From: lee@ms.uky.edu (Carl Lee)
Subject: Re: REVISING GEOMETRY CURRICU
Date: Tue, 8 Sep 1992 13:11:00 GMT
Organization: University Of Kentucky, Dept. of Math Sciences

Sheila,

Thanks for your insightful post on the high school geometry
curriculum.  Although I am not teaching high school geometry, I am
teaching a college geometry course for high school teachers, and have
also worked with COMAP on geometry workshops.  I found your list of 16
suggestions very appropriate, and wonder whether you are already using
Michael Serra's textbook (Key Curriculum Press) or something like it
(or is there something else like it?).  As one who took two years of
algebra before geometry, I favor this sequence because you can do so
much more in geometry given the algebraic tools.  But I would be even
a stronger advocate of an integrated curriculum as soon as such a
restructuring could occur.

For software I am using Geometer's Sketchpad (the one you mentioned
that is published by Key Curriculum Press) but have since seen Cabri
Geometry (published by Brooks Cole) which accomplishes just about the
same thing, costs 1/3 as much, and is available for PC's as well as
Macs.

Some of geometry I hope to convey this semester are:

Beautiful results in geometry that are not normally seen in high school.
Area and volume.
Transformations.
Symmetry.
Use of coordinates and the standard analytic model.
Dimension.
Axiomatics and non-Euclidean geometries.

I will probably have a better sense of this at the end of the
semester, but one inspirational book is T. F. Banchoff, Beyond the
Third Dimension:  Geometry, Computer Graphics, and Higher Dimensions,
Scientific American Library, New York, 1990.

To expose the students to topics that might not ordinarily surface
during the regular lectures, I have given the class the following, to
be done collectively.

A Geometry Scavenger Hunt

Your goal is to identify the following items.  Sometimes a sketch or
photograph will suffice.  In most cases a few sentences describing the
significance of the item will be necessary.  You are free to ask
anyone and everyone that you wish, but you should acknowledge your
sources in writing.  Results should be printed or typed and handed in
on 8 1/2 x 11 inch unlined paper, punched for a three-hole
notebook, suitable for photocopying.  At the end of the semester, the
entire collection will be copied for all the members of the class.

Example:  

Monge's Theorem.

Draw three disjoint circles with different radii.  For each pair of
circles, draw the pair of external tangent lines and mark their
intersection point.  In this way you will obtain three points, A,
B and C.
Monge's Theorem states that these three points will always lie on a
common line.  

This result was proposed by d'Alembert and proved by
Monge using the idea of viewing the problem in three dimensions.
Think of the three circles as three balls in space bisected by a plane P
in space.  Each pair of balls determines a cone.  The cone intersects
the plane in the original pair of tangent lines.  So the points A,
B and C all lie in this plane.  Now rest another plane Q on top of
the three spheres.  This plane is tangent to each of the three
spheres and to each of the three cones, so the points A, B and C
also lie in Q.  Therefore the three points lie in the intersection
of the planes P and Q, which is a straight line.

Reference:   I.A. Graham, Ingenious Problems and Methods, Dover,
New York, 1959.

The items to be found:

A quasi-crystal.

A virus for the "common cold."

The Witch of Agnesi.

A dissection of a square into four pieces that can be
reassembled into an equilateral triangle.

The text of "The Kiss Precise."

A painting by Dali that contains a dodecahedron.

A painting by Dali that contains an unfolded hypercube.

The mathematical name of a soccer ball.

An improperly drawn soccer ball.

The number of configurations of Rubik's cube.

A Chinese Rings puzzle.

A hexaflexagon.

A set of Soma Cubes.

Three works by Escher depicting impossible geometric figures.

A flower with 3-fold symmetry.  Similarly with 4-, 5-, 6-, 7-, 8-,
9-, and 10-fold symmetry.

A set of pentominoes.

A Borromean rings configuration and the name of
the beer company with which it is associated.

The formula for the number of ways of triangulating a convex
polygon.

A two-foot piece of string and a can containing three tennis balls.

A cube cut in half with a single slice yielding a regular
hexagonal cross-section.

A regular tetrahedron cut in half with a single slice yielding a
regular square cross-section.

A work by Escher containing glide-reflectional symmetry.

A pantograph.

The name of the shape of the St.~Louis arch.

Photographs of buildings with 3-, 4-, 5-, 6-, 7-, 8-, 9-, and 10-fold
symmetry.

A Penrose tiling.

A Koch snowflake.

A curve whose dimension lies strictly between 1 and 2.

A work by Escher depicting a tiling of the hyperbolic plane.

A tensegrity structure.

Pick's theorem.

Morley's theorem.

The inscription on Archimedes' tomb.

Kepler's conjecture regarding Platonic solids and planets.

A non-round manhole cover.

Three important geometric problems that were solved within the
last three years.

A method of constructing a regular pentagon with compass and
straight-edge.

A theorem sometimes attributed to Napoleon.

The coordinates of the vertices of the five Platonic solids.

The Banach-Tarski paradox.

The shape of a cell in a honey-bee comb, including the back
end.

A dragon design.

Heron's formula.

Where to place eight moonbases on the moon in order to keep
them mutually as far apart as possible.

The eleven regular and semiregular tilings.

The maximum number of regions into which space can be cut with seven
planes.

Euler's polyhedral formula.

A picture made with a Spirograph.

A ruled surface.

The name of the individual who spent ten years on the
construction of the regular polygon with 65537 sides, and where his
manuscript is to be found.

A Voronoi diagram.

Seven regions on a torus, each pair being somewhere
adjacent.

A planimeter.

A nine-point circle.

The tractrix.

The four-dimensional regular solids.

The shape of a sliding board giving the fastest slide.

A dissection of a cube into three congruent square-base
pyramids.

A dissection of a cube into five tetrahedra, one of which is
regular.

A dissection of a cube into six tetrahedra.

The smallest torus you can make using only equilateral
triangles.

A description of suitable shapes for swords and their
scabbards.

Verses in the Bible suggesting that pi equals 3.

States in which the government has tried to legislate the value
of pi.

The curve described by a point on the rim of a wheel of a moving
train.

A Towers of Hanoi puzzle.

The Argand plane.

Seven strip patterns (e.g., used as border patterns around the
top of a room) with different kinds of symmetry.

Peaucellier's inversor linkage.

A loxodrome.

The formula for the volume of a simple closed polygonal region
given the coordinates of its vertices.

The formulas for:

	The area enclosed by a circle.

	The circumference of a circle.

	The volume of a three-dimensional ball.

	The surface area of a three-dimensional ball.

	The volume of a four-dimensional ball.

	The surface area of a four-dimensional ball.

	The area of a triangle with base length b and height h.

	The volume of a pyramid with base area A and height h.

	The volume of the region common to two pipes of equal radius
	intersecting at right angles.

Cavalieri's Principle.

The number of vertices, edges, squares, and cubes in a
hypercube.

The curve describing the motion of the earth about the sun.

The reason we have seasons.

A map of the earth that preserves angles.  A map of the earth
that preserves areas.

A space-filling Archimedean solid.

A space-filling Archimedean dual.

A pair of enantiomorphic objects.

A Mascheroni construction.

The U.S. patent numbers for the Moebius strip.

A model of a flexible sphere.

An art gallery theorem.

A golden rectangle.

A plant that displays two terms in the Fibonacci sequence.

A pentagon that tiles the plane.

The statement of the Delian problem.

How to trisect an angle with a T-square or "tomahawk."

A published false "proof" of the four-color theorem.

Five significant problems in geometry that have not yet been solved.

A Buckyball.

An account of a Greek who determined the size of
the earth using a well in Syene.

A map of Fayette County with the locations of the public
elementary schools marked.

The isoperimetric problem.

A dissection of a squares into unequal squares.

The ham sandwich theorem.

A three-and-a-half story Easter egg in Canada.

How to obtain a parabola by curve-stitching.

The Mandelbrot set.

--------------------------------------------------------------------------------
Carl Lee
Department of Mathematics
University of Kentucky
Lexington, KY 40506
606-257-1405
lee@s.ms.uky.edu
lee@ukma.bitnet