Geometry and Imagination
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Description
Book Introduction
A treasure trove of mathematical ideas
A true masterpiece on how to express mathematics intuitively!
Geometry and Imagination, which laid the foundation for modern geometry teaching methods
This book was compiled and published by Hilbert's student, Stefan Cohn-Posen, based on lectures he gave at the university over the three years before his retirement.
Through this book, Hilbert hoped to expose many areas of geometry that could be seen and intuitively understood, and to make geometry more accessible to those studying it.
The efforts of many people have added various images, and by drawing and presenting rather complex or difficult-to-imagine stereoscopic projections, we have reduced the difficulty of having to mentally imagine various axioms and geometric assumptions.
Geometry and Imagination is one of the few classics that is still widely read, even after more than half a century since its publication.
Still fresh as time goes by, this book is full of clear and elegant ideas that explain mathematics well.
I hope that both beginners and experts in mathematics will enjoy swimming in this treasure trove of ideas.
A true masterpiece on how to express mathematics intuitively!
Geometry and Imagination, which laid the foundation for modern geometry teaching methods
This book was compiled and published by Hilbert's student, Stefan Cohn-Posen, based on lectures he gave at the university over the three years before his retirement.
Through this book, Hilbert hoped to expose many areas of geometry that could be seen and intuitively understood, and to make geometry more accessible to those studying it.
The efforts of many people have added various images, and by drawing and presenting rather complex or difficult-to-imagine stereoscopic projections, we have reduced the difficulty of having to mentally imagine various axioms and geometric assumptions.
Geometry and Imagination is one of the few classics that is still widely read, even after more than half a century since its publication.
Still fresh as time goes by, this book is full of clear and elegant ideas that explain mathematics well.
I hope that both beginners and experts in mathematics will enjoy swimming in this treasure trove of ideas.
- You can preview some of the book's contents.
Preview
index
introduction
Chapter 1: The Simplest Curves and Surfaces
1.1 Plane curve
1.2 Cylinders, cones, conic sections, and surfaces of revolution
1.3 Quadratic surface
1.4 Construction of ellipsoids using strings and isotropic quadratic surfaces
Chapter 1 Appendix
1.
Drawing the starting point of a conic section
2.
Directrix of a conic section
3.
Movable rod model of a hyperboloid
Chapter 2 Regular Dot System
2.1 Plane grid
2.2 Plane lattices in number theory
2.3 Lattice in three or more dimensions
2.4 Decision as a regular point system
2.5 Regular point system and discontinuous motion group
2.6 Planar motion and composition.
Classification of discontinuous groups of plane motion
2.7 Discontinuous group of plane motions with infinite fundamental domain
2.8 Crystallographic group of plane motion.
Regular dot system and regular directional dot system.
Divide the plane into congruent regions
2.9 Decision classes, motion groups in space, groups with left-right symmetry, and point systems
2.10 Regular polyhedron
Chapter 3 Projective Layout
3.1 Preliminary observations on the floor plan
3.2 (7£) and (8£) batches
3.3 (9£) batch
3.4 Perspective, imaginary elements, and duality in the plane
3.5 Virtual elements and the principle of duality in space.
Desargue's Theorem and Desargue's Placement Blank (10£)
3.6 Comparison of Pascal's theorem and Desargues's theorem
3.7 Preliminary observations on the layout of the space
3.8 Raye Deployment
3.9 Regular hyperhedra and projections in three and four dimensions
3.10 Counting Methods in Geometry
3.11 Schlafly's Twin Six
Chapter 4 Differential Geometry
4.1 Plane curves
4.2 Space Curve
4.3 Curvature of the surface.
Elliptic points, hyperbolic points, parabolic points.
Curvature lines and asymptotic curves.
navel point, microscopic curve, monkey saddle
4.4 Spherical surface and Gaussian curvature
4.5 Development surface, mother line surface
4.6 Twisting the space curve
4.7 Eleven Properties of the Ball
4.8 Bending while preserving the surface
4.9 Elliptical Geometry
4.10 Relationships between hyperbolic, Euclidean, and elliptic geometry
4.11 Transformations that preserve polar projection and circles.
Poincaré model of the hyperbolic plane
4.12 Appearance map, area-preserving map, geodesic map, continuous map, conformal map
4.13 Geometric function theory.
Riemann's theorem.
Conformal mapping in space
4.14 Conformal mapping of surfaces.
Minimal surface.
Plato's problem
Chapter 5 Kinematics
5.1 Linkage Device
5.2 Continuous rigid body motion of plane figures
5.3 Device for drawing ellipses and circles
5.4 Continuous motion in space
Chapter 6 Topology
6.1 Polyhedrons
6.2 Curved surfaces
6.3 Cross-section surface
6.4 Projective plane as a closed surface
6.5 Standard form of a surface with finite connectivity
6.6 Topological mapping of self-over-self on a surface.
Floating point.
Class of thought.
Universal cover surface of torus
6.7 Conformal mapping of torus
6.8 Problems of adjacent areas, real problems, and coloring problems
Chapter 6 Appendix
1.
Projective plane in four-dimensional space
2.
Euclidean plane in four-dimensional space
References
Translator's Note
Search
Chapter 1: The Simplest Curves and Surfaces
1.1 Plane curve
1.2 Cylinders, cones, conic sections, and surfaces of revolution
1.3 Quadratic surface
1.4 Construction of ellipsoids using strings and isotropic quadratic surfaces
Chapter 1 Appendix
1.
Drawing the starting point of a conic section
2.
Directrix of a conic section
3.
Movable rod model of a hyperboloid
Chapter 2 Regular Dot System
2.1 Plane grid
2.2 Plane lattices in number theory
2.3 Lattice in three or more dimensions
2.4 Decision as a regular point system
2.5 Regular point system and discontinuous motion group
2.6 Planar motion and composition.
Classification of discontinuous groups of plane motion
2.7 Discontinuous group of plane motions with infinite fundamental domain
2.8 Crystallographic group of plane motion.
Regular dot system and regular directional dot system.
Divide the plane into congruent regions
2.9 Decision classes, motion groups in space, groups with left-right symmetry, and point systems
2.10 Regular polyhedron
Chapter 3 Projective Layout
3.1 Preliminary observations on the floor plan
3.2 (7£) and (8£) batches
3.3 (9£) batch
3.4 Perspective, imaginary elements, and duality in the plane
3.5 Virtual elements and the principle of duality in space.
Desargue's Theorem and Desargue's Placement Blank (10£)
3.6 Comparison of Pascal's theorem and Desargues's theorem
3.7 Preliminary observations on the layout of the space
3.8 Raye Deployment
3.9 Regular hyperhedra and projections in three and four dimensions
3.10 Counting Methods in Geometry
3.11 Schlafly's Twin Six
Chapter 4 Differential Geometry
4.1 Plane curves
4.2 Space Curve
4.3 Curvature of the surface.
Elliptic points, hyperbolic points, parabolic points.
Curvature lines and asymptotic curves.
navel point, microscopic curve, monkey saddle
4.4 Spherical surface and Gaussian curvature
4.5 Development surface, mother line surface
4.6 Twisting the space curve
4.7 Eleven Properties of the Ball
4.8 Bending while preserving the surface
4.9 Elliptical Geometry
4.10 Relationships between hyperbolic, Euclidean, and elliptic geometry
4.11 Transformations that preserve polar projection and circles.
Poincaré model of the hyperbolic plane
4.12 Appearance map, area-preserving map, geodesic map, continuous map, conformal map
4.13 Geometric function theory.
Riemann's theorem.
Conformal mapping in space
4.14 Conformal mapping of surfaces.
Minimal surface.
Plato's problem
Chapter 5 Kinematics
5.1 Linkage Device
5.2 Continuous rigid body motion of plane figures
5.3 Device for drawing ellipses and circles
5.4 Continuous motion in space
Chapter 6 Topology
6.1 Polyhedrons
6.2 Curved surfaces
6.3 Cross-section surface
6.4 Projective plane as a closed surface
6.5 Standard form of a surface with finite connectivity
6.6 Topological mapping of self-over-self on a surface.
Floating point.
Class of thought.
Universal cover surface of torus
6.7 Conformal mapping of torus
6.8 Problems of adjacent areas, real problems, and coloring problems
Chapter 6 Appendix
1.
Projective plane in four-dimensional space
2.
Euclidean plane in four-dimensional space
References
Translator's Note
Search
Into the book
The purpose of this book is to present geometry visually and intuitively.
Geometric facts and problems on manifolds can be explained with the help of visual imagery, and in many cases the geometric outlines of research and proof methods can be grasped without rigorously defining concepts or going into detail about actual calculations.
For example, the fact that a ball with a hole in it can be bent—no matter how small the hole is—or that two different torus-shaped surfaces cannot wrap around each other while preserving their angles can be treated in a way that gives a sense of why and how, even for someone who doesn't want to follow the detailed analytic arguments.
Geometric facts and problems on manifolds can be explained with the help of visual imagery, and in many cases the geometric outlines of research and proof methods can be grasped without rigorously defining concepts or going into detail about actual calculations.
For example, the fact that a ball with a hole in it can be bent—no matter how small the hole is—or that two different torus-shaped surfaces cannot wrap around each other while preserving their angles can be treated in a way that gives a sense of why and how, even for someone who doesn't want to follow the detailed analytic arguments.
---p.5
So far we have discussed hypercurves, but now it is time to consider other curves formed by pairs of different types of surfaces within the same hypercurve family.
As we will discuss later, these curves have simple properties in differential geometry (see page 258).
These provide the first examples of curves that do not lie in a plane.
A curve formed by the intersection of any two quadratic surfaces in general positions cannot intersect any plane at more than five points unless the entire plane is contained within the same plane.
This is because it can be easily proved analytically that two conic sections that do not coincide and do not share an entire straight line cannot meet at more than five points when a plane intersects each surface (see page 221). This theorem is intuitively obvious.
So far we have discussed hypercurves, but now it is time to consider other curves formed by pairs of different types of surfaces within the same hypercurve family.
As we will discuss later, these curves have simple properties in differential geometry (see page 258).
These provide the first examples of curves that do not lie in a plane.
A curve formed by the intersection of any two quadratic surfaces in general positions cannot intersect any plane at more than five points unless the entire plane is contained within the same plane.
This is because it can be easily proved analytically that two conic sections that do not coincide and do not share an entire straight line cannot meet at more than five points when a plane intersects each surface (see page 221). This theorem is intuitively obvious.
As we will discuss later, these curves have simple properties in differential geometry (see page 258).
These provide the first examples of curves that do not lie in a plane.
A curve formed by the intersection of any two quadratic surfaces in general positions cannot intersect any plane at more than five points unless the entire plane is contained within the same plane.
This is because it can be easily proved analytically that two conic sections that do not coincide and do not share an entire straight line cannot meet at more than five points when a plane intersects each surface (see page 221). This theorem is intuitively obvious.
So far we have discussed hypercurves, but now it is time to consider other curves formed by pairs of different types of surfaces within the same hypercurve family.
As we will discuss later, these curves have simple properties in differential geometry (see page 258).
These provide the first examples of curves that do not lie in a plane.
A curve formed by the intersection of any two quadratic surfaces in general positions cannot intersect any plane at more than five points unless the entire plane is contained within the same plane.
This is because it can be easily proved analytically that two conic sections that do not coincide and do not share an entire straight line cannot meet at more than five points when a plane intersects each surface (see page 221). This theorem is intuitively obvious.
---p.42
Minkowski's successful proof of the theorem on lattices, despite its simplicity, solved many problems in number theory that could not be addressed otherwise.
For the sake of clarity, we will not describe this theorem in its most general form here, but will confine ourselves to a special case that is easy to formalize while still capturing the essence of the theorem.
The summary is as follows.
If a square with side length 2 is superimposed on any plane unit grid so that its center lies on a lattice point, there must be another lattice point inside or on the boundary of the square.
Minkowski's successful proof of the theorem on lattices, despite its simplicity, solved many problems in number theory that could not be addressed otherwise.
For the sake of clarity, we will not describe this theorem in its most general form here, but will confine ourselves to a special case that is easy to formalize while still capturing the essence of the theorem.
The summary is as follows.
If a square with side length 2 is superimposed on any plane unit grid so that its center lies on a lattice point, there must be another lattice point inside or on the boundary of the square.
For the sake of clarity, we will not describe this theorem in its most general form here, but will confine ourselves to a special case that is easy to formalize while still capturing the essence of the theorem.
The summary is as follows.
If a square with side length 2 is superimposed on any plane unit grid so that its center lies on a lattice point, there must be another lattice point inside or on the boundary of the square.
Minkowski's successful proof of the theorem on lattices, despite its simplicity, solved many problems in number theory that could not be addressed otherwise.
For the sake of clarity, we will not describe this theorem in its most general form here, but will confine ourselves to a special case that is easy to formalize while still capturing the essence of the theorem.
The summary is as follows.
If a square with side length 2 is superimposed on any plane unit grid so that its center lies on a lattice point, there must be another lattice point inside or on the boundary of the square.
---p.69
By simply adding reflection to the true motion, we obtain all the various crystal structures found in nature.
… … Instead of geometric methods, we can also find the discontinuous group of symmetry transformations using algebraic methods.
When this method is used on the plane, it reveals surprising relationships between complex numbers, and in space it becomes a method based on the hyper-complex number system.
Generalizing the current discussion to higher-dimensional spaces is an interesting problem.
We have discovered some results regarding the discontinuous group of symmetry transformations in high-dimensional spaces, and we also know the figures corresponding to regular polyhedra in spaces of arbitrary dimensions.
Higher-dimensional shapes are discussed further in the next chapter.
Moreover, Bieberbach proved that for all n, there are a finite number of n-dimensional crystallographic groups, and each of these groups contains n linearly independent translations.
By simply adding reflection to the true motion, we obtain all the various crystal structures found in nature.
… … Instead of geometric methods, we can also find the discontinuous group of symmetry transformations using algebraic methods.
When this method is used on the plane, it reveals surprising relationships between complex numbers, and in space it becomes a method based on the hyper-complex number system.
Generalizing the current discussion to higher-dimensional spaces is an interesting problem.
We have discovered some results regarding the discontinuous group of symmetry transformations in high-dimensional spaces, and we also know the figures corresponding to regular polyhedra in spaces of arbitrary dimensions.
Higher-dimensional shapes are discussed further in the next chapter.
Moreover, Bieberbach proved that for all n, there are a finite number of n-dimensional crystallographic groups, and each of these groups contains n linearly independent translations.
… … Instead of geometric methods, we can also find the discontinuous group of symmetry transformations using algebraic methods.
When this method is used on the plane, it reveals surprising relationships between complex numbers, and in space it becomes a method based on the hyper-complex number system.
Generalizing the current discussion to higher-dimensional spaces is an interesting problem.
We have discovered some results regarding the discontinuous group of symmetry transformations in high-dimensional spaces, and we also know the figures corresponding to regular polyhedra in spaces of arbitrary dimensions.
Higher-dimensional shapes are discussed further in the next chapter.
Moreover, Bieberbach proved that for all n, there are a finite number of n-dimensional crystallographic groups, and each of these groups contains n linearly independent translations.
By simply adding reflection to the true motion, we obtain all the various crystal structures found in nature.
… … Instead of geometric methods, we can also find the discontinuous group of symmetry transformations using algebraic methods.
When this method is used on the plane, it reveals surprising relationships between complex numbers, and in space it becomes a method based on the hyper-complex number system.
Generalizing the current discussion to higher-dimensional spaces is an interesting problem.
We have discovered some results regarding the discontinuous group of symmetry transformations in high-dimensional spaces, and we also know the figures corresponding to regular polyhedra in spaces of arbitrary dimensions.
Higher-dimensional shapes are discussed further in the next chapter.
Moreover, Bieberbach proved that for all n, there are a finite number of n-dimensional crystallographic groups, and each of these groups contains n linearly independent translations.
---p.133
We have seen that Desargues' theorem and Pascal's fourth theorem are similar in many ways.
Both theorems were proven through projections of three-dimensional figures.
Both theorems are quite similar in that they both give arrangements, are regular, self-dual, can be constructed by themselves, automatically satisfy the last associative condition, and can be considered as self-inscribed and self-circumscribed polygons.
Nevertheless, there is a fundamental difference between the two theorems.
The spatial figure used in the proof of Desargues' theorem could be constructed using only the axiom of associativity in space without adding any other axioms.
On the other hand, the Pascal-Brianchon configuration was obtained by studying quadratic surfaces.
Of course, the core of the proof seems to be a problem of considering purely the associative relationships between points, lines, and planes in a spatial hexagon, but a closer examination reveals that constructing such a hexagon is essentially the same as constructing a quadratic matrix surface, and the possibility of such a construction cannot be proven by the associative axiom alone.
We have seen that Desargues' theorem and Pascal's fourth theorem are similar in many ways.
Both theorems were proven through projections of three-dimensional figures.
Both theorems are quite similar in that they both give arrangements, are regular, self-dual, can be constructed by themselves, automatically satisfy the last associative condition, and can be considered as self-inscribed and self-circumscribed polygons.
Nevertheless, there is a fundamental difference between the two theorems.
The spatial figure used in the proof of Desargues' theorem could be constructed using only the axiom of associativity in space without adding any other axioms.
On the other hand, the Pascal-Brianchon configuration was obtained by studying quadratic surfaces.
Of course, the core of the proof seems to be a problem of considering purely the associative relationships between points, lines, and planes in a spatial hexagon, but a closer examination reveals that constructing such a hexagon is essentially the same as constructing a quadratic matrix surface, and the possibility of such a construction cannot be proven by the associative axiom alone.
Both theorems were proven through projections of three-dimensional figures.
Both theorems are quite similar in that they both give arrangements, are regular, self-dual, can be constructed by themselves, automatically satisfy the last associative condition, and can be considered as self-inscribed and self-circumscribed polygons.
Nevertheless, there is a fundamental difference between the two theorems.
The spatial figure used in the proof of Desargues' theorem could be constructed using only the axiom of associativity in space without adding any other axioms.
On the other hand, the Pascal-Brianchon configuration was obtained by studying quadratic surfaces.
Of course, the core of the proof seems to be a problem of considering purely the associative relationships between points, lines, and planes in a spatial hexagon, but a closer examination reveals that constructing such a hexagon is essentially the same as constructing a quadratic matrix surface, and the possibility of such a construction cannot be proven by the associative axiom alone.
We have seen that Desargues' theorem and Pascal's fourth theorem are similar in many ways.
Both theorems were proven through projections of three-dimensional figures.
Both theorems are quite similar in that they both give arrangements, are regular, self-dual, can be constructed by themselves, automatically satisfy the last associative condition, and can be considered as self-inscribed and self-circumscribed polygons.
Nevertheless, there is a fundamental difference between the two theorems.
The spatial figure used in the proof of Desargues' theorem could be constructed using only the axiom of associativity in space without adding any other axioms.
On the other hand, the Pascal-Brianchon configuration was obtained by studying quadratic surfaces.
Of course, the core of the proof seems to be a problem of considering purely the associative relationships between points, lines, and planes in a spatial hexagon, but a closer examination reveals that constructing such a hexagon is essentially the same as constructing a quadratic matrix surface, and the possibility of such a construction cannot be proven by the associative axiom alone.
---pp.185~196
By starting with the axioms of Euclidean plane geometry and determining whether each axiom is valid in elliptic geometry or needs to be replaced with a relaxed axiom, we can fully understand how elliptic geometry is related to Euclidean geometry.
We have already mentioned the conjunction axiom (p. 174) and the continuity axiom (p. 187, 188).
Euclidean plane geometry can be composed of five types of axiom groups: associative axiom group, ordered axiom group, congruent axiom group, parallel axiom group, and continuous axiom group.
Each axiom group contains several basic concepts, for example, the associative axiom is based on the concepts of point, line, and conjunction.
Among the additional concepts, some are possible only if there are certain axioms.
For example, the concept of a line segment or a ray is only possible if there is an order axiom.
Since the concept of line segment forms the basis of the axiom of congruence, a certain axiom of order must be assumed in advance to describe the axiom of congruence.
By starting with the axioms of Euclidean plane geometry and determining whether each axiom is valid in elliptic geometry or needs to be replaced with a relaxed axiom, we can fully understand how elliptic geometry is related to Euclidean geometry.
We have already mentioned the conjunction axiom (p. 174) and the continuity axiom (p. 187, 188).
Euclidean plane geometry can be composed of five types of axiom groups: associative axiom group, ordered axiom group, congruent axiom group, parallel axiom group, and continuous axiom group.
Each axiom group contains several basic concepts, for example, the associative axiom is based on the concepts of point, line, and conjunction.
Among the additional concepts, some are possible only if there are certain axioms.
For example, the concept of a line segment or a ray is only possible if there is an order axiom.
Since the concept of line segment forms the basis of the axiom of congruence, a certain axiom of order must be assumed in advance to describe the axiom of congruence.
We have already mentioned the conjunction axiom (p. 174) and the continuity axiom (p. 187, 188).
Euclidean plane geometry can be composed of five types of axiom groups: associative axiom group, ordered axiom group, congruent axiom group, parallel axiom group, and continuous axiom group.
Each axiom group contains several basic concepts, for example, the associative axiom is based on the concepts of point, line, and conjunction.
Among the additional concepts, some are possible only if there are certain axioms.
For example, the concept of a line segment or a ray is only possible if there is an order axiom.
Since the concept of line segment forms the basis of the axiom of congruence, a certain axiom of order must be assumed in advance to describe the axiom of congruence.
By starting with the axioms of Euclidean plane geometry and determining whether each axiom is valid in elliptic geometry or needs to be replaced with a relaxed axiom, we can fully understand how elliptic geometry is related to Euclidean geometry.
We have already mentioned the conjunction axiom (p. 174) and the continuity axiom (p. 187, 188).
Euclidean plane geometry can be composed of five types of axiom groups: associative axiom group, ordered axiom group, congruent axiom group, parallel axiom group, and continuous axiom group.
Each axiom group contains several basic concepts, for example, the associative axiom is based on the concepts of point, line, and conjunction.
Among the additional concepts, some are possible only if there are certain axioms.
For example, the concept of a line segment or a ray is only possible if there is an order axiom.
Since the concept of line segment forms the basis of the axiom of congruence, a certain axiom of order must be assumed in advance to describe the axiom of congruence.
---pp.319~320
So far, we have mainly studied 'fixed' objects in space, as these are the starting point for the study of geometry.
However, the concept of 'motion' is also used in the elements of geometry.
If two shapes can be superimposed using rigid body motion, they are congruent.
Furthermore, we studied movable hyperboloids (see page 23), determined the mother plane through a moving plane (pages 278-279), and even bent and twisted surfaces (Chapter 4).
Kinesiology is the systematic study of movement.
First, we will begin studying the linkage mechanism, which is closely related to the basic distance geometry of kinematics.
Afterwards, we discuss more general continuous motion using differential geometry methods.
So far, we have mainly studied 'fixed' objects in space, as these are the starting point for the study of geometry.
However, the concept of 'motion' is also used in the elements of geometry.
If two shapes can be superimposed using rigid body motion, they are congruent.
Furthermore, we studied movable hyperboloids (see page 23), determined the mother plane through a moving plane (pages 278-279), and even bent and twisted surfaces (Chapter 4).
Kinesiology is the systematic study of movement.
First, we will begin studying the linkage mechanism, which is closely related to the basic distance geometry of kinematics.
Afterwards, we discuss more general continuous motion using differential geometry methods.
However, the concept of 'motion' is also used in the elements of geometry.
If two shapes can be superimposed using rigid body motion, they are congruent.
Furthermore, we studied movable hyperboloids (see page 23), determined the mother plane through a moving plane (pages 278-279), and even bent and twisted surfaces (Chapter 4).
Kinesiology is the systematic study of movement.
First, we will begin studying the linkage mechanism, which is closely related to the basic distance geometry of kinematics.
Afterwards, we discuss more general continuous motion using differential geometry methods.
So far, we have mainly studied 'fixed' objects in space, as these are the starting point for the study of geometry.
However, the concept of 'motion' is also used in the elements of geometry.
If two shapes can be superimposed using rigid body motion, they are congruent.
Furthermore, we studied movable hyperboloids (see page 23), determined the mother plane through a moving plane (pages 278-279), and even bent and twisted surfaces (Chapter 4).
Kinesiology is the systematic study of movement.
First, we will begin studying the linkage mechanism, which is closely related to the basic distance geometry of kinematics.
Afterwards, we discuss more general continuous motion using differential geometry methods.
---p.363
A problem closely related to the adjacent region problem is the coloring problem.
This problem can be expressed as a practical cartographic problem as follows:
Let's say there are several areas drawn on a curved surface.
Color each area, and color the areas that border each other along the curve with different colors (if two areas meet only at an isolation point, they can be colored the same color).
The problem is to find the minimum number of colors sufficient to color the region for all possible segmentations on the surface without violating these rules.
A problem closely related to the adjacent region problem is the coloring problem.
This problem can be expressed as a practical cartographic problem as follows:
Let's say there are several areas drawn on a curved surface.
Color each area, and color the areas that border each other along the curve with different colors (if two areas meet only at an isolation point, they can be colored the same color).
The problem is to find the minimum number of colors sufficient to color the region for all possible segmentations on the surface without violating these rules.
This problem can be expressed as a practical cartographic problem as follows:
Let's say there are several areas drawn on a curved surface.
Color each area, and color the areas that border each other along the curve with different colors (if two areas meet only at an isolation point, they can be colored the same color).
The problem is to find the minimum number of colors sufficient to color the region for all possible segmentations on the surface without violating these rules.
A problem closely related to the adjacent region problem is the coloring problem.
This problem can be expressed as a practical cartographic problem as follows:
Let's say there are several areas drawn on a curved surface.
Color each area, and color the areas that border each other along the curve with different colors (if two areas meet only at an isolation point, they can be colored the same color).
The problem is to find the minimum number of colors sufficient to color the region for all possible segmentations on the surface without violating these rules.
---p.448
Publisher's Review
A treasure trove of mathematical ideas
A true masterpiece on how to express mathematics intuitively!
▶ Introduction
Geometry and Imagination, which laid the foundation for modern geometry teaching methods
This book was compiled and published by Hilbert's student, Stefan Cohn-Posen, based on lectures he gave at the university over the three years before his retirement.
The original title is 『Anschauliche Geometrie』, which would be translated directly into Korean as ‘Intuitive Geometry’, but the title of the English edition, 『Geometry and the Imagination』, was borrowed.
Although Hilbert is known as a mathematician who advocated formalism against intuitionism, his attitude of valuing intuition and imagination, which can be seen in this book, makes it difficult to simply label him as a formalist.
Through this book, Hilbert hoped to expose many areas of geometry that could be seen and intuitively understood, and to make geometry more accessible to those studying it.
The efforts of many people have added various images, and by drawing and presenting rather complex or difficult-to-imagine stereoscopic projections, we have reduced the difficulty of having to mentally imagine various axioms and geometric assumptions.
The fact that projections of shapes or real objects are indispensable in modern geometry is due to Hilbert's efforts.
The German mathematical community reached its peak during the Hilbert era, centered in Göttingen, and then began to decline with the rise of Nazism.
Traces of this unfortunate history remain in the bibliographic information of 『Geometry and Imagination』.
This is the case of Stefan Cohn-Posen, a student of Hilbert who participated in writing this book as a co-author.
Cohn-Posen was Jewish, and at the time, Germany did not allow Jews any rights due to Nazism, so he was not listed as a co-author at the time of publication. He was only recognized as a co-author after his death when the English version was translated.
Hilbert, a mathematician who left a mark on the history of 20th-century mathematics
It is no exaggeration to say that the mathematics of the first half of the 20th century was entirely the achievements of the German mathematical community.
At that time, the mathematical world was focused on organizing unestablished axioms and on topology and set theory, and Hilbert influenced all of these fields and established a trend in the philosophy of mathematics.
Hilbert's greatest achievements during his prime were in the fields of function theory and physics.
Hilbert broadened his understanding of infinite-dimensional spaces while studying the theory of integral equations.
Today, the term "Hilbert space" has become an inescapable term for both physicists and mathematicians.
In particular, he made contributions to the theory of spectral decomposition in this space, which became the basis of Heisenberg and Schrödinger's quantum mechanics in 1925.
Meanwhile, variational calculus, an important methodology in mathematical physics, is also a subject to which Hilbert devoted great effort, and this can be frequently seen in Chapter 4 of this book.
The importance of Hilbert's work can be gauged from the fact that Courant, in writing his book "Methodology of Mathematical Physics," quoted many of Hilbert's lectures and papers, and even listed Hilbert as a co-author, citing his decisive influence on mathematical research and education.
The German mathematical community reached its peak during the Hilbert era, centered in Göttingen.
However, in his later years, Hilbert himself was caught up in the turmoil of war and had to watch the world of mathematics he loved collapse.
Few of his colleagues were able to attend the funeral of this great mathematician, a brilliant researcher, writer, and lecturer.
What were the problems that Hilbert pondered so deeply?
"Geometry and Imagination" also provides an insight into the problems that interested Hilbert.
It is particularly interesting that Hilbert also took an interest in the speculative problem of "How many colors are needed to paint the borders of each country without overlapping?"
Moreover, this book is not only about the intuitive representation of geometry, but also shows very simply how geometric intuition can be translated into useful ideas in a wide range of fields, including algebra and kinematics.
For example, after considering the unit cell, we can easily derive the Leibniz series by adding a little number theory when necessary, or in the section on lattices in three and higher dimensions, we can deal with spherical packing problems, including the famous Kepler problem.
The influence of Hilbert and Cohn-Posen's books in this century cannot be overstated.
The most striking chapter is that on "projective arrangements," in which Hilbert and Cohn-Posen provide a brief introduction that concisely and clearly explains why geometers should be interested in projective geometry and how they came to the idea of describing complex structures with such a simple setup.
The chapter on kinematics contains a nice argument about the geometry of point and rod arrangements, which are in some ways constrained by their connection to linkages.
In geometry, this topic is becoming increasingly important in modern times, especially in robotics.
It's another example of how to incorporate rich geometry into a simple situation.
Geometry and Imagination is one of the few classics that is still widely read, even after more than half a century since its publication.
Still fresh as time goes by, this book is full of clear and elegant ideas that explain mathematics well.
I hope that both beginners and experts in mathematics will enjoy swimming in this treasure trove of ideas.
Classics of mathematical history being introduced to our reading world.
The rich inspiration contained in the original work remains intact!
Hilbert and Cohn-Posen's "Geometry and Imagination" is the second volume of the Sallim Math Classic series, published following Euler's "Elements of Algebra."
While we say that the history of mathematics is the foundation and basis for understanding the history of culture and thought, in reality, there has been almost no translation of classic works of the history of mathematics in our intellectual community.
In that sense, the Sallim Math Classics series will be a valuable project that fills a void in our publishing culture.
We ask for your warm interest and support for this project, which introduces masterpieces in the history of mathematics accessible to the general public, including Hilbert's "Foundations of Geometry."
A true masterpiece on how to express mathematics intuitively!
▶ Introduction
Geometry and Imagination, which laid the foundation for modern geometry teaching methods
This book was compiled and published by Hilbert's student, Stefan Cohn-Posen, based on lectures he gave at the university over the three years before his retirement.
The original title is 『Anschauliche Geometrie』, which would be translated directly into Korean as ‘Intuitive Geometry’, but the title of the English edition, 『Geometry and the Imagination』, was borrowed.
Although Hilbert is known as a mathematician who advocated formalism against intuitionism, his attitude of valuing intuition and imagination, which can be seen in this book, makes it difficult to simply label him as a formalist.
Through this book, Hilbert hoped to expose many areas of geometry that could be seen and intuitively understood, and to make geometry more accessible to those studying it.
The efforts of many people have added various images, and by drawing and presenting rather complex or difficult-to-imagine stereoscopic projections, we have reduced the difficulty of having to mentally imagine various axioms and geometric assumptions.
The fact that projections of shapes or real objects are indispensable in modern geometry is due to Hilbert's efforts.
The German mathematical community reached its peak during the Hilbert era, centered in Göttingen, and then began to decline with the rise of Nazism.
Traces of this unfortunate history remain in the bibliographic information of 『Geometry and Imagination』.
This is the case of Stefan Cohn-Posen, a student of Hilbert who participated in writing this book as a co-author.
Cohn-Posen was Jewish, and at the time, Germany did not allow Jews any rights due to Nazism, so he was not listed as a co-author at the time of publication. He was only recognized as a co-author after his death when the English version was translated.
Hilbert, a mathematician who left a mark on the history of 20th-century mathematics
It is no exaggeration to say that the mathematics of the first half of the 20th century was entirely the achievements of the German mathematical community.
At that time, the mathematical world was focused on organizing unestablished axioms and on topology and set theory, and Hilbert influenced all of these fields and established a trend in the philosophy of mathematics.
Hilbert's greatest achievements during his prime were in the fields of function theory and physics.
Hilbert broadened his understanding of infinite-dimensional spaces while studying the theory of integral equations.
Today, the term "Hilbert space" has become an inescapable term for both physicists and mathematicians.
In particular, he made contributions to the theory of spectral decomposition in this space, which became the basis of Heisenberg and Schrödinger's quantum mechanics in 1925.
Meanwhile, variational calculus, an important methodology in mathematical physics, is also a subject to which Hilbert devoted great effort, and this can be frequently seen in Chapter 4 of this book.
The importance of Hilbert's work can be gauged from the fact that Courant, in writing his book "Methodology of Mathematical Physics," quoted many of Hilbert's lectures and papers, and even listed Hilbert as a co-author, citing his decisive influence on mathematical research and education.
The German mathematical community reached its peak during the Hilbert era, centered in Göttingen.
However, in his later years, Hilbert himself was caught up in the turmoil of war and had to watch the world of mathematics he loved collapse.
Few of his colleagues were able to attend the funeral of this great mathematician, a brilliant researcher, writer, and lecturer.
What were the problems that Hilbert pondered so deeply?
"Geometry and Imagination" also provides an insight into the problems that interested Hilbert.
It is particularly interesting that Hilbert also took an interest in the speculative problem of "How many colors are needed to paint the borders of each country without overlapping?"
Moreover, this book is not only about the intuitive representation of geometry, but also shows very simply how geometric intuition can be translated into useful ideas in a wide range of fields, including algebra and kinematics.
For example, after considering the unit cell, we can easily derive the Leibniz series by adding a little number theory when necessary, or in the section on lattices in three and higher dimensions, we can deal with spherical packing problems, including the famous Kepler problem.
The influence of Hilbert and Cohn-Posen's books in this century cannot be overstated.
The most striking chapter is that on "projective arrangements," in which Hilbert and Cohn-Posen provide a brief introduction that concisely and clearly explains why geometers should be interested in projective geometry and how they came to the idea of describing complex structures with such a simple setup.
The chapter on kinematics contains a nice argument about the geometry of point and rod arrangements, which are in some ways constrained by their connection to linkages.
In geometry, this topic is becoming increasingly important in modern times, especially in robotics.
It's another example of how to incorporate rich geometry into a simple situation.
Geometry and Imagination is one of the few classics that is still widely read, even after more than half a century since its publication.
Still fresh as time goes by, this book is full of clear and elegant ideas that explain mathematics well.
I hope that both beginners and experts in mathematics will enjoy swimming in this treasure trove of ideas.
Classics of mathematical history being introduced to our reading world.
The rich inspiration contained in the original work remains intact!
Hilbert and Cohn-Posen's "Geometry and Imagination" is the second volume of the Sallim Math Classic series, published following Euler's "Elements of Algebra."
While we say that the history of mathematics is the foundation and basis for understanding the history of culture and thought, in reality, there has been almost no translation of classic works of the history of mathematics in our intellectual community.
In that sense, the Sallim Math Classics series will be a valuable project that fills a void in our publishing culture.
We ask for your warm interest and support for this project, which introduces masterpieces in the history of mathematics accessible to the general public, including Hilbert's "Foundations of Geometry."
GOODS SPECIFICS
- Date of publication: April 27, 2012
- Format: Hardcover book binding method guide
- Page count, weight, size: 472 pages | 944g | 152*225*30mm
- ISBN13: 9788952218384
- ISBN10: 8952218388
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