Table of Contents to
Explorations in Geometry

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The explorations were written so that they can be completed in any order.  While there is some interconnectedness between some of the topics, one exploration does not rely on another.  The short summary of the explorations are as follows.

  1. Symmetry: The four types of rigid motions are first explained and then how these relate to symmetry is explored.  Students are shown how symmetry is defined and how objects can be grouped by their type of symmetry.   
  2. Star Figures: The concepts of star figures and star polygons are given.  Students then explore a number of different properties of these shapes.  
  3. Regular Polyhedra: Students explore the five regular polyhedra.  They discover why there are only five.  They are asked to construct some of these solids and discover Euler’s formula.   
  4. The Pythagorean Theorem: Various visual proofs of the Pythagorean Theorem are explored.  Pythagorean triples are also investigated.  The exploration ends with a Pythagorean puzzle.  
  5. The Golden Ratio: A definition of the golden ratio is given.  It is shown how this number relates to the golden rectangle, the Fibonacci sequence, and the logarithmic spiral.  Golden triangles and the pentagram are also explored. 
  6. Drawing in Perspective: Students are shown how to construct a drawing in one-point, two-point, and three-point perspective.  They are shown that objects can look quite different depending on the viewing point.  
  7. Graph Theory: Five problems or puzzles are given at the beginning of this section.  After a number of definitions and examples are given, these puzzles are then solved by the students.  Along the way Euler paths and circuits are explored as well as Euler’s formula and a counting technique. 
  8. Tessellations: Regular and semi-regular tessellations are quickly explored.  It is shown how Escher-type tessellations can be made using a variety of techniques.  The nonperiodic tessellations (Penrose tiles and pinwheels) are also investigated.
  9. Fractals: Students construct and discover properties of the Koch snowflake, the Sierpinski triangle, and the dragon curve.  Fractal dimension is also explored.  
  10. Celtic Knots: Students are first shown how to draw Celtic knots and then are asked to explore various mathematical properties of these knots. 
  11. Shoelaces: We take a mathematical look at how shoes can be laced.  We explore how many different types of lacings of various sizes there are as well as the lengths of some of these lacings. 
  12. Möbius strips: We explore properties of Möbius strips and Möbius strip-like objects including Möbius shorts, trefoil knots, and Möbius rings. We also look at the properties of these objects when they are cut in a couple of ways.