Instruction 1. The project report should include the project construction and explanations. They must demonstrate the knowledge of the mathematical language by using standard notation, and...

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Instruction 1. The project report should include the project construction and explanations. They must demonstrate the knowledge of the mathematical language by using standard notation, and explaining certain steps (definitions, hypothesis, conclusions, etc.). You must give evidence of understanding of the topic by including enough steps so that the argument is clear, pointing out extraneous solutions (if any) and explaining why the solution was preferred. (Using the guide pdf (chapter 6 only) and lecture notes) 1. Here below is the Project 9- CONSTRUCTING TESSELLATIONS 1. Here is the link for the website for explorer geometry program http://homepages.gac.edu/~hvidsten/gex/ PowerPoint Presentation SYMMETRY In geometry, symmetry is defined as a balanced and proportionate similarity that is found in two halves of an object. It means one-half is the mirror image of the other half. The imaginary line or axis along which you can fold a figure to obtain the symmetrical halves is called the line of symmetry. If an object is symmetrical, it means that it is equal on both sides. Suppose, if we fold a paper such that half of the paper coincides with the other half of the paper, then the paper has symmetry.  Symmetry can be defined for both regular and irregular shapes. For example, a square is a regular (all sides are equal) and a rectangle is an irregular shape (since only opposite sides are equal). The symmetries for both shapes are different. 2 Symmetry in Mathematics Symmetry Math definition states that “symmetry is a mirror image”. When an image looks identical to the original image after the shape is being turned or flipped, then it is called symmetry. In Mathematics, a meaning of symmetry defines that one shape is exactly like the other shape when it is moved, rotated, or flipped. Consider an example, when you are told to cut out a ‘heart’ from a piece of paper, don’t you simply fold the paper, draw one-half of the heart at the fold and cut it out to find that the other half exactly matches the first half? The heart carved out is an example of symmetry. Symmetry exists in patterns. You may have often heard of the term ‘symmetry’ in day to day life. It is a balanced and proportionate similarity found in two halves of an object, that is, one-half is the mirror image of the other half. And a shape that is not symmetrical is referred to as asymmetrical. Symmetric objects are found all around us, in nature, architecture, and art. 3 Symmetrical shapes or figures are the objects where we can place a line such that the images on both sides of the line mirror each other.  The below set of figures form symmetrical shapes when we place a plane or draw the lines. For example, figure (b) has the symmetrical figures when we draw two lines of symmetry as shown below. 5 Line of Symmetry The imaginary line or axis along which you fold a figure to obtain the symmetrical halves is called the line of symmetry. It basically divides an object into two mirror-image halves. The line of symmetry can be vertical, horizontal or diagonal. There may be one or more lines of symmetry. 1 Line Symmetry Figure is symmetrical only about one axis. It may be horizontal or vertical. The word ATOYOTA has one axis of symmetry along the axis passing through Y. 2 Lines of Symmetry l with only about two lines. The lines may be vertical and horizontal lines as viewed in the letters H and X. 3 Lines Symmetry An example of three lines of symmetry is an equilateral triangle. Here, the mirror line passes from the vertex to the opposite side dividing the triangle into two equal right triangles. 4 Lines Symmetry  Four lines of symmetry can be seen in a square, that has all the sides equal.  Infinite lines Some figures have not one or two, but infinite lines passing through the centre, and the figure is still symmetrical. Example: a circle. Types of Symmetry Symmetry may be viewed when you flip, slide or turn an object. There are four types of symmetry that can be observed in various situations, they are: Translation Symmetry Rotational Symmetry Reflection Symmetry Glide Symmetry Translation Symmetry If the object is translated or moved from one position to another, the same orientation in the forward and backward motion is called translational symmetry. In other words, it is defined as the sliding of an object about an axis. This can be observed clearly from the figure given below, where the shape is moved forward and backward in the same orientation by keeping the fixed axis. Rotational Symmetry When an object is rotated in a particular direction, around a point, then it is known as rotational symmetry or radial symmetry. Rotational symmetry existed when a shape turned, and the shape is identical to the origin. The angle of rotational symmetry is the smallest angle at which the figure can be rotated to coincide with itself. In geometry, many shapes consist of rotational symmetry. For example, the figures such as circle, square, rectangle have rotational symmetry. Rotational symmetry can also be found in nature, for instance, in the petals of a flower. Below figure shows the rotational symmetry of a square along with the degree of rotation. 11 Reflexive Symmetry  Reflection symmetry is a type of symmetry in which one half of the object reflects the other half of the object. It is also called mirror symmetry or line of symmetry. Glide Symmetry The combination of both translation and reflection transformations is defined as the glide reflection. A glide reflection is commutative in nature. If we change the combination’s order, it will not alter the output of the glide reflection. Symmetrical Shapes The symmetry of shapes can be identified whether it is a line of symmetry, reflection or rotational based on the appearance of the shape.  The shapes can be regular or irregular. Based on their regularity, the shapes can have symmetry in different ways.  Also, it is possible that some shapes does not have symmetry. For example, a tree may or may not have symmetry. Example 1: If the figure below follows a reflexive or line of symmetry, then complete the figure. Given that, the figure has a line of symmetry. That means, the second half (i.e. missing part) of the figure will be exactly the same as the given. 14 Example 2: Identify the shapes which do not have rotational symmetry from the below figure. Solution: As we know, rotational symmetry is a type of symmetry, when we rotate a shape in a particular direction, the resultant shape is exactly the same as the original shape. Thus, from the given figure (a) and (c) do not have a rotational symmetry. A number of other kinds of symmetric types exist such as the point, translational, glide reflectional, helical, etc. which are beyond the scope of learning at this stage. 15 Recall that a group is a set of elements satisfying four properties. For a set of functions, these properties would be: 1.Given any two functions in the set, the composition of the two functions is again in the set. 2.The composition of functions is an associative operation. 3.The identity function is a member of the set. 4.Given any function in the set, its inverse exists and is an element of the set. Why is it important that the symmetries of a figure form a group?Groups are a fundamental concept in abstract algebra and would seem to have little relation to geometry. In fact, there is a very deep connection between algebra and geometry. As M. A. Armstrong states in the preface to Groups and Symmetry, “groups measure symmetry” [3]. Groups reveal to us the algebraic structure of the symmetries of an object, whether those symmetries are the geometric transformations of a pentagon, or the per-mutations of the letters in a word, or the configurations of a molecule. By studying this algebraic structure, we can gain deeper insight into the geom-etry of the figures under consideration. 16 Finite Plane Symmetry Groups We conclude that all rotations in a finite symmetry group are generated from a particular rotation having the smallest angle. Groups generated from a single element are called cyclic groups. The symmetry group of a snowflake is D6, a dihedral symmetry, the same as for a regular hexagon. (Wikipedia) Leonardo’s Theorem Recall that Leonardo’s theorem states that if a subset of the Euclidean plane has finitely many symmetries, then its symmetry group must be the cyclic group Cn or the dihedral group Dn for some positive integer n. Most of the mathematical content in the proof of Leonardo’s theorem is in the lemmas we proved above. All we need to do now is fit the pieces of the puzzle together as follows. Frieze Groups Let’s turn our attention now to infinite symmetry groups. Since we’ve already seen that a finite group of symmetries can’t contain a translation, an easy way to create an infinite symmetry group is to consider a single translation T. A symmetry group which contains T necessarily contains the infinitely many elements. A frieze pattern can certainly have other symmetries which aren’t translations, but there aren’t too many possibilities for such other symmetries. Actually, it shouldn’t be too difficult to see that they can only come in four types. H = a reflection in a horizontal mirror V = a reflection in a vertical mirror R = a 180 rotation G = a glide reflection along a horizontal axis So for each frieze pattern, we can give it an HVRG symbol, depending on which of these four symmetries it possesses. H = a reflection in a horizontal mirror V = a reflection in a vertical mirror R = a 180 rotation G = a glide reflection along a horizontal axis Theorem. There are exactly seven types of frieze pattern. translation τ = Tv rm (reflection across the midline m), ru (with u perpendicular to m), H (half-turn rotation about O on m) γ (glide reflection along m with glide vector equal to v ). Textbook Notation Wallpaper Groups In a simple language, the wallpaper group are patterns that cover the whole plane and are repetitive in two different directions. Just as we did for friezes, we can try to classify the types of wallpaper patterns —however, the game is much harder this time. It turns out that there are exactly seventeen different types of wallpaper patterns, although we won’t provide a proof here. We’ll merely content ourselves with knowing how to identify them. To each wallpaper pattern, we associate an RMG symbol consisting of the following three numbers R, M and G. R = the maximum order of a rotational symmetry M = the maximum number of mirrors which pass through a point G = the maximum number of proper glide axes (axis of a glide reflection which is not itself a mirror) which pass through a point We can classify lattices into five different kinds: If a lattice has a square fundamental region, it's called a square lattice. If it has a 60° rhombus as a fundamental region, it's called a hexagonal lattice. That's because in that case, the points in the lattice nearest any one point in the lattice are the vertices of a regular hexagon. (A rhombus is a parallelogram with equal sides.) If a lattice has a rhombus as a fundamental region, it's a rhombic lattice. (So square and hexagonal lattices are very special rhombic lattices.) If it has a rectangle as a fundamental region, it's a rectangular lattice. And in general, it's a parallelgrammatic lattice. If a pattern has a reflection as a symmetry, then its lattice has to be rhombic, rectangular, or square. If it has a 90° rotation, then the lattice must be square. But if it has a 60° rotation or a 120° rotation, the lattice must be hexagonal. Parallelogrammatic lattice The orange dots indicate the lattice points in the example. The light blue lines outline parallelograms. Besides the translations of a parallelogrammatic lattice, there are also half-turn symmetries, that is, 180° rotations. Some of the half-turns have lattice points as fixed points, but the fixed points of other half turns occur midway between two lattice points. It may be a little difficult to see these other half-turns, so their fixed points are covered with black dots in this image. In summary, a parallelogrammatic lattice's symmetry group (number 2, p2) has translations and half-turns, but there are neither reflections nor glide-reflections. The blue dots indicate the lattice points in the example. Besides the translations and half-turns that every lattice has, rectangular lattices have reflections. The axes of these reflections are drawn in red. Some of the reflection axes pass through the lattice points, but others lie midway between lattice points. The centers of the half-turns lie at the intersections of the reflection axes. There are no glide-reflections (except those which have the same axis as an axis of reflection). In summary, a rectangular lattice's symmetry group (number 6, pmm) has translations, half-turns, and reflections. Rectangular lattice Rhombic lattice Again, the blue dots indicate the lattice points. Rhombic lattices have translations, half-turns, and reflections, like the rectangular lattices do, but rhombic lattices also have glide reflections. The axes of the glide reflections are shown in green, and appear half-way between the axes of the reflections (shown in red, again). Some of the centers of the the half-turns occur at the intersections
Jan 27, 2023
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