Ideas for Projects

Plane Dissections

• Every plane polygon can be dissected and reassembled as a square. This is connected with puzzles and tessellations.
• Reference: Dissections: Plane and Fancy by Greg N. Frederickson (Cambridge)

Space Dissections

• In 3-space every polyhedron can NOT be dissected and reassembled as a cube. The Dehn invariant tells why.
• Reference: Numbers and Geometry by John Stillwell

Aperiodic Tessellations

• Roger Penrose discovered basic tile shapes that tile the plane without translation symmetry (kites and darts)
• Reference:Mile of Tiles by Charles Radin (MAA)
• Penrolse Tiles to Trapdoor Ciphers by Martin Gardner

Escher drawings with Patty Paper Geometry

• This approach uses folding and tracing to build symmetric "Escher" patterns.
• Reference: Patty Paper Geometry by Michael Serra (Key Curriculum Press)

Taxicab geometry - a simple non-Euclidean geometry

• Measure distance differently on the usual plane and get a new geometry. Some basic objects take on surprising appearances.
• Reference: Taxicab Geometry: An Adventure in Non-Euclidean Geometry by Eugene Krause (Dover)

Devices for drawing conics - ellipses, parabolas and hyperbolas

• There are a number of mechanical devices for drawing conics. Make some and explain how they work.
• Reference: Geometrical Models and Demonstrations by Bruyr (notes). Other books about curves.
• Reference: Geometry and the Visual Arts by Pedoe (Dover)

Devices for perspective drawing from the Renaissance

• Devices with rods and frames allow one to make correct perspective drawings. Make the device and demonstrate the geometry with it.
• Reference: Geometry and the Visual Arts by Pedoe (Dover)
• http://www.math.nus.edu.sg/aslaksen/projects/perspective/physicalmodel.htm
• http://www.physics.hku.hk/~tboyce/ss/assignments/ascent/perspective.html
• http://graphics.lcs.mit.edu/~fredo/ArtAndScienceOfDepiction/14_Perspective/perspective.html

Folding the circle to get solid models

• While this can get a bit cultish, there are some very intriguing models that can be made by folding circles (such as paper plates).
• Reference: The Geometry of wholemovement by Bradford Hansen-Smith (Wholemovement publications)
• Reference: Spherical Models by Angus Wenninger

The mathematics of polyhedra using Unit Origami

• Fold amazing shapes and explain why they work.
• Reference: Unfolding Mathematics with Unit Origami by Betsy Franco (Key Curriculum Press)
• Reference: Unit Origami by Tomoko Fuse

Constructions and transformations with transparent mirrors (e.g. Miras^TM)

• Take a different slant on constructions using mirrors.
• Reference: Geometry Constructions and Transformations by Iris Mack Dayoub and Johnny Lott (Dale Seymour)

Geometry of numbers - find number relations from connection between geometry and complex medals

• Reference: Geometry at Work Papers in Applied Geometry edited by Cathy Gorini (MAA)
• Reference: The Geometry of Numbers by Olds, Lax, Davidoff (MAA)

Advanced Polyhedra and Polyhedral Models and Connection with Design

• Reference: Shaping Space: A Polyhedral Approach - by Senechal and Fleck (MAA)
• Shapes, Space and Symmetry by Alan Holden (Dover)
• Space Structures by Arthur Loeb
• Connections: The Geometric Bridge between Art and Science by Jay Kapraff

Shape of Space

• Consider possible bounded universes from dimension 2 to 4
• Reference: The Shape of Space Book and Video - Jeff Weeks (Key Press)
• Reference: Video - Jeff Weeks lecture (Geometry Center)
• Reference: The Shape of Space - Jeff Weeks College Text

Construction with compasses only -

• Amazingly, straightedge and compass constructions don't need the straightedge
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Fourth Dimension

• The hypercube and higher dimensional analogs of the tetrahedron and other regular polyhedra
• Reference: Beyond the Third Dimension : Geometry, Computer Graphics, and Higher Dimensions (Scientific American Library Series) by Thomas F. Banchoff

Advanced straightedge and compass construction in traditional architecture (such as for cathedral windows)

• Source Book of Problems for Geometry by Mabel Sykes (Dale Seymour)

Finite Geometries

• Geometries exist with a finite number of points. They are interesting in their own right and useful for codes.
• Reference: A Course in Modern Geometries by Judith Cederberg
• Many other reference exist

The Golden Ratio

• This ratio appears in the golden rectangle, pentagons, the gold spiral, the isosahedron, etc.
• The Golden Section by Hans Walser (MAA)
• Connections: The Geometric Bridge between Art and Science by Jay Kapraff

Geometry of Maps

• There are many ways of mapping the sphere onto a plane page. They all have different geometric properties.

Geodesic Domes

• Invented by Buckminster Fuller, these domes use the geometry of spherical triangles.

History of Math

• The Greeks
• China and India
• The Renaissance and the Invention of Perspective
• Non-Euclidean Geometry (Reference: Euclidean and Non-Euclidean Geometries : Development and History -- by Marvin Jay Greenberg)
• Euclid and his Critics (Reference: Companion to Euclid by Robin Harshorne)

Explore geometry or geometry teaching further with Sketchpad Go deeper with into one of these Sketchpad books

• Pythagoras Plugged In by Dan Bennett
• Exploring Conic Sections by Daniel Scher
• Geometry Turned On (multiple authors), edited by Doris Schattschneider and J. King