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- Denise Wood

**Loe Fejis & Christoph Bartneck **

**Published Online: ****Dec 15, 2009**

**Abstract | References | Full Text: HTML, PDF (464 KB)**

**Abstract**

** **

**We propose a new method of teaching the principles of geometry to design students. The students focus on a field of design in which geometry is the design: tessellation. We review different approaches to geometry and the field of tessellation before we discuss the setup of the course. Instead of employing 2D drawing tools, such as Adobe Illustrator, the students define their tessellation in mathematical formulas, using the Mathematica software. This procedure enables them to understand the mathematical principles on which graphical tools, such as Illustrator are built upon. But we do not stop at a digital representation of their tessellation design we continue to cut their tessellations in Perspex. It moves the abstract concepts of math into the real world, so that the students can experience them directly, which provides a tremendous reward to the students.**

**Keywords**:* Design, math, tessellation, Escher, geometry*

**Introduction**

**One of design’s main goals is to give form to products and communication. However, most design students approach form intuitively, neglecting the understanding of the underlying geometry. Adobe Illustrator, for example, gives powerful tools to students to design geometrical shapes, such as Bézier curves (see Figure 1).**

** **

** **

**The mathematical principles of Bézier curves and other geometrical functions remain hidden. A quadratic Bézier curve, for example, is the path traced by the function B( t), given the points P_{0}, P_{1}, and P_{2}:**

**Teaching such abstract and technical topics to design students is challenging and we experimented with new approaches for industrial design students (Vlist et al., 2008). In this study, we describe a new method of teaching geometry to industrial design students, but we believe that it can be used for other design disciplines as well, such as graphic design or fashion design.**

**Geometry is of course a topic characteristic of most high school maths education and therefore we focus on slightly more advanced topics than, for example, the Pythagorean theorem. When it comes to geometrical principles of design, the golden ratio is frequently brought forward. It is surprising that a book entitled “ Geometry of Design” (Elam, 2001) is limited to the Golden Ratio, root squares and the human proportion. Moreover, the authors approach to directly apply these classic proportioning systems to design classics indicates how much design is concerned with the concretization and how little it is concerned with abstraction (Bartneck & Rauterberg, 2007). We capitalized on this move towards the concrete by focusing on a field of design in which geometry is the design: tessellation.**

**A tessellation is a collection of plane figures that fills the plane with no overlaps and no gaps. Tessellations are used in industrial design, textile design and interior design. Industrial design applies tessellations, amongst others, to tiles for bathrooms and kitchens and to paving stones. The principle of tessellation has also found great attention in manufacturing, since it minimizes the amount of wasted material when punching pieces from sheet metal (Bigalke & Wippermann, 1994). These application areas focus on the outer shape of the tiles. Sometimes tiles of different color are combined to introduce more variations.**

**Textile design and interior design focus even more on the inside of the tiles. They add decorative patterns onto the tiles, leaving the principle of the tessellation in the background. Throughout the centuries, fabrics and wallpapers utilized tessellation to create patterns. Maurice Cornelis Escher (1898-1972) took tessellation to the next level by adding meaning to the tiles. The graphical elements on the tiles are no longer just decoration, but they describe, for example, animals (see the fish in Figure 2). Schattschneider (1997) explored the different combinations in depth.**

**Before describing the process of the course, we will first give a short introduction to the different types of geometry, including group theory, matrices, and topology.**

**Geometry**

*Group theory: the mathematics of symmetry*

**Before going into the particularities of the mathematics used, we discuss a number of distinct views on geometry. Roughly the views were developed historically from concrete to abstract and from solutions of specific problems to powerful tools. In the context of design and education it is also important to pay attention to the mediums used. Each distinct view upon geometry naturally comes with its own preferred tools for representing and manipulating the constructions. We distinguish the following views upon geometry:**

** **

**Euclidian geometry,****analytical geometry,****transformational geometry.**

** **

**If we are to assign one mathematician as the main or most famous person behind each of these three views, then we have to name Euclid, Descartes and Klein. To simplify matters we call them the inventors of Euclidian geometry, analytical geometry and transformational geometry, respectively. Euclid (Greek: Εὐκλείδης) lived around 300BC and his main work is known as “ The Elements” (Euclid, Heath, & Densmore, 2002). René Descartes (1596-1650) published his main filosophical work “Discours de la méthode” (Descartes & Clarke, 1999) and the mathematical “La Géométrie” (Descartes et al., 1659) both in 1637. Felix Klein (1849-1925) presented his influential “Erlangen Program” in 1872 (Sharpe, 1997).**

**Euclidian geometry studies points, lines, circles, angles, lengths, areas, etc. It is the heart of the geometry taught at secondary schools with nowadays aspects of the other views blended in. Euclidian geometry can be studied on a purely axiomatic basis, and although the mathematics community honors Euclid for this, modern primary and secondary school teaching of geometry relies heavily on links to practical applications and examples from optics, physics and everyday live. Particularly in the Netherlands, this is the case (Feijs, 2005). Figure 3 shows one of the oldest fragments of a textbook on geometry and indeed the text is about lines, segments, rectangles, sections etc.[1]**

**The ideal media for this type of geometry are books and classrooms using pencil and paper, with tools such as straightedge (ruler) and compass. It also goes very well with traditional blackboard and chalk or modern white board and marker. Although things can be done using computers, usually it means that someone has to program most of it first, using some form of analytical geometry. A typical example is The Geometry Applet by Joyce (1996). Plane geometry fits well with today’s flat computer screens and ink jet printers. Important results include Pythagoras’ law, functions such as sine, cosine, tangent and their inverses. These results are indispensable when designing tessellations.**

** Now we turn to the second view, analytical geometry. Descartes connected geometry and algebra by coding the points of geometry as number pairs. Each point is given an x and a y coordinate. In Figure 4 this approach is shown in action.**

**The circle with radius 2 around the origin is described by an equation x^{2 }+ y^{2 }= 4. The circle can be shifted, for example, the substituting x → x–2 and y → y–2 and a bit of algebra give the equation of the circle around {2,2}, which is x^{2 }+ y^{2 }– 4x – 4y + 4 = 0. Straight lines, ellipses, parabolas etc. each have their own type of equation. Geometry is replaced by algebra. Deep geometric results can be obtained by unleashing the power of algebra, for example to prove that one cannot construct a square whose area is equal to that of a given circle using straight ruler and compass needs Galois theory (Galois, 1830a, 1830b), which is algebraic in nature. The ideal medium for analytical geometry is anything that helps to solve equations. First pen and paper, later log tables, calculators and now of course computers. When using computers, numerical procedures are useful, symbolic procedures are useful and their combination is best.**

**Finally we address the third view on geometry, transformational geometry. As we shall see, this is about symmetries, including the symmetries encountered in art, from ancient times to Escher and beyond. It also is useful for modern physics, for example quantum physics, but that is outside the scope of our teaching. In order to explain the basic idea, we refer to Figure 5 (snapshot of Microsoft PowerPoint).**

**Consider geometric transformations such as rotation, translation and reflection, for example those shown in the menu of Figure 5. The two-arrowed block shape can be rotated left over 90 ^{°} or right over 90^{° }but in both cases it becomes a different shape. When flipped horizontally or vertically it turns into the same shape again. The same applies rotation over 180^{°} (twice 90^{°}). Therefore we can say that the two-arrowed block shape is a symmetric shape. More precisely, each of the transformations which leave the figure the same, is a symmetry of this figure. Formulated somewhat more abstractly, the underlying mathematics is called group theory. Two operations can be composed, doing one and then the other, which is a transformation again. For example rotate left over 90° and then (again) left over 90° has the combined effect of the rotate 180° transformation[2].**

**In two dimensions one can create patterns filling the entire plane such that the pattern remains the same under certain transformations. For a specific pattern, the set of transformations forms a so-called group. Next to translations, reflections and translations one needs an extra transformation called glide-reflection. Group theory is in the heart of studying regular patterns. There are three possibilities:**

** **

**If there are no translations, the patterns are so-called rosettes. There exist two sets of rosette groups, either with or without reflections. They have applications in art and architecture, most notably the rose windows in Gothic cathedrals. A number of rosettes of logos, each with different symmetries, are in Figure 6. The logos are Opel (180° rotation), Crosiers (180° rotation and reflections), Mitsubishi (120° rotation and reflection), Yamaha (idem), and NATO (90° rotation, no reflection).****If there are only translations in one direction, the patterns are so-called friezes. There exist seven distinct frieze groups. They have numerous applications in art and architecture such as friezes of ancient temples and court buildings.****If there are translations in two or more directions, the patterns are so-called wall-paper patterns. There exist seventeen distinct wallpaper groups. They appear on wallpaper indeed, in brickwork, in floor tessellations, in Escher’s work and so on.**

**There exist also non-periodic tessellations such as invented by Penrose (1974), which have only been noted a few decades ago (folk wisdom of crystallography assumed such patterns would not be possible). In our teaching we used the rosette and the frieze patterns as easy forerunners of the more complex wallpaper patterns. We did so both in the lectures and in the assignments given to the students. As literature we provided Chapter I of the book Regular Figures by L. Fejes Toth (1964). The pedagogic idea behind this choice is that the book is mathematically quite rigorous[3] and that it is a good exercise for the students to develop their skill of getting information from such a book directly, instead of low-threshold and colorful Internet resources such as the Wikipedia Wallpaper group article [ http://en.wikipedia.org/wiki/Wallpaper_group S_{2}^{–1}S_{1}S_{2 }and interpret them as action sequences to be programmed. ]. **

**Another reason is that the idea of one transformation being transformed by another turns out useful later, when implementing transformations. It is very practical being able to read formulas like and interpret them as action sequences to be programmed.**

**What is the ideal medium for studying and using this type of geometry? Again, pen with paper and computer algebra are useful, but both the development of intuition and the practical applications call for ways to make the transformations happen in physical reality and before one’s eyes, or even better, in one’s hands. Shifting pieces of paper one over another, transparent sheets, sliding and rotating images and semi-transparent images in picture editors, physical tiles, paper and scissors are all very useful. For certain Heesch tilings (see next sections for a detailed discussion of the Heesch tilings) there exist a trick called “The Envelop Method” based on slicing 3D objects (envelopes) into tile-able ground forms (Feijs, 2008). It is important to realize that the student can not “see” the transformations in the same way he or she can see the points and lines of Euclidean geometry.**

*Matrices: implementing transformations*

**In this section we briefly outline a technique for programming and executing transformations in practice. The same technique is used in computer graphics, computer games, etc. The technique was explained to our students, examples being coded in Mathematica (Wolfram Research, 2009, Version 7) and in this form it was also used to make the tessellations. Stephen Wolfram conceived Mathematica and is a standard software package used in scientific, engineering, and mathematical fields and other areas of technical computing. It is also the foundation of Wolfram’s work on cellular automata (Wolfram, 2002). Points, lines and other geometric objects are moved around the plane by using Descartes’ trick: manipulating their coordinates. To move a point with coordinates {x,y} into a new position one needs four parameters: how the old x contributes to the new x, how the old x contributes to the new y and so on. It is customary to group those four numbers in a two by two block called a matrix. There are calculation rules how to apply a matrix to a coordinate pair. There are also rules how to multiply two matrices to yield a new one, representing the composed transformation. In Figure 7 four example matrices are shown. They represent the identity transformation, flipping along a diagonal, counter clockwise rotation over 90^{°} and 45^{°}, respectively.**

*Mathematica* has very useful built-in notations and algorithms to work with matrices. It takes another trick to move from the level of points to the level of lines and curves (sets of coordinate pairs) and to the higher levels of complex shapes and complete figures (for example birds, fish, etc. as appearing in Esher’s work). The trick deploys “mapping”, using the Mathematica operator /@. This operator is well known to programmers in special languages such as LISP, ML or Clean but no such thing exists in C, C++ or Java.

*Topology: from patterns to tilling.*

**It is tempting to assume that each of the 17 regular wallpaper patterns gives rise to precisely one schema of regular tiling. However, the relation is not one-to-one. What is still missing from the wallpaper patterns is: where are the cutting lines between the tiles? This question has been explored by Escher, aiming at his beautiful art. It also has been explored very thoroughly by Heesch and Kienzle (1963) aiming at industrial application and standardization. Heesch and Kienzle (1963) were well aware of group theory, but they added an analysis of the networks to be formed by the cutting lines between the tiles. Essentially this is about the edges (cutting line segments) and the vertices (nodes where three or more edges come together). For example, consider the structure of a honeybee comb where each tile has n = 6 edges and each vertex is joining three edges, see Figure 8.**

** **

** **

*This is a network of tile edges. This network has 32 vertices, 41 edges and 11 faces (including the outer space). Now there exists a mathematical result about such networks named after Leonhard Euler (1707-1783). It belongs to topology, that is the geometry of properties which are preserved by “rubber” transformations. Euler found that **V** *–* E** + F** = 2, where V** is the number of vertices, E** the number of edges and F** the number of faces. Indeed, 32 *– 41 + 11 = 2. Heesch (1963) characterizes this network as type 333333, meaning that the six consecutive vertices when going round one tile, each connect 3 edges. The above network has eight tiles, so a first very rough guess could be that is has 10 x *n* vertices but of course most vertices are then counted three times instead of once. And the first guess of 10 x *n* edges has to be corrected because edges are counted twice. Going from 10 tiles to *N* tiles and inserting the corrections we find that *N*.(⅓ + ⅓ + ⅓ + ⅓ + ⅓ + ⅓) – *N*.½.*n* + *N* = 2. For very large *N*, taking the limit, we find ⅓ + ⅓ + ⅓ + ⅓ + ⅓ + ⅓ = ½.*n* – 1. This is for the given honeybee network, but the same reasoning can be applied to find all networks of triangles (n = 3). Let the three vertices have *i*, *j* and *k* edges. Now 1/*i* + 1/*j* + 1/*k* = ½.3 – 1. This is only possible for certain numbers of *i*, *j* and *k*. For example ¼ + ⅛ + ⅛ = ½, so *i* = 4, *j* = *k* = 8 is a possibility. Thus the network type 488 is found. Working further in this way only 11 networks are possible. Finally Heesch (1963) integrates this theory with wallpaper theory to get 28 tiling types. He describes them in a prescriptive style (in German). This is what we gave to the students (actually we gave access to all of Heesch’s book, but we have no indication that they read anything except the prescriptions). An example is Figure 9. The network type is 43433. The tile itself is given a type code CC_{4}C_{4}C_{4}C_{4} where C refers to 180° rotational symmetry and C_{4} to 90° symmetry. These prescriptions are very practical to use. They could be used without any group theory or network theory, but we did present the essentials of those theories nevertheless.

** **

** **

**Execution of the course**

**Several steps need to be taken in advance of the course. First, the material needs to be ordered, ideally already cut to the dimensions of what the laser cutter can process. We use a Speedy300 laser cutter that can cut plates up to 40 x 70 cm. We used Perspex in at least three different colors to allow color variations. Dark colors are preferable since they bring out the engravings. Transparent Perspex is also available, but it requires additional attention during mounting.**

**Software licenses for Mathematica, Illustrator and Corel Draw must be available and the laser cutter needs to be booked. The duration of the cutting varies, in particular depending on the amount of engravings. In our experience, the works of six students can be cut in one day.**

*Workflow*

* *

* *

**The students prepare their tessellation in Mathematica. The lines they draw define the paths that the laser cutter will use for the cutting and engraving. The colours of the lines define if a certain line is used for cutting or engraving. For the engraving, the line thickness defines the thickness of the engraving. For the cutting, an ideal value of 0.001 mm is used. It is important that the students do not simply create a tile and repeat it, since this would create overlapping lines. The laser cut would cut this line multiple times, which takes far too much time and can reduce the overall quality of the tiles. The students export the Mathematica graphic as an encapsulated post script (EPS) file and final adjustments in Illustrator. The illustrator file is then imported into CorelDraw and transferred to the Laser Cutter control software through a virtual printer driver. Figure 10 shows the workflow that the students go through. First, they create a tile (a) which is then repeated using rotations and translations in Mathematica (b) before the necessary colours and thicknesses are assigned (c). The graphic in (c) is then sent to the laser cutter. Last, the students create a coloured version of (d), which makes it easier to puzzle the pieces together (see Figure 11).**

**The tiles are then mounted to foam board or another plate of Perspex. Transparent Perspex is more difficult to mount, since the mounting plate and the glue shines through. Additional examples of the students’ work are available at http://www.bartneck.de/2008/05/21/the-golden-ratio-course/.**

** **

** **

**Conclusions**

** **

**We described a new approach for bringing the mathematical foundations of geometry into the focus of design work. The approach is new to the best of our knowledge. One possible reason is that the tools used are relatively new and moreover the community of mathematicians and the community of industrial designers are usually very separated. **

** The students did not use common 2D design software packages, such as Adobe Illustrator, to create a tessellation, but we asked them to define the tessellation in the abstract language of math. This truly brought mathematical foundations of geometry into the focus of the course and Mathematica is one of the most accessible tools for design students to express their ideas in math. Of course the students could have stopped their effort after the production of a digital version of their tessellation, but going the extra mile of cutting real tiles with a laser cutter had a tremendous effect on the motivation of the students. The final tessellations have been hung at the walls of our department and it represents a great reward to our students. The Perspex tiles moved the abstract geometrical principles out of the computer and into the real world. The tessellations are physical objects to which the students can relate. They are not just an abstract ideas, but products that can be touched and experienced. The students responded that they appreciated the brushing-up of their math, but most of all it was fun. They truly enjoyed understanding the mathematical principles of vector graphics, as it is used by Adobe Illustrator and other design software. Moreover, it empowered them to create their own tessellations more easily. The patterns used in the currently fashionable handbags (e.g. Louis Vuitton) are now trivial to them and they can create advance tessellations for the design of textiles, bathroom tiles and visual design in general. Our approach facilitates design because of four reasons. First it empowers the design students because they become familiar with Mathematica and related tools, which allows them to address other modeling problems as well. Secondly it allows the students to design patterns whose aesthetic qualities can be added or embedded in other works of design. Thirdly it gives them additional knowledge about patterns as they appear in nature and in technical domains. Fourth it invites them to learn more about Escher, Japanese art and similar cultural topics relevant for design students.**

**We can also conclude that the technology for creating tessellations has developed rapidly. The math tools and graphic design tools allow the creation of tessellations within minutes and the remaining limiting factor is the creativity of the designer. The development of affordable laser cutters also has a tremendous effect on tessellation: the cutting line is negligible. It is no longer necessary to calculate in the width of the cutting line and true tessellations can be cut in one run. In the future we intend to extend the module to include aperiodic patterns, such as Penrose tilling. We are also further exploring the usage of semi-transparent Perspex that can then be glued against windows, creating an effect similar to windows in cathedrals. **

**References**

** **

**Bartneck, C., & Rauterberg, M. (2007). HCI Reality – An Unreal Tournament. International Journal of Human Computer Studies, 65(8), 737-743. | DOI: 10.1016/j.ijhcs.2007.03.003**

**Bigalke, H. G., & Wippermann, H. (1994). Regulaere Parkettierungen : mit Anwendungen in Kristallographie, Industrie, Baugewerbe, Design und Kunst Mannheim: Bibliographisches Institut.**

**Descartes, R., Beaune, F. d., Schooten, F. v., Witt, J. d., Bartholin, E., Elzevir, L., et al. (1659). Geometria. Amstelaedami [Amsterdam]: Apud Ludovicum & Danielem Elzevirios.**

**Descartes, R., & Clarke, D. M. (1999). Discourse on method and related writings. London: Penguin Books.**

**Elam, K. (2001). Geometry of design : studies in proportion and composition. New York: Princeton Architectural Press.**

**Escher, M. C. (Artist). (1938). Regular division of the plane drawing #20 [Ink, pencil, watercolor and gold painting].**

**Euclid. (75-125 A.D.). The Elements. excravated in 1896-97.**

**Euclid, Heath, T. L., & Densmore, D. (2002). Euclid’s Elements : all thirteen books complete in one volume : the Thomas L. Heath translation. Santa Fe, N.M.: Green Lion Press.**

**Feijs, E. (2005). Constructing a Learning Environment That Promotes Reinvention. In R. Nemirovsky, J. Rosebery & B. Solomon (Eds.), Everyday Matters in Science and Mathematics: Studies of Complex Classroom Events (pp. 241-266). Philadelphia: Lawrence Erlbaum Associates.**

**Feijs, E. (2008). De envelopjesmethode. In L. Feijs (Ed.). Eindhoven, [personal communication].**

**Fejes Tóth, L. (1964). Regular figures. New York,: Macmillan.**

**Galois, É. (1830a). Analyse d’un Mémoire sur la résolution algébrique des équations. Bulletin des Sciences mathématiques, 13(271).**

**Galois, É. (1830b). Note sur la résolution des équations numériques. Bulletin des Sciences mathématiques, 13(413).**

**Heesch, H., & Kienzle, O. (1963). Flächenschluss; System der Formen lückenlos aneinanderschliessender Flachteile. Berlin,: Springer.**

**Joyce, D. E. (1996). The geometry Applet. Retrieved February 25th, 2009, from http://aleph0.clarku.edu/~djoyce/java/Geometry/Geometry.html**

**Microsoft. (2007). Symmetry option in Office 2007.**

**Penrose, R. (1974). Role of aesthetics in pure and applied research. Mathematics Today – Bulletin of the Institute of Mathematics and its Applications, 10, 266.**

**Schattschneider, D. (1997). Escher’s Combinatorial Patterns. Electronic Journal of Combinatorics, 4(2), #17.**

**Sharpe, R. W. (1997). Differential geometry : Cartan’s generalization of Klein’s Erlangen program. New York: Springer.**

**Vlist, B. v. d., Westelaken, R., Bartneck, C., Jun, H., Ahn, R., Barakova, E., et al. (2008). Teaching Machine Learning to Design Students. In Z. Pan, X. Zhang, A. E. Rhalibi, W. Woo & Y. Li (Eds.), Technologies for E-Learning and Digital Entertainment, LNCS (Vol. 5093/2008, pp. 206-217). Berlin: Springer. | DOI: 10.1007/978-3-540-69736-7_23**

**Wolfram Research. (2009). Mathematica.**

**Wolfram, S. (2002). A new kind of science. Champaign, IL: Wolfram Media.**

**Biographical statement**

** Dr. Christoph Bartneck is an assistant professor in the Department of Industrial Design at the Eindhoven University of Technology. He has a background in Industrial Design and Human-Computer Interaction, and his projects and studies have been published in various journals, newspapers, and conferences. His interests lie in the fields of social robotics, Design Science, and Multimedia Applications. He has worked for several companies including the Technology Centre of Hannover (Germany), LEGO (Denmark), Eagle River Interactive (USA), Philips Research (Netherlands), and ATR (Japan). Christoph is an associate editor of the International Journal of Social Robotics.**

**Email: c.bartneck@tue.nl**

* *

*Prof. dr. ir. Loe Feijs* (1954) studied Electrical Engineering at TU/e where he graduated in 1979 in the group Information and Communication Theory. He joined Philips Telecommunications Industry, later AT&T-Philips Telecom and in 1984 Feijs changed to the Philips Natuurkundig Laboratorium In 1990 he obtained a Ph.D. in Computer Science of TU/e. In 1994 Feijs was appointed part-time professor at TU/e Mathematics and Computer Science. From 1998 to 2001 he was scientific director of the Eindhoven Embedded Systems Institute and in 2001 he was appointed full professor for the chair Industrial Design of Embedded Systems.

** Email: l.m.g.feijs@tue.nl**

**Abstract | References | Full Text: HTML, PDF (464 KB)**

[1] To give an impression what we mean with “quite rigorous” we give a short quotation from Fejes Toth (pages 11 and 12), the paragraph where it is explained why a rosette group can only have rotations around a single centre:

*Such a group contains only rotations about a single centre. For, if there were two rotations S*

_{1}, and S_{2}with distinct centres O_{1}and O_{2}, the transformation S_{1}^{–1}S_{2}^{–1}S_{1}S_{2}of the group would be a degenerate rotation, by the additivity theorem for angles of rotation. It cannot be the identity, since it displaces O_{1}, into the image of O_{1}, under S_{2}^{–1}S_{1}S_{2}, i.e. under S_{1}transformed by S_{2}. But the centre of this transformed rotation is the image of O_{1}under S_{2}i.e. a point different from O_{1}and therefore S_{2}^{–1}S_{1}S_{2 }effects a change in O_{1}. Hence S_{1}^{–1}S_{2}^{–1}S_{1}S_{2}would be a non-degenerate translation, contrary to our assumption.[2] A collection of such transformations is said to form a

*group*if the following properties hold: Closure: if two transformations are in the group, then so is their composition; Associativity: composing T

_{1}with “T

_{2}composed with T

_{3}” is the same as “T

_{1}composed with T

_{2}” composed with T

_{3}; Identity: there is a special transformation called the identity (doing nothing) which can be composed with any T with the effect of T; Inverses: for any transformation T in the group there is a transformation denoted as T

^{–1}such that T composed with T

^{–1}equals the identity and also T

^{–1}composed with T equals the identity transformation.

[3] translation:

*If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half”,*

*[5]*

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