematically pure.  Again both the fish and birds can ultimately be whittled down into simple quadrilaterals, which tessellate on their own. 

            The notion of altering a fundamental geometric shape to achieve more intricate patterns can be seen by comparing figures three and four.  Figure 3 shows how an equilateral diamond (all four sides the same length) can be placed so that no gaps are formed.  If their tops are curved in and right angles are added, the same deformations also occur in the bottoms of the diamonds above.  With a little study, it should not be too difficult to see how this becomes the repeating shapes in the panel in Figure 4.  Escher discovered this property and exploited it in such a refined way as to create tessellations of amazing complexity.  His 1922 woodcut titled Eight Heads is just such an example. 

            Although the level of mathematical analysis certainly must have varied among ancient artists, it is clear that they were adept at creating symmetry.  Another medium where it has been found for countless centuries is in weaving.  This is for good reason, since he warp and weft (vertical and horizontal fibers) create a natural grid in which to create the designs. 

            The machi in Figure 5 displays several dimensions of symmetry.  If the entire colored pattern is considered as a single figure, then it displays both horizontal and vertical reflective symmetry.  Additionally, each row of patterns, except for the center, is a frieze.   Meticulous effort is maintained to keep exacting symmetry.  The process is so time consuming in fact, that the production of a machi can be used as repayment to a family for a wrongful death. 

            If the Navajo wedding basket in Figure 6 is analyzed, one might outline many elements of rotational and reflective symmetry.  However, if it is scrutinized closely, the 5-point star in the center nullifies rotational symmetry, and the longer point being offset by a few degrees also negates reflective symmetry.  In fact, because the basket is actually made by wrapping fibers around a thicker coil spiraled outward, the foundation for symmetry found in the warp and weft of flat weaving is not present.   Yet in actuality, the non-symmetry is intentional, and is present as a contrived flaw in most all of Navajo traditional works of art.  It is to assert that the artist, though possessing great skill, is not perfect; a status reserved only for the gods. 

            Symmetry seems to have an innate place in the human mind.  Besides the fact that it is pervasive in human creations since the dawn of civilization, there is also a tendency for the brain to look for and even create symmetry where it is merely suggested.  A far more complicated element of symmetry is produced when the third dimension is added.  Curved surfaces bend and distort parallel and perpendicular lines, change angles, and expand or contract distances.  All of these are crucial elements of achieving the symmetry discussed so far.   The mathematical dissections of three-dimensional surfaces take on a non-Euclidain nature and are usually topics for advanced studies.  But somehow the human mind tends to overlook these irregularities and fit them into the previous categories.   Therefore a review of the symmetry on pottery can be considered appropriate, and adds an interesting array of examples. 

            One such example can be seen on modern day pottery from the pueblo of Acoma.  This ancient city has seen a revival within the last century of its traditional fine-lined black on white pottery.  Although there are numerous examples of pictorial creations that display little or no true symmetry, a large percentage do, and they achieve wonderful geometry. 

            The vases in figures seven and eight actually contains none of the symmetry discussed so far because the transformations that produced them are defined for two-dimensional surfaces.  But if we consider the projections of these images onto a plane, such as is produced with a photograph, with a little fudging we can continue the discussion. 

            Consider one of the crossed-diamond figures inscribed in an oval.  If one from the center of Figure 7 is compared with one from the side, the outer is clearly distorted.  Yet if it is viewed from the top as in Figure 8, the rotational symmetry achieved by the artist is obvious.  

            By looking at another element, the white crosses inside the diamonds, another system of rotational symmetry is seen.  Each branch is carefully lined up in such a way as to produce the illusion of a continuous spiral from the center, one set rotating clockwise, and the other counter clockwise.  

            Dorithy Torivio’s vase displays an excellent example a less-often-seen element of transformations.  Each of her oval figures is dilated.  As is readily apparent, they are not only reduced in size but also they are brought closer towards the center.  Both are necessary for a true mathematical dilation.  Although difficult to obtain from just photographs, it would be possible to take measurements from the actual vase and compute the dilation factors.  It is the author’s opinion that they would very closely match those of a true dilation. 

Implementation 

The following is a set of lesson plans that might be used to present the previous background material.   They are given merely as a collection of ideas to be modified in any way the teacher may see fit.  They do rely on the use of Geometer’s Sketchpad; a dynamic graphing program published by Key Curriculum Press, as well as other electronics presentation means.  If none are available, the lesson plans can still be implemented by doing the activities by hand.  The differences will be less accuracy in the constructions and a restriction on their complexity.  However, students will still be just as able to express their creativity in a mathematical setting, and they may become more intimate with the transformations’ properties. 

Lesson 1 – Introduction to Transformations  (1-3 Days) 

The materials needed are cardstock, fine grid graph paper, rulers, protractors, and dice.  An overhead may be helpful for explaining procedures and giving examples.  When finished, the students should have a clear understanding of the more precise definitions of the four linear transformations and how to produce them on a grid system.  Additionally, they should be aware of the “shape-preserving” properties of linear transformations and how to recognize them.  Assessment can be done by evaluating the finished products using the attached rubric for this lesson.  Additionally, if a journal is kept, a student’s explanation of the goals just mentioned can be recorded.   This lesson addresses state standards 1.E.1, 2.E.1, 8.B, 8.F.1, 8.G.1, and 8.G.2. 

            Begin the lesson by modeling isometries with a shape cut or torn out of cardstock.  Slide, flip, and spin it on the blackboard, defining each as it is accomplished.  A dilation is more difficult to show, but present its notion by comparing enlargements and reductions from a copy machine.

              Then with graph paper, construct each transformation in the following ways.  Have the students draw a scalene triangle on their graph paper.   Roll a die twice.  The first number will be the rise of the transformation and the second its run.  Translate each vertex of the triangle in the prescribed manner and connect them to form the triangle.   This can be repeated for other polygons and different translation directions if desired. 

            For a reflection, have the students draw another scalene triangle.  Somewhere on the same half of the paper as the triangle, have them draw a line.  For each vertex have them draw a line perpendicular to the reflection line through the vertex.   Measure the distance of the vertex to the reflection line and then reproduce the point on the opposite side, the same distance away.   Have them connect the vertices to complete the reflection. 

            Draw another scalene triangle for a rotation.  Mark a point of rotation clearly on the paper.  It can be on or inside the triangle if desired.  Using the numbers from a dice roll (or other random or arbitrary method), determine an angle.  From the point of rotation to a vertex, draw a ray.  Draw another ray to form the angle determined previously.   Mark a point on the new ray that is the same distance to the angle vertex as is the triangle vertex.  Do this for all three vertices and connect them to form the rotation. 

            Determine a point of dilation near the paper’s center, and then draw a scalene triangle.  Randomly select a dilation factor between 0.2 and 2.  Draw a ray from the dilation point through a vertex.  Multiply the distance between the two points by the dilation factor and make a new point that distance from the dilation point.  Repeat for the other two vertices and complete the image triangle. 

            Other polygons can be transformed as well.  With the random generation of transformation variables, an array of different results can be produced by the class.  Display them on a wall by transformation types and have students make generalized comments. 

            Modifications for enriched students might include higher order polygons, the inclusion of arcs and other curves, and composite transformations.  Special needs students might benefit by being given set transformation parameters, instead of those randomly generated. 

Lesson 2 – Investigation of M.C. Escher  (2-5 Days) 

The resources needed are access to the Internet, library, or other research avenue, and a computer with a slide show program such as Microsoft Power Point or Claris Works.  When finished, the students should have an elementary understanding of how Escher used transformations to produce some of his tessellation pieces.   Additionally, they should know how to use the basic features of the chosen slide show program.  Assessment can be done by evaluating the finished products using the attached rubric for this lesson.  Additionally, if a journal is kept, a student’s explanation of the first goal just mentioned can be recorded.  This lesson addresses state standards 2.E.1, 8.F., and 8.F.1.

              If possible, a good way to introduce this lesson is for the teacher to present some of the works of Escher in a slide show presentation of his or her own.  This serves the purposes of exposing the students to some of the concepts of symmetry in Escher’s works, but also of giving an example of a finished product expected for the lesson.  The requirements are for each student, or designated groups, to make their own slide shows, highlighting several works and explaining the transformations involved.  If the slide shows are to be presented to the class, it might be advisable to keep the number of slides to around three, to keep redundancy to a minimum and to allow for a timely presentation schedule. 

            A suggestion for enriched students is for them to critique non-linear works such as Fishes and Scales or the Circle Limit series, or three-dimensional works like Depth or Cubic Space Division, and compare and contrast those to two-dimensional linear transformations. For special needs students, an analysis of one of Escher’s symmetry drawings, such as what he did for Swans might be appropriate.   They include only the tessellating figures without the artistic interpretations and orientations added to the final works. 

Lesson 3 – Creation of Tessellations  (1-2 Days) 

The main resource needed is access to computers downloaded with Geometer’s Sketchpad.   If that is not available, this lesson can be done mechanically by having the students make tessellation templates from cardstock.   When finished, the students should have a clear understanding of many of the properties of tessellations, but specifically how a simple square or other tessellating polygon can be modified to produce more complex shapes, and the particular transformations used to produce the symmetry achieved.  They should also be aware of how to perform basic manipulations of geometric figures in Geometer’s Sketchpad.  Assessment can be done by evaluating the finished products using the attached rubric for this lesson.  Additionally, if a journal is kept, a student’s explanation of the goals just mentioned can be recorded.   This lesson addresses state standards 1.E.1, 2.D.2, 8.B, and 8.F.1. 

            Familiarity of Geometer’s Sketchpad by the instructor is assumed.  If that is not the case, Key Curriculum Press has tutorials on their web site, and the instruction manual that comes with the program is very informative.  The entire lesson will be done on the computer.  Begin by having the students make a five cm square.  It needs to be fixed (non-dynamic), so start with a point, translate it five cm at zero degrees.  Connect the two points using the segment button with no more than seven segments.  More will not work with some versions.  Then translate that entire figure five cm at negative 90º.   Notice that moving one point in the top section causes the same displacement of the corresponding point in the bottom.  Now connect the left two points using seven more segments, and translate just the segments and points in between (NOT the corner points) five cm at zero degrees.  Carefully select each point in the figure consecutively in a clockwise manner.  Then construct a polygon interior.  Select just the interior and translate it five cm at zero degrees.  Change its color to distinguish it from the pre-image.  Continue this process, including alternating color changes in both vertical and horizontal directions until the student is satisfied with the results.  Six by ten is usually sufficient.  Larger is not a problem because any sized figure can be rescaled and printed on a single page.  The students can now go back and drag points and segments on the original figure to produce changing tessellating shapes.  When the desired affect is achieved, hide the points and segments and print the tessellations.  The results can be compiled on a wall, and, especially if a colored printer is available, the effect is very pleasing.

              Requiring the above lesson using a regular triangle or hexagon may challenge gifted students.  They may also be asked to produce a familiar figure and add details to accentuate it, or even research an ornamental tessellation from architecture, textiles, or pottery and reproduce it as accurately as possible.  Depending on the level, special needs students may be given fewer segments to manipulate, or any or all of the construction can be done in advance by the instructor and copied to the appropriate terminals.  Then the students can manipulate the points and see the desired affects. 

Lesson 4 – Production of a Brochure  (2-5 Days) 

The resources needed are access to the Internet, library, or other research avenue, and a computer with a publication program such as Microsoft Publisher or Adobe PageMaker.  When finished, the students should be familiar with a work of art or collection that displays a number of degrees of symmetry.  They should be able to distinguish between single and composite transformations and their effects as related to their subjects.  Additionally, they should be familiar with a few aspects of the culture from where the work came.  Assessment can be done by evaluating the finished products using the attached rubric for this lesson.  Additionally, if a journal is kept, a student’s explanation of the goals just mentioned can be recorded.  This lesson addresses state standards 8.F.1, and 8.G.1. 

            The aim of this lesson is for students to produce a brochure or flyer as though they were travel, museum, or sales agents attempting to entice mathematicians to view or purchase a particular work or collection of art.  After the instructor reviews properties of composite transformations, the students need to research different possibilities for their topics, and then compile an information base as a reference to complete the final product.  The brochure needs to include specific examples of the mathematics involved as well as graphics to support the statements made.  The final copies can be displayed on a wall or presented to the class. 

           Gifted students might be required to delve deeper into the culture by having to present the brochure from the perspective of the artist who produced the work.  Or they might have to role-play the part of the agent and convince a skeptic of the merits of the piece they represented.  Special needs modifications might include a list of works and resources from which to choose, or abbreviated requirements on graphics, explanations, and cultural references.  

Lesson 5 – Using Transformations to Create Art (2-5 Days) 

The main resource needed is access to computers downloaded with Geometer’s Sketchpad.   If that is not available, this lesson can be done mechanically by having the students make tessellation templates from cardstock, though they will be much less complex.  When finished, the students should have a clear understanding of many of the properties of tessellations, but specifically what visual and aesthetic effects the particular transformations used to produce the symmetry achieved.  They should begin to become more broadly aware of the symmetry present in the world around them.  They should also be more comfortable with how to perform further manipulations of geometric figures in Geometer’s Sketchpad.  Assessment can be done by evaluating the finished products using the attached rubric for this lesson.  Additionally, if a journal is kept, a student’s explanation of the goals just mentioned can be recorded.  This lesson addresses state standards 1.E.1, 8.B, 8.F.1, and 8.G.1. 

            Using Geometer’s Sketchpad, have students produce a “foundation” polygon.  It should be relatively complex, and it is preferable that it not be symmetrical.  Create a number of diagonals and intersecting segments so that a polychrome figure can be achieved.   Have the students be creative.  Save this figure as Foundation Polygon.  With a new copy, draw a segment independent of the foundation polygon and mark it as the translation vector.  Translate the foundation polygon in a systematic way to cover the page.  Drag an endpoint of the translation vector until a desired pattern is achieved.  Save this as Translation.  Now, with a fresh copy of the foundation polygon, create a point of rotation and rotate the polygon seven times around the point at 45º intervals.  Drag the point of rotation to vary the effects and decide on a final form.  Save it as Rotation.  With another fresh copy of the foundation polygon, draw seven unique segments.   Choose one as a line of reflection and reflect the polygon over it.  Then reflect that image over the next segment and continue until it has been reflected over all the lines.   By dragging the reflection segments and their endpoints, endless combinations are generated.  This set of transformations has a definite element of randomness that should be exploited, however the symmetry will still be produced.  Save this as Reflection.  Create a dilation point on a fresh copy of the foundation polygon.  Choose a dilation factor less than one, and dilate the polygon seven times.  Drag the dilation point and notice the results.  Save the desired form as Dilation.  Now set the students free to create a final composite transformation.  Have them experiment until they have achieved their “ultimate” work of art.  Save this as Composite. 

            There are numerous ways that the final products can be presented.  All of the saved works can be printed and displayed on a hallway wall.   Each student can publish his or her own book with explanations and descriptions for each transformation.  Or the entire collection from the class can be compiled into a single book.  If the class or school has a web site, the transformations can be saved as picture files and each student or the class can publish their work on the Web. 

            This project is very open ended and because of the large number of variables within the control of the teachers and students, countless modification can be easily made.  For example, the numbers of rotations or reflections given above were merely suggestions for starting points.  The complexity of the foundation figure can be varied greatly.  The specifics of the composite transformations can also be changed in a vast number of ways.  Simplification or complication for appropriate students is readily available to each instructor. 

Documentation

Supplemental Resources

(Note:   All figures were photographed, or generated by the author using Geometer’s Sketchpad, Adobe Photo Deluxe, or Microsoft Word.)

Standard	Description
1.E.1.	Use appropriate tools (e.g., manipulatives, calculators, and computers) to demonstrate mathematical properties and relationships from numeric, algebraic, and geometric perspectives.
2.D.2	Learn to use computers and graphing calculators to simulate situations, analyze data, and solve complicated problems in an effective way.
2.E.1	Demonstrate that translating a real-life problem to a mathematical equation results in an easier and organized method of solution.
8.B	Deduce properties of figures using transformations and using coordinates.
8.F	Identify congruent and similar figures using transformations.
8.F.1	Understand the effect that translating, reflecting, rotating, or dilating has on a figure and use this information to identify congruencies or similarities between figures.
8.G	Analyze properties of Euclidean transformations and relate transformations to vectors.
8.G.1	Understand the effect that translating, reflecting, rotating, or dilating has on a figure and use this information to identify the results of transforming vectors.
8.G.2	Use vectors as tools to transform figures into congruent or similar figures.

(New Mexico state standards applicable to unit.)                   Table 1.

Translation only.		Hop
Vertical reflection.		Sidle
Horizontal reflection		Jump
Horizontal and vertical reflections.		Spinning jump
Rotation (180º).		Spinning hop
Horizontal glide reflection.  (Foot reflected horizontally then translated)		Step
Vertical reflection then 180º rotation.		Spinning sidle

 (Seven transformation possibilities in frieze patterns.)                                       Table 2.

 

CATEGORY	4	3	2	1
Neatness and Attractiveness	Exceptionally neat, and attractive. A ruler and graph paper are used.	Relatively neat and attractive. A ruler and graph paper are used.	Points and lines are somewhat neat but not enough to demonstrate the exactness of the transformations.	Appears messy. Lines are visibly crooked.
Accuracy of Plots	All points and shapes are plotted very precisely and are easy to see.	All points and shapes are plotted correctly and are easy to see.	Most points and shapes are plotted fairly accurately.	Most points are not plotted correctly OR extra points and segments were included.
Labeling	Has a clear, neat label for each figure, image and pre-image and describes the transformation correctly.	Has label for most figures, image and pre-image and describes the transformation correctly.	Has a label for most figures, but might not designate the image and pre-image, or incorrectly describes the transformations.	Not labeled, very incomplete, or over 50% incorrect.

(Rubric for Lesson 1.)                                                                                          Table 3.

 

CATEGORY	4	3	2	1
Content	Covers topic in-depth with details and examples. Subject knowledge is excellent.	Includes essential knowledge about the topic. Subject knowledge appears to be good.	Includes some essential information about the topic but there are 1-2 factual errors.	Content is minimal OR there are several factual errors.
Mechanics	No misspellings or grammatical errors.	Three or fewer misspellings and/or mechanical errors.	Four misspellings and/or grammatical errors.	More than 4 errors in spelling or grammar.
Requirements	All requirements are met and exceeded.	All requirements are met.	One requirement was not completely met.	More than one requirement was not completely met.
Presentation	Well-rehearsed with smooth delivery that holds audience attention.	Rehearsed with fairly smooth delivery that holds audience attention most of the time.	Delivery not smooth, but able to maintain interest of the audience most of the time.	Delivery not smooth and audience attention often lost.
Attractiveness	Makes excellent use of font, color, graphics, effects, etc. to enhance to presentation.	Makes good use of font, color, graphics, effects, etc. to enhance to presentation.	Makes use of font, color, graphics, effects, etc. but occasionally these detract from the presentation content.	Use of font, color, graphics, effects etc. but these often distract from the presentation content.
Sources	Source information collected for all graphics, facts and quotes. All documented in desired format.	Source information collected for all graphics, facts and quotes. Most documented in desired format.	Source information collected for graphics, facts and quotes, but not documented in desired format.	Very little or no source information was collected.

(Rubric for Lesson 2.)                                                                                          Table 4.

 

CATEGORY	4	3	2	1
Content	Tessellations are exact with no gaps or overlap. Coloration illustrates the pattern.	Tessellations are exact with no gaps or overlap. Coloration does not completely illustrate the pattern.	Tessellations are not exact with some gaps or overlap. Coloration does not completely illustrate the pattern.	The notion of tessellations is not evident.
Originality	Product shows a large amount of thought. Ideas are creative and inventive, within the requirements stipulated.	Product shows required understanding. Ideas are within the requirements stipulated.	Product shows some understanding. Some ideas are outside of the requirements stipulated.	Product shows little or no understanding. Few ideas follow the requirements stipulated.
Requirements	All requirements are met and exceeded.	All requirements are met.	One requirement was not completely met.	More than one requirement was not completely met.
Attractiveness	Makes excellent use of font, color, graphics, effects, etc. to enhance the presentation.	Makes good use of font, color, graphics, effects, etc. to enhance the presentation.	Makes use of font, color, graphics, effects, etc. but occasionally these detract from the presentation content.	Use of font, color, graphics, effects etc. but these often distract from the presentation content.

(Rubric for Lesson 3.)                                                                                          Table 5.

 

CATEGORY	4	3	2	1
Writing - Organization	Each section in the brochure has a clear beginning, middle, and end.	Almost all sections of the brochure have a clear beginning, middle, and end.	Most sections of the brochure have a clear beginning, middle, and end.	Less than half of the sections of the brochure have a clear beginning, middle, and end.