Transformation Geometry: An Introduction to SymmetrySpringer Science & Business Media, 20.12.1996 - 240 Seiten 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. |
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Transformation Geometry: An Introduction to Symmetry George E. Martin Keine Leseprobe verfügbar - 1982 |
Transformation Geometry: An Introduction to Symmetry George E. Martin Keine Leseprobe verfügbar - 1982 |
Häufige Begriffe und Wortgruppen
2-center AABC affine transformation Algebra axes axis base called Chapter circle collinear collineation concurrent congruent consider contains conversely cube defined definition Desargues determined dilation directed angle distance distinct divides edges elements equations exactly example Exercise faces fact Figure finite group fixed point four geometry give given glide reflection halfturn Hence hexagon identity images infinite integer intersect introduced inverse involution line of symmetry linear mathematics midpoint multiplicity Name noncollinear nonidentity Note odd isometries opposite pattern perpendicular perpendicular bisector plane polygon positive possible preserves proof prototile Prove Prove or disprove ratio regular respectively result rotation rotation group sides similarity solid space square stretch Suppose symmetry group Table taking tessellation Theorem tiling translation triangle unique vertex vertices wallpaper group
Beliebte Passagen
Seite 3 - GREEK ALPHABET Letters Names Letters Names Letters Names A a Alpha I t Iota...
Seite 13 - This is often a help in computation, since the value of e is known ( approximately 2.718). A problem related to that of derangement is the famous problem of Latin squares, first studied by Euler. A Latin square is anxn square in which objects 1,2, . . ., n are placed so that each object appears exactly once in each row and exactly once in each column. Two Latin squares are called orthogonal if, when one square is superimposed on the other, each object of the first square occurs once and only once...
Seite 167 - The bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. 144. Theorem. The bisector of an exterior angle of a triangle divides the opposite side produced into segments proportional to the other two sides.
Seite 11 - Thm.X.4.17 below gives as a special case the familiar result, (RS)- = (S-'XR-1), that the inverse of a product is the product of the inverses in reverse order.
Seite 67 - Leonardo da Vinci engaged in systematically determining the possible symmetries of a central building and how to attach chapels and niches without destroying the symmetry of the nucleus. In abstract modern terminology, his result is essentially our above table of the possible finite groups of rotations (proper and improper) in two dimensions.
Seite 149 - This proves the first part of the theorem. For the second part, suppose xe л,Л/0(8) is v \ -torsion.
Seite 3 - Pi Q 0) Omega 3 there exists V for ail CAPACITY 1 centimetre 10 millilitres = 1 centilitre 1 decimetre 10 centilitres = 1 decilitre 1 metre 10 decilitres = 1 litre 1 decametre 10 litre = 1 decalitre 1 hectometre 10 dekalitres = 1 hectolitre...
Seite 52 - Show that the composition of the reflections in the three angle bisectors of a triangle is a reflection in a line that is perpendicular to one side of the triangle.
Seite 159 - ... of the third medial line. Deduced algebraically from the preceding. 687. The sum of the squares of the three greater segments of the medial lines of a triangle is equivalent to one-third the sum of the squares of the sides of a triangle. Deduced algebiaically from the preceding. 688. The lines from the vertices of a triangle to the points of tangency of the inscribed circle intersect in a common point SUG'S. DC is parallel to AF, BD = BF = 4, CF = CP = c, AD = AP = a, DC = d, FC = e. OF = * c...
Seite 6 - In the plane, the locus of all points equidistant from two points A and B is the perpendicular bisector of A and B, which is a line through the midpoint of Zff and perpendicular to ÄB.
Verweise auf dieses Buch
Group Theoretical Methods and Their Applications E. Stiefel,A. Fässler Keine Leseprobe verfügbar - 1992 |