A dynamic visualization of the hyperbolic geometry

szilassilajos@t-online.hu, Oct 29, 2018

“I have created a new, different world from nothing...” (János Bolyai) Most of the primary and secondary school curriculum is based on Euclidean geometry. Our students are somewhat familiar with the life and works of János Bolyai’s, but his geometry, what is more, non-Euclidean geometries in general are completely unknown even for most secondary or university students. Since it is impossible to study the topic precisely in an axiomatic way, teachers do not even exploit any opportunities for abstraction. In fact, if we can make non-Euclidean geometry visual, we can even enlighten the basic concepts of the “well-known” Euclidean geometry, including some simpler statements of it. We shall see that the well-known Euclidean geometry and Bolyai’s geometry, being visualized here, have common roots: all concepts before defining the axiom of parallels are valid in both geometries. In this way we can use the model presented here to provide a new approach to teach elementary geometry in secondary school as well. This GeoGebra Book uses only the basics from secondary school in order to define the main concepts of hyperbolic geometry. We do not want to explain general knowledge on non-Euclidean geometries. Instead, a visual picture book is presented to invite the reader to learn more on the topic. Let us see then now, how Bolyai’s “new, different world” looks like! [b]Note: This book is the English translation of a [url=https://www.geogebra.org/m/NSQ9meGe]Hungarian GeoGebra book on the same topic[/url]. The translation is an on-going work, so please expect several missing translations. However, all applets are already translated. In case the English text is missing, you may be interested in the original Hungarian text and try to translate it automatically in English, e.g. with [url=https://translate.google.com/]Google Translator[/url].[/b]

Table of Contents

  1. 01 Building up a geometry system with axioms
    1. 0101 A system of axioms in geometry as introduced in the geometry class
  2. 02 Models in geometry
    1. 0201 The model: the Poincaré model of hyperbolic geometry
    2. 0202 Tools in the P-model
  3. 03 Absolute geometry—hyperbolic geometry
    1. 0301 Absolute geometrical relations in the P-model
    2. 0302 Displaying reflection axioms in the P-model
  4. 04 Triangles
    1. 0401 Remarkable points and lines in a triangle
  5. 05 Problems (Congruence)
    1. 0501 Challenges
    2. 0502 Circle given with its center and radius
    3. 0503 Circle given with its diameter
    4. 0504 Measuring two segments
    5. 0505 Relationship among the sides of a triangle
    6. 0506 Point reflection
    7. 0507 Rotation
    8. 0508 Integers on the number line
    9. 0509 Translation parallel to a line by a multiple of the unit
    10. 0510 Translation parallel to a line by a given segment
    11. 0511 Concatenation of two reflections
  6. 06 Measuring segments and angles
    1. 0601 Procedures for measuring in the P-model
  7. 07 Problems (Measuring)
    1. 0701 Challenges
    2. 0702 Defect of a triangle
    3. 0703 Copying angles
    4. 0704 Isosceles triangles
    5. 0705 Defining a number line with a function
    6. 0706 Relationship between sides and angles in a triangle
    7. 0707 Is Thales' circle theorem valid in hyperbolic geometry?
    8. 0708 Regular polygons
    9. 0709 Constructing ultraparallel lines
    10. 0710 Angle and distance of parallelism
  8. 08 Trigonometry
    1. 0801 Law of sines and cosines
    2. 0802 Basic cases of constructing a triangle
  9. 09 Tesselations
    1. 0901 A tesselation with triangles
    2. 0902 Circles in a triangular grid
    3. 0903 A triangular tesselation
    4. 0904 M.C. Escher: Circle limit I
    5. 0905 A tesselation of squares in the P-model
  10. 10 Pencils and cycles
    1. 1001 Pencils and cycles
    2. 1002 Plotting a graph in the P-model
  11. 11 Advanced problems
    1. 1101 Problems and constructions without tools of measurement
    2. 1102 Constructing tangents to a circle through an external point
    3. 1103 Cyclic quadrilaterals
    4. 1104 Parallelogram
    5. 1105 Midpoints of a triangle
    6. 1106 Triangles with equal areas
    7. 1107 A contest problem
    8. 1108 A lemma
    9. 1109 Circles that are tangent to each other and to a line
    10. 1110 Three circles
    11. 1111 Constructing lines that are perpendicular to a given line and tangent to a given circle
    12. 1112 Common tangents of two circles

1.

01 Building up a geometry system with axioms

  • 1. 0101 A system of axioms in geometry as introduced in the geometry class

1.1.

–

2.

02 Models in geometry

  • 1. 0201 The model: the Poincaré model of hyperbolic geometry
  • 2. 0202 Tools in the P-model

2.1.

–

2.2.

–

3.

03 Absolute geometry—hyperbolic geometry

  • 1. 0301 Absolute geometrical relations in the P-model
  • 2. 0302 Displaying reflection axioms in the P-model

3.1.

–

3.2.

–

4.

04 Triangles

  • 1. 0401 Remarkable points and lines in a triangle

4.1.

–

5.

05 Problems (Congruence)

  • 1. 0501 Challenges
  • 2. 0502 Circle given with its center and radius
  • 3. 0503 Circle given with its diameter
  • 4. 0504 Measuring two segments
  • 5. 0505 Relationship among the sides of a triangle
  • 6. 0506 Point reflection
  • 7. 0507 Rotation
  • 8. 0508 Integers on the number line
  • 9. 0509 Translation parallel to a line by a multiple of the unit
  • 10. 0510 Translation parallel to a line by a given segment
  • 11. 0511 Concatenation of two reflections

5.1.

–

5.2.

–

5.3.

–

5.4.

–

5.5.

–

5.6.

–

5.7.

–

5.8.

–

5.9.

–

5.10.

–

5.11.

–

6.

06 Measuring segments and angles

  • 1. 0601 Procedures for measuring in the P-model

6.1.

–

7.

07 Problems (Measuring)

  • 1. 0701 Challenges
  • 2. 0702 Defect of a triangle
  • 3. 0703 Copying angles
  • 4. 0704 Isosceles triangles
  • 5. 0705 Defining a number line with a function
  • 6. 0706 Relationship between sides and angles in a triangle
  • 7. 0707 Is Thales' circle theorem valid in hyperbolic geometry?
  • 8. 0708 Regular polygons
  • 9. 0709 Constructing ultraparallel lines
  • 10. 0710 Angle and distance of parallelism

7.1.

–

7.2.

–

7.3.

–

7.4.

–

7.5.

–

7.6.

–

7.7.

–

7.8.

–

7.9.

–

7.10.

–

8.

08 Trigonometry

  • 1. 0801 Law of sines and cosines
  • 2. 0802 Basic cases of constructing a triangle

8.1.

–

8.2.

–

9.

09 Tesselations

  • 1. 0901 A tesselation with triangles
  • 2. 0902 Circles in a triangular grid
  • 3. 0903 A triangular tesselation
  • 4. 0904 M.C. Escher: Circle limit I
  • 5. 0905 A tesselation of squares in the P-model

9.1.

–

9.2.

–

9.3.

–

9.4.

–

9.5.

–

10.

10 Pencils and cycles

  • 1. 1001 Pencils and cycles
  • 2. 1002 Plotting a graph in the P-model

10.1.

–

10.2.

–

11.

11 Advanced problems

  • 1. 1101 Problems and constructions without tools of measurement
  • 2. 1102 Constructing tangents to a circle through an external point
  • 3. 1103 Cyclic quadrilaterals
  • 4. 1104 Parallelogram
  • 5. 1105 Midpoints of a triangle
  • 6. 1106 Triangles with equal areas
  • 7. 1107 A contest problem
  • 8. 1108 A lemma
  • 9. 1109 Circles that are tangent to each other and to a line
  • 10. 1110 Three circles
  • 11. 1111 Constructing lines that are perpendicular to a given line and tangent to a given circle
  • 12. 1112 Common tangents of two circles

11.1.

–

11.2.

–

11.3.

–

11.4.

–

11.5.

–

11.6.

–

11.7.

–

11.8.

–

11.9.

–

11.10.

–

11.11.

–

11.12.

–

Information