2012 fall maa meeting (md dc va section) ppt

Roanoke Valley Reef Project

MAA MD-DC-VA Section MeetingVMI - October 2012

Jan Minton Caren DiefenderferRoanoke College Hollins University

From

To

The Institute For Figuring Hyperbolic Crochet Coral Reef, with urchins by Christine Wertheim and sea slug by Marianne Midelburg.Photo © The Institute For Figuring (by Alyssa Gorelick)

with Mathematics

along the way

Smithsonian Museum of Natural History

Euclidean Elliptical Hyperbolic

Types of Geometry

Plane Geometry (standard high school geometry)

Spherical Geometry is a special case of Elliptical Geometry. Geometry on a sphere.

This is the question of the evening.

“For God’s sake, please give it up. Fear it no less than the sensual passions, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.”

- Wolfgang Bolyai (1775-1856) to his son, Janos Bolyai, regarding the study of hyperbolic geometry.

A point and its immediate neighboood

Disk (Flat) Cap Saddle

Like a Pringle Potato Chip!

Euclidean Spherical Hyperbolic

Euclidean Spherical Hyperbolic

Euclidean plane is flat

Spherical plane is shaped like a ball.

Hyperbolic plane??An infinite Pringle?

What about lines?

Euclidean Spherical Hyperbolic

Straight lines

Circles (great circles)

Curves (arcs or circles)

What about parallel lines?

Euclidean – under control

Spherical – Every pair of great circles intersect and so there are no parallel lines in spherical geometry.

What about parallel lines?

Hyperbolic A Strange New World

The Sum of Angles in a Triangle

Euclidean Spherical Hyperbolic

Exactly 180 degrees

More than 180 degrees

Less than 180 degrees

Two Models of Hyperbolic Geometry

The Poincaré Disk Lines

The Open Half Plane Two Types of Lines

Poincaré Disk Open Half Plane

Triangles

Poincaré Disk Open Half Plane

Violating the Parallel Postulate

Hyperbolic Distance in the Poincaré Disk

Definition: Given points P and Q in the Poincare Disk, the hyperbolic distance from P to Q is defined as

|ln((PM/PN)/(QM/QN)| = |ln([(PM)(QN)]/[(PN)(QM)])where M and N are the points on the boundary of the disk on the line that goes through P and Q.

Hyperbolic Distance in the Open Half Plane

Definition: Given points P and Q in the Open Half plane, the hyperbolic distance from P to Q when P=(x1,y1) and Q=(x2,y2) is

arcosh(1+((x2-x1)^2 (y2-y1)^2)/(2y1y2)).

How about some Hands On models?

{3,7} Polyhedral Model

Hyperbolic Soccer Ball

Paper Annuli

DainaTaiminaCrocheting Adventures with Hyperbolic Planes

Bookseller/Diagram Prize for Oddest Title of the Year for 2009

2012 Euler Book Prize

Crocheting Adventures with Hyperbolic Planes is a novel approach to geometry that has brought a whole new audience to mathematics. In this respect it has greater outreach potential than any book we have previously considered. It is truly a book where art, craft, science, and mathematics come together in perfect harmony.

To Figure:To form or shape, to trace, to reckon or calculate, to represent in a diagram or picture, to ornament or adorn with a design or pattern.

The mission of the Institute For Figuring is to contribute to the public understanding of scientific and mathematical themes through innovative programming that includes exhibitions, lectures, workshops, and participatory, community based projects.

Margaret Wertheim

TED talk: The Beautiful Math of Coral

US Satellite Reef Projects in

Arizona MaineCalifornia New YorkFlorida North CarolinaIllinois UtahIndiana VirginiaDist. of Columbia

And Around the World

AustraliaEngland

GermanyIrelandJapanLatvia

I made a mistake. What should I do?

It doesn’t

matter …

keep

going!

Roanoke Valley ReefRoanoke College

Olin Gallery January 25 – March 1, 2013

Opening LecturePaul Snelgrove, Biological Oceanographer

Memorial University of Newfoundland

Echo Sounding Hollins University

Eleanor D Wilson MuseumJanuary 10 – March 2, 2013

Institute for Figuring Hyperbolic Crochet Coral Reef

www.crochetcoralreef.orgRoanoke College Project

www.roanoke.edu/coralreefEmail

coralreef@roanoke.edu

Roanoke Valley Reef on facebook!