Continuous Symmetry: From Euclid to Klein"This text is for a one-semester undergraduate course on geometry. It is richly illustrated and contains hundreds of exercises."--BOOK JACKET. |
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Page 1
... denote the set of real numbers with the symbol R. Other important number systems are Z , the set of integers ; N , the set of natural numbers ( non - negative integers ) ; and Q , the set of rational numbers ( quotients of integers ) ...
... denote the set of real numbers with the symbol R. Other important number systems are Z , the set of integers ; N , the set of natural numbers ( non - negative integers ) ; and Q , the set of rational numbers ( quotients of integers ) ...
Page 6
... denotes a unique real number because the real number system is complete ! Suppose x1 , x2 , x3 , ... is a sequence of digits ( integers between 0 and 9 ) . For each positive integer n define the two finite decimals an.X1X2 ... Xn and bn ...
... denotes a unique real number because the real number system is complete ! Suppose x1 , x2 , x3 , ... is a sequence of digits ( integers between 0 and 9 ) . For each positive integer n define the two finite decimals an.X1X2 ... Xn and bn ...
Page 8
... denote as R2 , is simply the set of all ordered pairs of real numbers : R2 = { ( x , y ) | x , y ≤R } . A line in this model is the set of all points ( x , y ) that satisfy an equation of the form ax + by + c = 0 for some set of real ...
... denote as R2 , is simply the set of all ordered pairs of real numbers : R2 = { ( x , y ) | x , y ≤R } . A line in this model is the set of all points ( x , y ) that satisfy an equation of the form ax + by + c = 0 for some set of real ...
Page 10
... denote as P2 , is the open unit disk 2 p2 = { ( x , y ) | x , y Є R and x2 + y2 < 1 } . There will be two types of lines in this model , both shown in Figure 2.5 . One will be any ordinary straight line containing the origin ( 0,0 ) ...
... denote as P2 , is the open unit disk 2 p2 = { ( x , y ) | x , y Є R and x2 + y2 < 1 } . There will be two types of lines in this model , both shown in Figure 2.5 . One will be any ordinary straight line containing the origin ( 0,0 ) ...
Page 18
... denoted by several different notations : d ( p , q ) , pq , or simply pq . Let ExƐ denote the collection of all ordered pairs ( p , q ) of two points in the plane . Then the distance d is a function assigning a real number to each pair ...
... denoted by several different notations : d ( p , q ) , pq , or simply pq . Let ExƐ denote the collection of all ordered pairs ( p , q ) of two points in the plane . Then the distance d is a function assigning a real number to each pair ...
Contents
XXXVI | 235 |
XXXVII | 240 |
XXXVIII | 251 |
XL | 256 |
XLI | 266 |
XLII | 276 |
XLIII | 287 |
XLIV | 292 |
XIII | 62 |
XIV | 70 |
XV | 84 |
XVI | 94 |
XVII | 110 |
XVIII | 115 |
XIX | 119 |
XX | 121 |
XXI | 135 |
XXII | 146 |
XXIII | 156 |
XXIV | 161 |
XXV | 165 |
XXVII | 181 |
XXVIII | 188 |
XXIX | 191 |
XXX | 199 |
XXXI | 206 |
XXXII | 211 |
XXXIII | 217 |
XXXIV | 224 |
XXXV | 231 |
XLV | 298 |
XLVI | 309 |
XLVII | 315 |
XLVIII | 322 |
XLIX | 340 |
L | 347 |
LII | 356 |
LIII | 363 |
LIV | 375 |
LV | 376 |
LVI | 399 |
LVII | 416 |
LVIII | 439 |
LIX | 459 |
LXI | 467 |
LXII | 482 |
LXIII | 487 |
LXIV | 505 |
LXV | 520 |
LXVI | 531 |
LXVII | 533 |
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Common terms and phrases
AABC ABCD Axiom circle collinear composition compute congruent conjugacy conjugacy classes conjugate consider containing coordinate system define Definition denote desired dilation factor directed angle measure distance Dp,s equal equilateral equivalent Euclidean geometry example Exercise exists finite fixed point frieze groups frieze pattern glide reflection group G Hence Hint integer interior angles intersect invariant isometry group L-Jordan measurable l₁ lattice line segment maps medial triangle midpoint non-split non-trivial orientation-preserving orientation-reversing orthic triangle parallel lines Parallel Postulate parallelogram perpendicular bisector point group point inversion polygonal regions preserves proof properties Proposition Prove quadrilateral radius real number rectangle result rhombic rotation angle shown in Figure side length split groups square Structure Theorem Suppose symmetry group transformations translation group translation orbit translation subgroup triangle ABC uniform dilation unique verify vertex vertices wallpaper groups y₁
Popular passages
Page xiii - That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Page 58 - The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Page 48 - The SAS Postulate: Given a correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.
Page 4 - the best way to explain it is to do it." (And, as you might like to try the thing yourself some winter day, I will tell you how the Dodo managed it.) First it marked out a race-course, in a sort of circle, ("the exact shape doesn't matter," it said,) and then all the party were placed along the course, here and there. There was no "One, two...
Page 61 - If two sides of one triangle are equal to two sides of a second triangle and the included angle of the first is greater than the included angle of the second, then the third side of the first triangle is greater than the third side of the second.
Page 294 - A median of a triangle is a line which joins a vertex of the triangle to the midpoint of the opposite side. Let ABC be any triangle, and let a, b and c be position vectors, of A, B and C relative to some fixed origin.
Page 114 - The radius of a circle is a line segment from the center of the circle to a point on the circle.
Page 48 - If two sides of a triangle are congruent, then the angles opposite these sides are congruent.
Page 68 - The sum of the measures of the interior angles of any triangle is 180°.
Page 120 - Suppose a correspondence between two triangles is such that two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle.