In the morning session, we will explore the non-Euclidean hyperbolic plane, historically and concretely in nature. The participants will make their own hyperbolic surfaces either from paper or yarn. Along with this we will explore how geometries came into human life through varied experiences.

In the afternoon, we will explore and discuss how experiences with geometries on the sphere and hyperbolic plane can strengthen students' understandings of plane Euclidean geometry.

• Hyperbolic surface template

September 20: The Mathematics and Geometry of Voting, presented by William Zwicker and Davide Cervone

• Why do different voting rules lead to different election results? What makes one rule more fair than another?
• Is there a single voting rule that best represents the collective will of the electorate?
• What do squares, cubes and hypercubes tell us about “yes-no” voting? How can we slice these shapes with lines, planes, and hyperplanes to represent different voting rules?
• Suppose Sue casts more votes than Peter, according to some yes-no voting rule. Does she necessarily have more influence? Can a voter cast several votes, yet have absolutely no influence at all?
• What do hexagons and truncated octahedra tell us about multicandidate elections?
• How can each voter’s “pull” be represented by a rubber band? Can a scheming voter use his rubber band to manipulate the election outcome?
• What happens when pulleys and weights replace the rubber bands?

Whoever gets the most votes wins, so voting is simple, right? Not quite! In yes-no voting, legislators cast ballots in favor of, or against, some proposed new law. A voting rule may build in more influence for some voters than for others; for example, Germany is more populous than Portugal, and its representative casts more votes in the Council of Ministers of the European Union. In the US Federal system, the president can veto legislation, and this gives her more influence than a senator. In multicandidate voting, a variety of rules have been proposed, some quite different from the one most commonly used in the US. In some of these alternative rules a voter casts a ballot listing the candidates in descending order of preference; the rule may then take account of a voter’s full ranking, rather than his first choice only, and this offers some advantages.

We’ll use examples from both types of voting to show that:

• Voting is more interesting and subtle than it may first seem
• Some beautiful mathematics can be used to describe and analyze voting rules
• Geometry provides surprising insights into the mathematics of voting

• Voting and the Geometry of Means and Medians
• Who Should Win this Election?
• Legislative Fairness

August 15: Projective Geometry and Perspective in Art, presented by Tony Phillips

Perspective is the art and science of representing a 3-dimensional subject on a 2-dimensional surface. Painters in antiquity were aware of the problem, but a complete and systematic solution was not worked out until the 15th century, in Italy. (Brunelleschi, Alberti, Piero della Francesca).

Participants will learn how to carry out the "pavement construction" and why it works. Participants will also learn the "inverse construction," which allows the location of the ideal viewing point for an existing perspective image.

Participants will apply this technique to reproductions of several Renaissance works of art.

Projective geometry is a mathematical discipline which grew out of perspective studies. Moving forward in history about 200 years, we will explore Pascal's "mystical hexagram" Theorem (1640), an amazing fact about conic sections and hexagons.

• Menelaus' Theorem and its Converse
• Pascal's Theorem for Circles
• Assorted Conics
• The Secant Theorem
• The Invention of Infinity: Mathematics and Art in the Renaissance

 August 14: Using Symmetries, presented by Doris Schattschneider

The morning presentation will show how the Dutch graphic artist M.C. Escher used distance-preserving transformations (translation, rotation, glide-reflection) to classify and create tiles that he knew could fill the plane so that every tile would be surrounded in the same way. 

The talk will focus on tiles that use rotations and translations to fill the plane. The presentation will also demonstrate how a dynamic geometry program like The Geometer's Sketchpad can use the action of transformations to create tiles and tilings. Participants will create (by hand) some of their own Escher-like tiles using Escher's techniques.

The afternoon session will focus on the proof that there are only 7 frieze groups. As preparation for this, participants will demonstrate that the only symmetries of an infinite strip of uniform width are translations (parallel to the edge of the strip), reflections (in midline or perpendicular to the edge of strip), 180-degree rotations, and glide-reflections (with glide vector on the midline).

Then participants will develop the composition table for these five types of symmetries of the strip. Finally we'll do the proof by elimination that there are only 7 possible symmetry types of periodic frieze patterns, and prove by example that all seven types are possible. Participants will make their own examples of all seven types.

• The five types of Conway Criterion polygon tile
• Table of compositions of pairs of symmetries for a frieze pattern and a completed version of the same table
• Escher's Hexagon
• Short crystallographic notation for frieze patterns
• References

 August 13: Seeing Symmetries, presented by Heidi Burgiel

Heidi Burgiel will give a visual presentation of material in the first third of her new book, The Symmetries of Things, co-authored with John Conway and Chaim Goodman-Strauss.

After demonstrating the notion of symmetry, she will show how Bill Thurston's orbifold notation makes it simple to write down a description of the symmetry group of an abstract pattern or a concrete object. Is it possible to list all the types of symmetry we might see around us? Yes!

If time permits, Heidi will outline a proof of this, paying particular attention to details accessible to high school geometry and algebra students.

The post-talk session will focus on the question, "What are the symmetries of the different types of quadrilateral, and how can we use them to describe different quadrilaterals?"

• Symmetry of Quadrilaterals Handout
• Symmetry of Plane Patterns
• Classification of Convex Quadrilaterals Based on an Idea of Branko Grunbaum

Ongoing: Teachers’ Learning Project
The Teachers’ Learning Project is the component of the PDO that bridges teachers’ mathematical experiences in thePDO sessions with the experiences they plan for their students.

A Learning Project tells a story of what happens when you plan for student learning and then look carefully and honestly at what happens for your students. It generally consists of the following components:

1) A plan for student learning
2) A depiction of what happened when the plan was carried out
3) Examples of student learning
4) An analysis or reflection on students’ learning in light of original intentions and plans

• Learning Project Handout