Geometric Structures (Notes)

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

The materials below were developed for both graduate and undergraduate courses taught under the title "Geometric Structures." The philosphy of these courses is to cover as broad a range of topics with a geometrical flavor as possible. It it generally easier to get a deeper knowledge of a subject X where one has seen the key ideas and some of the major results than it is to start off reading in a book devoted only to the content of topic X.

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Syllabus

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A brief essay (with exercises) about the value of teaching geometry using manipulatives. This essay raises the issue of how different kinds of manipulatives leads to different kinds of geometrical questions.

Manipulatives

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This essay/activity addresses issues about the value and use of definitions in geometry. Different people will use informal language to describe things they see in different ways. The activity challenges students to look at a geometric object and list as many properties of the object as they can.

Careful Looking

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This assignment asks for nets of a special truncated pyramid. Its purpose is to raise issues about how to represent 3-dimensional objects in the plane, as well as the difficulty of proving the completeness of lists in ennumeration problems.

Homework 1

You might find reading this material helpful with this assignment:

Essay About Nets


A diagram showing the 11 nets of the 3-dimensional cube can be found here

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This assignment probes questions related to quadrilaterals, their properties and classification. Also see Practice 2 and 3 below.

Homework 2

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This assignment has questions related to the real projective plane, using triples of real numbers to represent points (homogenous coordinates). There are mechanical exercises about writing down the line thought two points, and finding the point of intersection of two lines. Remember, every pair of (distinct) lines must intersect.

Homework 3

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This assignment has questions related to traversabiltiy problems in graphs. Finding Eulerian circuits or solving the chinese postman problem for a graph has applications to such settings as snow removal, street sweeping and garbage collection. Finding Hamiltonian circuits could be interpreted as inspecting corners to see if traffic lights are working properly.

Homework 4

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Here are some take home problems that can serve as a midterm examination.

Take Home Problems

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Course Project for those wanting or required to do one.

Project

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Reiview for the final examination.

Final Examination Review


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Sample final examination:

Sample Final Examination

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This section offers brief notes about different sessions for a Geometric Structures Course.

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This session was devoted to issues about how to define geometry; thinking of geometry as a branch of mathematics; thinking of geoemtry as part of physics; the role of definitions in geometry; using our visual system to find properties of geometric objects.

Session 1


Some notes following up on issues raised in Session 1.

Followup Session 1


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This session was devoted to some aspects of axiomatics, and to illustrating the way that one gets three different kinds of geometry depending on whether or not one has unique, no, or many parallels through a point to a given line.

Session 2


Some notes following up on issues raised in Session 2.

Followup Session 2
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This session was devoted in part to an exposition of the real projective plane in a model where triples of coordinates are used to describe points.

Session 3


Some notes following up on issues raised in Session 3. In particular it disucsses the idea of coordinates for lines and equations for points.

Followup Session 3

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This session was devoted in part to an exposition of basic graph theory ideas including degree sequence and traversability.

Session 4

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This session was devoted in part to a further exposition of basic graph theory ideas including ideas related to Eulerian Circuits.

Session 5


For those individuals who are rusty on work involving congruences (modular arithmetic) this very skeletal account may be of use.

Congruence Primer

In conjunction with further work on finite planes here is material related to finite arithmetics, finite fields, and finite geometries.

Finite Fields 1


Finite Fields 2

Material about Eulerian circuits and the associated Chinese Postman Problem and the use of these ideas in solving various urban operations research problems can be found here.

Some basic ideas about finite geometries, and finite arithmetics (fields) which can be used to construct them (in an historical framework) can be found here.

Another essay about modular arithmetic and finite fields (integers mode a prime and the Galois fields) covers some of the same ground in the other essays but with additional examples.

Finite Arithmetics.

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This session was devoted in part to an exposition of how to use finite arithmetics to construct finite affine and finite projective planes. This is done in analogy to how the Euclidean was represented by Fermat-Descartes, and extended to the real projective plane via homogeneous coordinates. Also there is a discussion of how to use mathematical inducation to prove that a tree wtih n vertices (graph theory) has (n-1) edges. The concept of graph distance is discussed and some basic ideas about facility location are developed.

Session 6

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Sess1on 7 was devoted to the important ideas of planarity and the drawing of a graph on a particular surface in this case the plane. Plane graphs have not only vertices and faces, but also faces. However, isomorphic plane graphs can have different face structures. Interest in graph theory was stimulated for nearly 100 years by attempts to solve the four color problem.

Here is a link for some notes about plane, planar, and non-planar graphs a

Planar, Plane, and Non-Planar Graphs


Some brief notes about the four color problem (solved by Haken and Appel in 1976) are linked below:

Brief Account of the Four Color Problem


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This section offers practice problems and exercises, sometimes including additional new ideas and directions to consider.

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Some practice problems related to polygons and some ideas related to diagonals for polygons.

Practice 1: Polygons

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Some practice problems and notes related to classifying quadrilaterals. Issues of how to classify quadrilaterals and their properties are raised.

Practice 2: Quadrilaterals

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Some practice problems related to quadrilaterals. Issues of how to classify quadrilaterals and their properties are raised.

Practice 3: Quadrilaterals

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Some practice problems involving using determinants to solve problems in the real projective plane. Determinants can be used to check if three points are collinear or if three lines are concurrent.

Practice 4: Determinants, Points and Lines

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This practice includes some brief remarks about Desargues' Theorem and then offers some exercises to practice verifying this theorem for some specific triangles in the real projective plane.

Practice 5: Desargues Theorem in the Real Projective Plane

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This practice includes involves work with congruences and asks for multiplication and addition tables for some finite algebraic structures.

Practice 6: Congruences

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This practice includes involves work with homogeneous coordinates selected from a finite field with 5 elements. Problems here involve writing down equations of lines passing through two points and finding the coordinates of the point where two lines intersect.

Practice 7: Finite Projective Plane (Analytical version)



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This practice includes work involving the analogue for a finite field of constructing the complex numbers from the real numbers.

Practice 8: Field Extension of a Finite Field

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This collection of exercises involves the concept of plane graph. There is also some additional opportunity to practice with the concepts of Eulerian circuit and Hamiltonian circuit.

Practice 9: Plane Graphs

Note: If G is a plane graph its medial graph is the graph obtained from G by placing a vertex on each edge of G and joining two of these new vertices by an edge if the edges they represent have a vertex in common and lie in the same face.

Here are some questions about the isomorphism of very simple graphs to test your understanding of when graphs are "the same" or "different."

Graph Theory Problems

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This practice involves the concept of an induced subgraph of a graph. This an important concept because many characterizations of graph properties "Q" involve determination if a particular graph (or set of graphs) occurs as an induced subgraph of the graph being tested for the property "Q" invovled.

Practice 9: Induced Subgraphs

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For teachers looking for projects for their students (high school or beyond), here is a collection of ideas related to tetrahedra that involve connections with other areas of geoemetry and ideas from non-geometrical parts of mathematics.

Student Projects Dealing With Tetrahedra

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Geometry Resources of Various Kinds



0. One of the big themes of the geometric structures course is that there are many parts and aspects of geometry. Some years ago, as an "exercise" I tried listing as many of these different aspects as I could. Thouugh many of these aspects overlap heavily I found it useful to list even heavily overlaping parts separately. The result, with a now out of date bibliography which supports learning about these different parts is available via my "Geometry in Utopia" project.

Geometry in Utopia

The list of parts, without the bibliographic "noise" is given below.

Parts of Geoemtry

1. An essay attempting to put Euclid's work in an understandable context.

Essay about Euclid's contribution to Geometry


2. Information about the New York Area Geometry Seminar.

Information about the New York Area Geometry Seminar at the Courant Institute

3. David Eppstein has wonderful materials about geometry as part of his Geometry Junkyard. This particular page is about polyhedra but the link to the Junkyard is at the bottom.

David Eppstein's Polyhedra and Polytope page in the Geometry Junkyard

4. David Eppstein also has a great page dealing with the applicability of Geometry, "Geometry in Action."

Geometry in Action (Applications of Geometry)

5. Here are some interactive programs for working with polyhedra, and other geometrical ideas such as the convex hull of a plane set of points.


Polyhedron Explorer


6. Enthousiasts of "advanced" theorems in the domain of Euclidean geometry will find this journal devoted to such theorems of interest:

Advanced Euclidean Geometry




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