a gentle introduction to hyperbolic...
TRANSCRIPT
![Page 1: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/1.jpg)
A Gentle Introduction to Hyperbolic Geometry
Kevin P. Knudson
Director of the Honors Program and Professor of MathematicsUniversity of Florida
http://www.math.ufl.edu/∼kknudson/
April 12, 2011
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 2: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/2.jpg)
Let’s recall basic high school geometry.
Euclid’s postulates form the basis for classical plane geometry. There isone that stands out though:
The Parallel Postulate For any given line ` and a point P not on `, thereis exactly one line through P that does not intersect `.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 3: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/3.jpg)
Let’s recall basic high school geometry.
Euclid’s postulates form the basis for classical plane geometry. There isone that stands out though:
The Parallel Postulate For any given line ` and a point P not on `, thereis exactly one line through P that does not intersect `.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 4: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/4.jpg)
Let’s recall basic high school geometry.
Euclid’s postulates form the basis for classical plane geometry. There isone that stands out though:
The Parallel Postulate For any given line ` and a point P not on `, thereis exactly one line through P that does not intersect `.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 5: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/5.jpg)
Here’s how we usually look at it:
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 6: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/6.jpg)
For centuries, mathematicians attempted to prove that the ParallelPostulate followed from the other four postulates, but were unable to doso.
Mathematicians being mathematicians, they began to wonder what wouldhappen if they tried to drop the postulate, or replace it with a differentversion.
Hence, non-Euclidean geometries were born.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 7: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/7.jpg)
For centuries, mathematicians attempted to prove that the ParallelPostulate followed from the other four postulates, but were unable to doso.
Mathematicians being mathematicians, they began to wonder what wouldhappen if they tried to drop the postulate, or replace it with a differentversion.
Hence, non-Euclidean geometries were born.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 8: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/8.jpg)
For centuries, mathematicians attempted to prove that the ParallelPostulate followed from the other four postulates, but were unable to doso.
Mathematicians being mathematicians, they began to wonder what wouldhappen if they tried to drop the postulate, or replace it with a differentversion.
Hence, non-Euclidean geometries were born.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 9: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/9.jpg)
So how would you change the parallel postulate? Two possibilities:
1. Given a line ` and a point P not on `, there are no lines through Pthat do not intersect `.
2. Given ` and P, there exist multiple lines through P that do notintersect `.
Of course, this means that we have to decide what a “line” is.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 10: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/10.jpg)
So how would you change the parallel postulate? Two possibilities:
1. Given a line ` and a point P not on `, there are no lines through Pthat do not intersect `.
2. Given ` and P, there exist multiple lines through P that do notintersect `.
Of course, this means that we have to decide what a “line” is.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 11: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/11.jpg)
So how would you change the parallel postulate? Two possibilities:
1. Given a line ` and a point P not on `, there are no lines through Pthat do not intersect `.
2. Given ` and P, there exist multiple lines through P that do notintersect `.
Of course, this means that we have to decide what a “line” is.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 12: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/12.jpg)
So how would you change the parallel postulate? Two possibilities:
1. Given a line ` and a point P not on `, there are no lines through Pthat do not intersect `.
2. Given ` and P, there exist multiple lines through P that do notintersect `.
Of course, this means that we have to decide what a “line” is.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 13: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/13.jpg)
It gets pounded into us at an early age that the shortest path between twopoints in the plane is a straight line, and that any two points determine aunique line. Well, that’s how mathematicians define it:
Given two points P and Q in some space, a line joining them is theshortest path in the space from P to Q.
In the usual two-dimensional plane, this is exactly what we think of, but inother contexts it might be something else.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 14: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/14.jpg)
It gets pounded into us at an early age that the shortest path between twopoints in the plane is a straight line, and that any two points determine aunique line. Well, that’s how mathematicians define it:
Given two points P and Q in some space, a line joining them is theshortest path in the space from P to Q.
In the usual two-dimensional plane, this is exactly what we think of, but inother contexts it might be something else.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 15: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/15.jpg)
It gets pounded into us at an early age that the shortest path between twopoints in the plane is a straight line, and that any two points determine aunique line. Well, that’s how mathematicians define it:
Given two points P and Q in some space, a line joining them is theshortest path in the space from P to Q.
In the usual two-dimensional plane, this is exactly what we think of, but inother contexts it might be something else.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 16: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/16.jpg)
Example:
On the surface of a sphere (like our planet), the shortest path between twopoints isn’t a straight line, but rather an arc of a longitude through thetwo points.
This is an example of elliptic or spherical geometry. In this case, everyline through a point not on a given line intersects the line.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 17: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/17.jpg)
Example:
On the surface of a sphere (like our planet), the shortest path between twopoints isn’t a straight line, but rather an arc of a longitude through thetwo points.
This is an example of elliptic or spherical geometry. In this case, everyline through a point not on a given line intersects the line.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 18: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/18.jpg)
Notice also that the sum of the angles of a triangle add up to more than180◦ in this case. Since we are so small relative to the size of the earth, wedon’t really notice this, and we generally observe that the shortest distancebetween points is a straight line in the usual sense.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 19: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/19.jpg)
What about the other case–where more than one line can exist?
This leads to hyperbolic geometry, and examples exist in nature.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 20: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/20.jpg)
What about the other case–where more than one line can exist?
This leads to hyperbolic geometry, and examples exist in nature.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 21: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/21.jpg)
Coral reefs:
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 22: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/22.jpg)
Lettuce:
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 23: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/23.jpg)
Pringles:
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 24: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/24.jpg)
The Pringle is a realization of a hyperbolic paraboloid:
Note that if we draw a triangle on this surface, the angles add up to lessthan 180◦.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 25: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/25.jpg)
The Pringle is a realization of a hyperbolic paraboloid:
Note that if we draw a triangle on this surface, the angles add up to lessthan 180◦.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 26: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/26.jpg)
The most popular model of the hyperbolic plane is the Poincare discmodel. To build it we begin with the interior of the unit circle and declarethat the following paths are straight lines:
1. diameters of the circle
2. circles perpendicular to the boundary circle
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 27: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/27.jpg)
The most popular model of the hyperbolic plane is the Poincare discmodel. To build it we begin with the interior of the unit circle and declarethat the following paths are straight lines:
1. diameters of the circle
2. circles perpendicular to the boundary circle
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 28: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/28.jpg)
The most popular model of the hyperbolic plane is the Poincare discmodel. To build it we begin with the interior of the unit circle and declarethat the following paths are straight lines:
1. diameters of the circle
2. circles perpendicular to the boundary circle
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 29: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/29.jpg)
In this geometry, it is possible for there to be infinitely many lines passingthrough a given point “parallel” to a given line:
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 30: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/30.jpg)
In this geometry, it is possible for there to be infinitely many lines passingthrough a given point “parallel” to a given line:
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 31: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/31.jpg)
Once you define the notion of “line” you then have a notion of distance(the length of the line between two points). And once you have distanceyou have area. Let’s look at this tiling of the hyperbolic plane:
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 32: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/32.jpg)
Once you define the notion of “line” you then have a notion of distance(the length of the line between two points). And once you have distanceyou have area. Let’s look at this tiling of the hyperbolic plane:
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 33: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/33.jpg)
Every red region has the same area in hyperbolic space. Note that theylook “smaller” as you go out towards the boundary circle. What thismeans is that the boundary circle is infinitely far away in this space.
Note also that the number of red (or greeen or blue) regions increasesexponentially as you head toward the boundary circle.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 34: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/34.jpg)
Every red region has the same area in hyperbolic space. Note that theylook “smaller” as you go out towards the boundary circle. What thismeans is that the boundary circle is infinitely far away in this space.
Note also that the number of red (or greeen or blue) regions increasesexponentially as you head toward the boundary circle.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 35: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/35.jpg)
This model of hyperbolic space is most famous for inspiring the Dutchartist M. C. Escher. Here are two examples of wood cuts he producedfrom this theme.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 36: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/36.jpg)
This model of hyperbolic space is most famous for inspiring the Dutchartist M. C. Escher. Here are two examples of wood cuts he producedfrom this theme.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 37: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/37.jpg)
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 38: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/38.jpg)
It’s this notion that the regions really all have the same area that createsthe unusual structures we observe in lettuce and coral. If we try to embedthis geometry into ordinary Euclidean space, we run into trouble. So thespace is forced to curve to preserve the areas and parallel lines.
But I’ll leave it to the biologists to explain why this is advantageous forthe organisms.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 39: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/39.jpg)
It’s this notion that the regions really all have the same area that createsthe unusual structures we observe in lettuce and coral. If we try to embedthis geometry into ordinary Euclidean space, we run into trouble. So thespace is forced to curve to preserve the areas and parallel lines.
But I’ll leave it to the biologists to explain why this is advantageous forthe organisms.
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry
![Page 40: A Gentle Introduction to Hyperbolic Geometryufdcimages.uflib.ufl.edu/AA/00/00/14/23/00001/Knudson.pdf · A Gentle Introduction to Hyperbolic Geometry. This model of hyperbolic space](https://reader033.vdocuments.site/reader033/viewer/2022042808/5f8b0d2c566e9e4fb31aab8f/html5/thumbnails/40.jpg)
Fin
Kevin P. Knudson University of Florida
A Gentle Introduction to Hyperbolic Geometry