- Any Books by Martin Gardner
- On the web: Math Forum
- On the web: Cut-the-Kno

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:
*Miles of Tiles*by Charles Radin (MAA) *Penrolse Tiles to Trapdoor Ciphers*by Martin Gardner

Euclidean Geometry - Intermediate to Advanced

- Mathematical Gems (I, II, III) and other books by Ross Honsberger

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 D. 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 D. 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 numbers

- 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* *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 alone

- Amazingly, straightedge and compass constructions don't need the straightedge
- Reference (bibliography at this link): Cut-the-Knot

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