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1982 | Buch

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Transformation Geometry

An Introduction to Symmetry

verfasst von: George E. Martin

Verlag: Springer New York

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Transformation geometry is a relatively recent expression of the successful venture of bringing together geometry and algebra. The name describes an approach as much as the content. Our subject is Euclidean geometry. Essential to the study of the plane or any mathematical system is an under­ standing of the transformations on that system that preserve designated features of the system. Our study of the automorphisms of the plane and of space is based on only the most elementary high-school geometry. In particular, group theory is not a prerequisite here. On the contrary, this modern approach to Euclidean geometry gives the concrete examples that are necessary to appreciate an introduction to group theory. Therefore, a course based on this text is an excellent prerequisite to the standard course in abstract algebra taken by every undergraduate mathematics major. An advantage of having nb college mathematics prerequisite to our study is that the text is then useful for graduate mathematics courses designed for secondary teachers. Many of the students in these classes either have never taken linear algebra or else have taken it too long ago to recall even the basic ideas. It turns out that very little is lost here by not assuming linear algebra. A preliminary version of the text was written for and used in two courses-one was a graduate course for teachers and the other a sophomore course designed for the prospective teacher and the general mathematics major taking one course in geometry.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
There are only seventeen mathematically different types of wallpaper patterns. This intriguing result, which we shall encounter as an application of our study of transformations, was empirically known to the Moors of fifteenth-century Spain but was unknown to the Greek mathematician Euclid. Euclid’s Elements first placed mathematics on an axiomatic basis. Written shortly after 300 B.C. in Alexandria, then a Greek city at a mouth of the Nile, the Elements proved to be the world’s most successful secular book. According to one story, Euclid had to tell King Ptolemy, who was not the last to find the work difficult, that there was no royal road to geometry. Today, Euclid would revise that comment and present much of his mathematics through transformations. Transformation geometry is a stretch along the royal road to geometry. The name describes an approach as much as the content of our study. Our subject is Euclidean geometry and, until Chapter 16 where we consider three-space, we suppose we are talking about the Euclidean plane.
George E. Martin
Chapter 2. Properties of Transformations
Abstract
The identity transformation i is defined by i(P) = P for every point P. No other transformation is allowed to use this Greek letter iota. As you can see, i is in some sense actually the least exciting of all the transformations. If I is in set G of transformations, then G is said to have the identity property. We continue below to look at properties of a set G of transformations that make G algebraically interesting.
George E. Martin
Chapter 3. Translations and Halfturns
Abstract
The methods of this chapter are from analytic geometry. To emphasize the use of numbers, we often call this approach coordinate geometry. In either case, we start with the translations as they are important and so easily studied by these methods. Later, we shall not hesitate to use the older synthetic methods from the geometry of Euclid when they are most convenient. Then, as we progress, we shall be using more and more group-algebraic methods. At times it may even be hard to say whether a particular part is essentially synthetic, analytic, or algebraic, but this will be completely unimportant.
George E. Martin
Chapter 4. Reflections
Abstract
Suppose house H and barn B are on the same side of the Mississippi River m as indicated in Figure 4.1. The problem is to go from the house to the barn by way of the river along the shortest possible path. You may wish to stop and try to solve this problem before reading further. Supposedly, we are after a pail of water from the river. Anyway, it is clear that we do not walk along the river once we get there. Suppose the shortest path touches the river at point R. So R is the point on m such that HR+RB is minimal. As you may have guessed, a solution to this problem has something to do with reflections. If we let H’ be the mirror image of H reflected in line m, then surely HR = H’R. So we want R on m such that H’R + RB is minimal. This problem is very easy. Obviously R is the intersection of m and BH’. The path obtained is the path of a ray of light traveling from H to B that is reflected in mirror m, as the angle of incidence at m is congruent to the angle of reflection.
George E. Martin
Chapter 5. Congruence
Abstract
The halfturns generate the group H . What group of isometries does the set of reflections generate? Since a reflection is its own inverse, every element in this group must be a product of reflections (Theorem 2.4). A product of reflections is clearly an isometry; in this section we show that every isometry is a product of reflections. Thus, we shall see that the reflections generate all of H. The reflections are the building blocks for the symmetries of the plane.
George E. Martin
Chapter 6. The Product of Two Reflections
Abstract
Every isometry is a product of at most three reflections (Theorem 5.6). So each isometry is of the form σl, σmσl or σrσmσl. In this section the case σmσl is examined. Since a reflection is an involution, we know σlσl = l for any line l. Thus we are concerned with the product of two reflections in distinct lines l and m. There are two cases: either l and m are parallel or else l and m intersect at a unique point. We shall show first that if l and m are parallel lines then the product σmσl is the translation through twice the directed distance from l to m.
George E. Martin
Chapter 7. Even Isometries
Abstract
A product of two reflections is a translation or a rotation. By considering the fixed points of each, we see that neither a translation nor a rotation can be equal to a reflection. Thus, σnσm ≠ σl for lines l,m,n When a given isometry is expressed as a product of reflections, the number of reflections is not invariant; indeed, we can always add 2 to the number of reflections in a given product by inserting “σlσl” into the product. Although a product of two reflections cannot be a reflection, we know that in some cases a product of three reflections is a reflection. We shall see this is possible only because both 3 and 1 are odd integers. In mathematics, parity refers to the property of an integer being either even or odd. An isometry that is a product of an even number of reflections is said to be even; an isometry that is a product of an odd number of reflections is said to be odd. Since an isometry is a product of reflections, then an isometry is even or odd. However, the definition will be useful only if we show that no isometry is both even and odd. Of course no integer can be both even and odd, but is it not conceivable some product of ten reflections could be equal to some product of seven reflections? To show this is impossible, we first show that a product of four reflections is always equal to a product of two reflections.
George E. Martin
Chapter 8. Classification of Plane Isometries
Abstract
We have classified all the even isometries as translations or rotations. An odd isometry is a reflection or a product of three reflections. Only those odd isometries σc σb σa where a, b, c are neither concurrent nor have a common perpendicular remain to be considered. Although it seems there might be many cases, depending on which of a, b, c intersect or are parallel to which, we shall see this turns out not to be the case. However, we begin with the special case where a and b are perpendicular to c. Then σb σa is a translation or glide and σc is, of course, a reflection. If a and b are distinct lines perpendicular to line c, then σc σb σa is called a glide reflection with axis c. We might as well call line m the axis of σm as the reflection and the glide reflection then share the property that the midpoint of any point P and its image under the isometry lies on the axis. To show this holds for the glide reflection, suppose P is any point. See Figure 8.1. Let line l be the perpendicular from P to c. Then there is a line m perpendicular to c such that σb σa = σm σl. If M is the intersection of m and c, then P and M are distinct points such that σcσbσa(P)=σcσmσl(P)=σcσm(P)= σM(P)≠P.
George E. Martin
Chapter 9. Equations for Isometries
Abstract
The equations for a general translation were incorporated in the definition of a translation. Equations for a reflection were determined in Theorem 4.2. We now turn to rotations. Equations for the rotation about the origin through a directed angle of Θº are considered first. Let ρ(0,),Θ =where l is the X-axis. Then one directed angle from l to m has directed measure Θ/2. From the definition of the trigonometric functions, we know (cos(Θ/2), sin(Θ/2)) is a point on m. So line m has equation (sin(Θ/2))X - (cos(Θ/2))Y+ 0 = 0. Hence σm has equations
$$\begin{gathered} x' = x - \frac{{2\left( {\sin \left( {\Theta /2} \right)} \right)x - \left( {\cos \left( {\Theta /2} \right)} \right)y]}}{{{{\sin }^2}\left( {\Theta /2} \right) + {{\cos }^2}\left( {\Theta /2} \right)}} \hfill \\ = \left[ {1 - 2{{\sin }^2}\left( {\Theta /2} \right)} \right]x + \left[ {2\sin \left( {\Theta /2} \right)\cos \left( {\Theta /2} \right)} \right]y \hfill \\ = \left( {\cos \Theta } \right)x + \left( {\sin \Theta } \right)y, \hfill \\ y' = y + \frac{{2\left( {\cos \left( {\Theta /2} \right)} \right)\left[ {\left( {\sin \left( {\Theta /2} \right)x - \left( {\cos } \right)\left( {\Theta /2} \right)} \right)y} \right]}}{{{{\sin }^2}\left( {\Theta /2} \right) + {{\cos }^2}\left( {\Theta /2} \right)}} = \left( {\sin \Theta } \right)x - \left( {\cos \Theta } \right)y \hfill \\ \end{gathered} $$
George E. Martin
Chapter 10. The Seven Frieze Groups
Abstract
Around the frieze of an older building there is often a pattern formed by the repetition of some figure or motif over and over again. The essential property of an ornamental frieze pattern is that it is left fixed by some “smallest translation.” We can call AB the length of translation τA,B and say τA,B is shorter than τC,D if AB < CD. Other symmetries in addition to translations are often apparent in a frieze as well. Of course, there is infinite variety in the subject for such patterns. However, by discounting the scale and subject matter and by considering only the symmetries under which such patterns are left invariant, we shall see that there are only seven possible types of ornamental frieze patterns.
George E. Martin
Chapter 11. The Seventeen Wallpaper Groups
Abstract
The ornamental groups of the plane are the rosette groups, the frieze groups, and the wallpaper groups. The rosette groups are the finite groups of isometries, which by Leonardo’s Theorem are the groups Cn and Dn. A frieze group is a group of isometries whose subgroup of translations is generated by one translation. Frieze groups were treated in Chapter 10. We now turn to the last of the ornamental groups of the plane by considering groups whose subgroup of translations is generated by two translations.
George E. Martin
Chapter 12. Tessellations
Abstract
Almost everyone has at one time or another been intrigued by mosaic patterns. Figure 11.31 illustrates seventeen tilings of the plane. A tiling of the plane is also called a mosaic, tessellation, or paving of the plane. Tiling as an art predates human history and, perhaps, reached its zenith in the Moorish forts, palaces, and mosques near the end of what Westerners call the Middle Ages. Except for an initial study by the astronomer Johannes Kepler (1571–1630), little formal mathematical investigation of tilings took place before the end of the last century. Much of what has been done is the work of chemists and crystallographers. Today, mathematicians are taking more interest in this ancient topic.
George E. Martin
Chapter 13. Similarities on the Plane
Abstract
This chapter may be read following Chapter 9.
George E. Martin
Chapter 14. Classical Theorems
Abstract
For each of the mathematicians that lends his name to the title of this section, there is at least one famous theorem that bears his name. As we shall see, the Alexandrian Greek mathematician Pappus can be paired with either of the seventeenth-century French mathematicians Desargues or Pascal. However, the Alexandrian Greek mathematician Menelaus and the seventeenthcentury Italian mathematician Ceva are invariably mentioned together. Menelaus’ Theorem, which involves a test for the collinearity of three points, and Ceva’s Theorem, which involves a test for the concurrency of three lines, are frequently called the Twin Theorems. These theorems should have been discovered together, and it is not insignificant that such a long period separates Menelaus and Ceva. During the 1500 years that separate the two there was little development in mathematics.
George E. Martin
Chapter 15. Affine Transformations
Abstract
Up to this point we have studied in modern format mostly the geometry of Euclid. We now turn to transformations that were first introduced by the great mathematician Leonhard Euler (1707–1783). (Euler was introduced in the preceding chapter following Theorem 14.16.) From the meaning of the word affine, we must define an affine transformation as a collineation on the plane that preserves parallelness among lines. So, if l and m are parallel lines and α is an affine transformation, then lines α (l) and α(m) are parallel. However, if β is any collineation and l and m are distinct parallel lines, then β(l) and β(m) cannot contain a common point β(P) as point P would then have to be on both l and m. Therefore, every collineation is an affine transformation. Hence, affine transformations and collineations are exactly the same thing for the Euclidean plane.
George E. Martin
Chapter 16. Transformations on Three-space
Abstract
This chapter may be read following Chapter 13.
George E. Martin
Chapter 17. Space and Symmetry
Abstract
If a convex polyhedron has υ vertices, e edges, and/faces, then υ ‒ e + f = 2. This simple but elegant theorem is known as Euler’s Formula, even though Descartes had stated the equation over 100 years before Euler gave the result in 1752. A polyhedron is convex if all the vertices not on any given face lie on one side of the plane containing that face. To prove the famous formula, imagine that all the edges of a convex polyhedron are dikes, exactly one face contains the raging sea, and all other faces are dry. We break dikes one at a time until all the faces are inundated, following the rule that a dike is broken only if this results in the flooding of a face. Now, after this rampage, we have flooded f ‒ 1 faces and so destroyed exactly f ‒ 1 dikes. Noticing that we can walk with dry feet along the remaining dikes from any vertex P to any other vertex along exactly one path, we conclude there is a one-to-one correspondence between the remaining dikes and the vertices excluding P. Hence, there remain exactly υ ‒ 1 unbroken dikes. So e = (f υ 1) +(υ υ 1) and we have proved Euler’s Formula.
George E. Martin
Backmatter
Metadaten
Titel
Transformation Geometry
verfasst von
George E. Martin
Copyright-Jahr
1982
Verlag
Springer New York
Electronic ISBN
978-1-4612-5680-9
Print ISBN
978-1-4612-5682-3
DOI
https://doi.org/10.1007/978-1-4612-5680-9