Visualizing Hyperbolic Geometry

Home , Hyperbolic geometry, Nikolai Lobachevsky, Spherical geometry

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Evelyn Lamb University of Utah May 8, 2014

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

1 Euclidean Geometry

2 A Spherical Interlude

3 Hyperbolic geometry

4 Models

Visualizing Hyperbolic Geometry 1st example of axiomatic approach to mathematics 23 deﬁnitions, 5 axioms, 5 postulates (This is part of the Pythagorean theorem in Oliver Byrne’s 1847 edition of the text.)

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Euclid’s Elements

c. 300 BCE

Visualizing Hyperbolic Geometry 23 deﬁnitions, 5 axioms, 5 postulates (This is part of the Pythagorean theorem in Oliver Byrne’s 1847 edition of the text.)

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Euclid’s Elements

c. 300 BCE 1st example of axiomatic approach to mathematics

Visualizing Hyperbolic Geometry (This is part of the Pythagorean theorem in Oliver Byrne’s 1847 edition of the text.)

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Euclid’s Elements

c. 300 BCE 1st example of axiomatic approach to mathematics 23 deﬁnitions, 5 axioms, 5 postulates

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Euclid’s Elements

c. 300 BCE 1st example of axiomatic approach to mathematics 23 deﬁnitions, 5 axioms, 5 postulates (This is part of the Pythagorean theorem in Oliver Byrne’s 1847 edition of the text.)

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

A few statements that are equivalent to the 5th postulate

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.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

A few statements that are equivalent to the 5th postulate

The Pythagorean Theorem

Visualizing Hyperbolic Geometry There exists a triangle whose angles add up to 180◦. All triangles have the same sum of angles. 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.

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

A few statements that are equivalent to the 5th postulate

The sum of interior angles in a triangle is 180◦.

Visualizing Hyperbolic Geometry All triangles have the same sum of angles. 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.

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

A few 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◦.

Visualizing Hyperbolic Geometry 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.

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

A few 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 sum of angles.

Visualizing Hyperbolic Geometry There exists a pair of triangles that are similar but not congruent. There exists a quadrilateral in which all angles are right angles.

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

A few statements that are equivalent to the 5th postulate

Visualizing Hyperbolic Geometry There exists a quadrilateral in which all angles are right angles.

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

A few 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 sum of angles. There is no upper limit to the area of a triangle. There exists a pair of triangles that are similar but not congruent.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

A few statements that are equivalent to the 5th postulate

Visualizing Hyperbolic Geometry Spoiler alert: they were wrong.

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

For 2000 years, mathematicians tried to “prove” the fifth postulate using the other four. In other words, they wanted to show that in order for the first four postulates to hold, the fifth had to as well.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Spoiler alert: they were wrong.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Alternative Versions of the Fifth Postulate If the ﬁfth postulate didn’t hold, what could happen?

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.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

How Could We Change the Fifth Postulate?

If the ﬁfth postulate doesn’t hold, what does that mean?

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.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

How Could We Change the Fifth Postulate? If the ﬁfth postulate doesn’t hold, what does that mean? 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.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

A spherical interlude These alternatives to Playfair’s Axiom might seem bizarre, but if you’ve ever ﬂown on a plane, you’re familiar with one of them already.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

A spherical interlude

Yes, we live on a sphere, so the shortest distance between two points doesn’t “look like” a straight line on a map.

Visualizing Hyperbolic Geometry None of them work for the Earth.

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Remember these statements that are equivalent to the 5th postulate?

The sum of interior angles in a triangle is 180◦. All triangles have the same sum of angles. There is no upper limit to the area of a triangle.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Remember these statements that are equivalent to the 5th postulate?

The sum of interior angles in a triangle is 180◦. All triangles have the same sum of angles. There is no upper limit to the area of a triangle. None of them work for the Earth.

Visualizing Hyperbolic Geometry Spherical geometry is sitting right under our feet, but it was harder to discover hyperbolic geometry.

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

What about the other version?

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

What about the other version?

Spherical geometry is sitting right under our feet, but it was harder to discover hyperbolic geometry.

Visualizing Hyperbolic Geometry (“For God’s sake, I beseech you, give it up. Fear it no less than sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life”-Farkas Bolyai, a few years earlier, urging his son J´anosto stop studying non-Euclidean geometry)

J´anosBolyai and Nikolai Lobachevsky independently developed (discovered?) hyperbolic geometry in the 1820s.

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Bolyai and Lobachevsky

“When the time is ripe for certain things, they appear at diﬀerent places in the manner of violets coming to light in early spring.”-Farkas Bolyai, urging his son J´anos to publish his work on non-Euclidean geometry

Visualizing Hyperbolic Geometry J´anosBolyai and Nikolai Lobachevsky independently developed (discovered?) hyperbolic geometry in the 1820s.

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Bolyai and Lobachevsky

(“For God’s sake, I beseech you, give it up. Fear it no less than sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life”-Farkas Bolyai, a few years earlier, urging his son J´anosto stop studying non-Euclidean geometry)

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Bolyai and Lobachevsky

J´anosBolyai and Nikolai Lobachevsky independently developed (discovered?) hyperbolic geometry in the 1820s.

Visualizing Hyperbolic Geometry The sum of angles in a triangle is strictly less that 180◦.

The angles of a triangle determine its side lengths. (No such thing as similar but not congruent.)

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

How hyperbolic geometry is diﬀerent from Euclidean geometry

Given a line L and a point P not on L, there are inﬁnitely many lines through P that do not intersect L.

Visualizing Hyperbolic Geometry The angles of a triangle determine its side lengths. (No such thing as similar but not congruent.)

Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

How hyperbolic geometry is diﬀerent from Euclidean geometry

Given 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 angles in a triangle is strictly less that 180◦.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

How hyperbolic geometry is diﬀerent from Euclidean geometry

Given 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 angles in a triangle is strictly less that 180◦.

The angles of a triangle determine its side lengths. (No such thing as similar but not congruent.)

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

So what does it look like?!

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

So what does it look like?!

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

So what does it look like?!

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

So what does it look like?!

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

So what does it look like?!

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Poincar´eDisk

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. Like a map of the Earth, the PoincaréDisk tells truths and lies. It tells the truth about angles and lies about area.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Circle Limit I

M.C. Escher

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Circle Limit I

Doug Dunham

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Circle Limit I

Doug Dunham

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Circle Limit I

Doug Dunham

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Circle Limit I

Escher Dunham

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Circle Limit I

Doug Dunham

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Circle Limit III

M.C. Escher

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Circle Limit III

Doug Dunham

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Circle Limit III

Doug Dunham

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry 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.

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Soccer Balls

Euclidean “soccer ball”

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Soccer Balls

Spherical soccer ball

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Soccer Balls

Hyperbolic “soccer ball”

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Hyperbolic Fish

Katie Mann

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Hyperbolic crochet

Gabriele Meyer

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Hyperbolic Crochet

Daina Taimina

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

Hyperbolic Blanket

Design by Helaman Ferguson, construction by Jeﬀ Weeks

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

3D Printing

Visualizing Hyperbolic Geometry Outline Euclidean Geometry A Spherical Interlude Hyperbolic geometry Models

How to play along at home More on Escher and math: Doug Dunham, Bill Casselman, Doris Schattschneider Hyperbolic geometry maze (either Poincarédisk or Beltrami-Klein model): David Madore Instructions to build your own hyperbolic soccer ball: Keith Henderson, Cabinet Magazine Template for hyperbolic fish: Katie Mann (Teaching) How to crochet hyperbolic shapes: Daina Taimina, Institute for Figuring Instructions for other models of hyperbolic space: David Henderson (Experiencing Geometry), Jeff Weeks (Geometry Games) 3D printed model: Henry Segerman, Shapeways Or you could grow some kale

Visualizing Hyperbolic Geometry