
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.
 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.
 Star Figures:
The concepts of star figures and star polygons are given.
Students
then explore a number of different properties of these
shapes.
 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.
 The
Pythagorean Theorem: Various visual proofs of the Pythagorean
Theorem
are explored. Pythagorean triples are also investigated.
The
exploration ends with a Pythagorean puzzle.
 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.
 Drawing
in Perspective: Students are shown how to construct a drawing in
onepoint, twopoint, and threepoint perspective. They are shown
that
objects can look quite different depending on the viewing
point.
 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.
 Tessellations: Regular and
semiregular tessellations are quickly explored. It is
shown how Eschertype tessellations can be made using a variety of
techniques. The nonperiodic tessellations (Penrose tiles and
pinwheels) are also investigated.
 Fractals:
Students construct and discover properties of the Koch snowflake, the
Sierpinski triangle, and the dragon curve. Fractal dimension is
also
explored.
 Celtic
Knots: Students are first shown how to draw Celtic knots and
then are
asked to explore various mathematical properties of these knots.
 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.
 Möbius
strips: We explore properties of Möbius strips and
Möbius striplike
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.
