Download Visualizing Hyperbolic Geometry

Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Visualizing Hyperbolic Geometry
Evelyn Lamb
August 22, 2016
Evelyn Lamb
Visualizing Hyperbolic Geometry
Physical Models
Outline
Euclidean Geometry
A spherical interlude
1 Euclidean Geometry
2 A spherical interlude
3 Hyperbolic geometry
4 Mathematical Models
5 Physical Models
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Euclid’s Elements
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Euclid’s Elements
Built on 23 definitions, 5 axioms, 5 postulates
Evelyn Lamb
Visualizing Hyperbolic Geometry
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
One of these postulates is not like the others
A straight line segment can be drawn joining any two points.
A straight line segment can be extended indefinitely in a
straight line.
Given any straight line segment, a circle can be drawn having
the segment as a radius and one endpoint as center.
All right angles are congruent.
If two lines are drawn which intersect a third in such a way
that the sum of the inner angles on one side is less than two
right angles, then the two lines inevitably must intersect each
other on that side if extended far enough.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
One of these postulates is not like the others
A straight line segment can be drawn joining any two points.
A straight line segment can be extended indefinitely in a
straight line.
Given any straight line segment, a circle can be drawn having
the segment as a radius and one endpoint as center.
All right angles are congruent.
If two lines are drawn which intersect a third in such a way
that the sum of the inner angles on one side is less than two
right angles, then the two lines inevitably must intersect each
other on that side if extended far enough.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
One of these postulates is not like the others
A straight line segment can be drawn joining any two points.
A straight line segment can be extended indefinitely in a
straight line.
Given any straight line segment, a circle can be drawn having
the segment as a radius and one endpoint as center.
All right angles are congruent.
If two lines are drawn which intersect a third in such a way
that the sum of the inner angles on one side is less than two
right angles, then the two lines inevitably must intersect each
other on that side if extended far enough.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
One of these postulates is not like the others
A straight line segment can be drawn joining any two points.
A straight line segment can be extended indefinitely in a
straight line.
Given any straight line segment, a circle can be drawn having
the segment as a radius and one endpoint as center.
All right angles are congruent.
If two lines are drawn which intersect a third in such a way
that the sum of the inner angles on one side is less than two
right angles, then the two lines inevitably must intersect each
other on that side if extended far enough.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
One of these postulates is not like the others
A straight line segment can be drawn joining any two points.
A straight line segment can be extended indefinitely in a
straight line.
Given any straight line segment, a circle can be drawn having
the segment as a radius and one endpoint as center.
All right angles are congruent.
If two lines are drawn which intersect a third in such a way
that the sum of the inner angles on one side is less than two
right angles, then the two lines inevitably must intersect each
other on that side if extended far enough.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
If two lines are drawn which intersect a third in such a way that
the sum of the inner angles on one side is less than two right
angles, then the two lines inevitably must intersect each other on
that side if extended far enough.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Equivalently,
Playfair’s axiom: Between a line L and a point P not on L, there is
exactly one line through P that does not intersect L.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Equivalently,
Playfair’s axiom: Between a line L and a point P not on L, there is
exactly one line through P that does not intersect L.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
A few other statements that are equivalent to the 5th
postulate
The sum of interior angles in a triangle is 180◦ .
There exists a triangle whose angles add up to 180◦ .
All triangles have the same interior angle sum.
There is no upper limit to the area of a triangle.
There exists a pair of triangles that are similar but not
congruent.
There exists a quadrilateral in which all angles are right angles.
The Pythagorean Theorem: a2 + b 2 = c 2 .
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
A few other statements that are equivalent to the 5th
postulate
The sum of interior angles in a triangle is 180◦ .
There exists a triangle whose angles add up to 180◦ .
All triangles have the same interior angle sum.
There is no upper limit to the area of a triangle.
There exists a pair of triangles that are similar but not
congruent.
There exists a quadrilateral in which all angles are right angles.
The Pythagorean Theorem: a2 + b 2 = c 2 .
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
A few other statements that are equivalent to the 5th
postulate
The sum of interior angles in a triangle is 180◦ .
There exists a triangle whose angles add up to 180◦ .
All triangles have the same interior angle sum.
There is no upper limit to the area of a triangle.
There exists a pair of triangles that are similar but not
congruent.
There exists a quadrilateral in which all angles are right angles.
The Pythagorean Theorem: a2 + b 2 = c 2 .
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
A few other statements that are equivalent to the 5th
postulate
The sum of interior angles in a triangle is 180◦ .
There exists a triangle whose angles add up to 180◦ .
All triangles have the same interior angle sum.
There is no upper limit to the area of a triangle.
There exists a pair of triangles that are similar but not
congruent.
There exists a quadrilateral in which all angles are right angles.
The Pythagorean Theorem: a2 + b 2 = c 2 .
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
A few other statements that are equivalent to the 5th
postulate
The sum of interior angles in a triangle is 180◦ .
There exists a triangle whose angles add up to 180◦ .
All triangles have the same interior angle sum.
There is no upper limit to the area of a triangle.
There exists a pair of triangles that are similar but not
congruent.
There exists a quadrilateral in which all angles are right angles.
The Pythagorean Theorem: a2 + b 2 = c 2 .
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
A few other statements that are equivalent to the 5th
postulate
The sum of interior angles in a triangle is 180◦ .
There exists a triangle whose angles add up to 180◦ .
All triangles have the same interior angle sum.
There is no upper limit to the area of a triangle.
There exists a pair of triangles that are similar but not
congruent.
There exists a quadrilateral in which all angles are right angles.
The Pythagorean Theorem: a2 + b 2 = c 2 .
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
A few other statements that are equivalent to the 5th
postulate
The sum of interior angles in a triangle is 180◦ .
There exists a triangle whose angles add up to 180◦ .
All triangles have the same interior angle sum.
There is no upper limit to the area of a triangle.
There exists a pair of triangles that are similar but not
congruent.
There exists a quadrilateral in which all angles are right angles.
The Pythagorean Theorem: a2 + b 2 = c 2 .
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Pythagorean Theorem
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
2000 years of...
Mathematicians (paraphrased): Ugh, I hate that postulate! It is
not as self-evident as the other postulates! Can I prove it from the
other postulates?
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
Evelyn Lamb
Visualizing Hyperbolic Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Alternatives to the Parallel Postulate
Playfair’s Axiom: Between a line L and a point P not on L, there
is exactly one line through P that do not intersect L.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Alternatives to the Parallel Postulate
Playfair’s Axiom, elliptic style: Between a line L and a point P not
on L, there are no lines through P that do not intersect L.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Alternatives to the Parallel Postulate
Playfair’s Axiom, hyperbolic style: Between a line L and a point P
not on L, there are infinitely many lines through P that do not
intersect L.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
A spherical interlude
Evelyn Lamb
Visualizing Hyperbolic Geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Playfair’s Axiom, elliptic style
Between a line L and a point P not on L, there are no lines
through P that do not intersect L.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Playfair’s Axiom, elliptic style
Between a line L and a point P not on L, there are no lines
through P that do not intersect L.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Statements equivalent to the Euclidean parallel postulate
The sum of interior angles in a triangle is 180◦ .
All triangles have the same interior angle sum.
There is no upper limit to the area of a triangle.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Statements equivalent to the Euclidean parallel postulate
The sum of interior angles in a triangle is 180◦ .
All triangles have the same interior angle sum.
There is no upper limit to the area of a triangle.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Statements equivalent to the elliptic parallel postulate
The sum of interior angles in a triangle is 180◦ .
The sum of interior angles in a triangle is greater than 180◦ .
All triangles have the same interior angle sum.
The interior angle sum of a triangle depends on its area.
There is no upper limit to the area of a triangle.
There is a largest triangle.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Here ends the spherical interlude
Evelyn Lamb
Visualizing Hyperbolic Geometry
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Playfair’s axiom, hyperbolic style
Evelyn Lamb
Visualizing Hyperbolic Geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Statements equivalent to the hyperbolic parallel postulate
The sum of interior angles in a triangle is 180◦ .
The sum of interior angles in a triangle is less than 180◦ .
All triangles have the same interior angle sum.
The interior angle sum of a triangle depends on its area.
There is no upper limit to the area of a triangle.
There is a largest triangle.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
So what does it look like?!
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
So what does it look like?!
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
So what does it look like?!
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
So what does it look like?!
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
So what does it look like?!
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Poincaré Disk
{z = x + iy ∈ C : |z| < 1}
Inner product: If
(a, b), (c, d) are tangent
vectors at z, then
4(ac + bd)
(a, b) · (c, d) =
(1 − |z|2 )2
Norm:
1
2(a2 + b 2 ) 2
||(a, b)|| =
1 − |z|2
Metric:
2|z2 − z1 |
tanh d(z1 , z2 ) =
|1 − z1 z2 |
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Poincaré Disk
All the lines shown are “straight”
from the point of view of
hyperbolic geometry because
distance is defined differently
here than it is in the Euclidean
plane.
Truth: angles
Lies: area, straightness, distance
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Circle Limit I
Image: M.C. Escher
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Circle Limit I
Image: Doug Dunham
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Circle Limit I
Image: Doug Dunham
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Circle Limit I
Image: Doug Dunham
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Circle Limit I
Image: Doug Dunham
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Circle Limit III
Image: M.C. Escher
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Circle Limit III
Image: Doug Dunham
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Circle Limit III
Image: Doug Dunham
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Tile Yourself
Tool: Malin Christersson
Background: Firth of Forth
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Upper half-plane model
{z = x + iy ∈ C : y > 0}
Inner product: If (a, b) and (c, d) are tangent vectors at
ac + bd
z = x + iy , then (a, b) · (c, d) =
.
y2
Evelyn Lamb
Visualizing Hyperbolic Geometry
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Upper half-plane model and Poincaré disk
Evelyn Lamb
Visualizing Hyperbolic Geometry
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Now with more lizards
Image: M.C. Escher
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Revisiting the hyperbolic parallel postulate equivalents
Between a line L and a point P not on L, there are infinitely
many lines through P that do not intersect L.
The sum of interior angles in a triangle is less than 180◦ .
The interior angle sum of a triangle depends on its area.
There is a largest triangle.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Physical Models
In hyperbolic geometry, area increases more quickly than in
Euclidean geometry. A circle of radius 1 has area larger than π.
Many physical models of the hyperbolic plane put “too much area”
around vertices or edges of flat shapes.
Evelyn Lamb
Visualizing Hyperbolic Geometry
Outline
Euclidean Geometry
A spherical interlude
Soccer Balls
Euclidean soccer ball
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Soccer Balls
Spherical soccer ball
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Soccer Balls
Hyperbolic soccer ball
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Soccer Balls
Truths: area, distance
Lie: angles
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic Fish
Image by Katie Mann
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Hyperbolic crochet
Images by Gabriele Meyer
Evelyn Lamb
Visualizing Hyperbolic Geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic Crochet
Image by Daina Taimina
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Hyperbolic Blanket
Design by Helaman Ferguson, construction by Jeff Weeks
Evelyn Lamb
Visualizing Hyperbolic Geometry
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic Plane Skirt
Andrea Hawksley
Evelyn Lamb
Visualizing Hyperbolic Geometry
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
3D Printing
Designed by Henry Segerman and printed by Shape ways
Evelyn Lamb
Visualizing Hyperbolic Geometry
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
3D Printing
Designed by Henry Segerman and printed by Shapeways
Evelyn Lamb
Visualizing Hyperbolic Geometry
Physical Models
Outline
Euclidean Geometry
Edibles
Evelyn Lamb
Visualizing Hyperbolic Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
Outline
Euclidean Geometry
A spherical interlude
Hyperbolic geometry
Mathematical Models
Physical Models
How to play along at home
Escher and math: Doug Dunham, Bill Casselman, Doris
Schattschneider
Picture tiling tool: Malin Christersson
Instructions for building your own hyperbolic soccer ball:
Keith Henderson, Cabinet Magazine
How to crochet hyperbolic things: Daina Taimina, Institute
for Figuring
3D printed models: Henry Segerman, Shapeways
Instructions for other models of hyperbolic space: David
Henderson (Experiencing Geometry), Jeff Weeks (Geometry
Games)
How to grow kale: almanac.com
Hyperbolic geometry maze: David Madore
HyperRogue game
Evelyn Lamb
Visualizing Hyperbolic Geometry