Fano Plane
Fano Plane

Fano Graph

Fano Graph = (3,6)-cage

Joseph Malkevitch

Department of Mathematics/Computing
York College (CUNY)
Jamaica, NY 11451-0001

Office: 2C07d, Academic Core Building


Personal Activities:

I am a member (Professor, Emeritus) of the Mathematics and Computing Department at York College (CUNY-The City University of New York). I am also a member (Professor, Emeritus) of the mathematics doctoral faculty of CUNY's Graduate Center.

I am a big fan of the Mathematics and Computer Club of York College. Refreshments are usually served. Nearly all the talks are self-contained (i.e. require no prior mathematics or computing background). My colleague T. C. Wu maintains a web page for the Club In conjunction with talks that I have given at the Mathematics and Computing Club and examples I talk about in my classes I have developed a collection of Mathematical Tidbits. Here is a list of talks given at the Mathematics and Computer Club over the last few years.

I serve as co-editor of the Feature Column, a web based column sponsored by the American Mathematical Society. I would welcome feedback about the columns that I have written and suggestions for future columns. An annotated list of my columns is here. The list of my columns in chronological order appears here.

My three year term as the Governor for the Metropolitan New York Section of the Mathematical Association of America (MAA) ended June 30, 2002. However, suggestions concerning improved ways of having the Section serve the metropolitan New York area mathematics community can still be addressed to me.

In May, 2000 I co-organized (with Amir Assadi, Dorothy Washburn, and Robert Wilson) a Geometry Conference (which has already taken place) to honor my thesis advisor Donald Crowe. Those interested in looking at titles and abstracts and photos should consult the conference web page. Donald W. Crowe died on July 4, 2022 at the age of 94. He was an extraodrinary person.

Mathematical Interests:

My mathematical interests include geometry (polytopes, graph theory, tilings, etc.), discrete mathematics and mathematical modeling (especially models dealing with fairness and equity issues and codes of all kinds). Most of the research I have done concerns using combinatorial methods (graph theory) to solve problems about polyhedra. You can find a list of my papers and my doctoral students at CUNY's Graduate Center here. For the benefit of CUNY Graduate Center students, here is some information about my "research program."

In conjunction with my mathematical and mathematics educational interests I have written and edited a variety of materials (often jointly with other people). References to these materials can be found here.

I believe that the teaching of geometry is significantly under represented in the curriculum. I think this has come about because mathematicians and the public identify geometry with the axiomitc geometry taught in high school years ago. Geometry, in fact, is a vast subject covering a wide variety of different arenas. To promote this point of view I have prepared an ennumeration of the different parts of geometry that arise when geometry is viewed in broad perspective. I include suggestions for references where one can get started in learning about these different parts of geometry. I refer to the way geometry might be viewed as Geometry in Utopia. I have developed a small collection of problems which show how geometric ideas and/or tools arise in applications oriented settings. Due to the narrow definition of geometry that is often used, many people would not consider some of this geometry.

Mathematics Education Interests:

I have a keen interest in mathematics education, specifically curriculum development matters, at all levels (primary school to graduate school). Many exciting new developments are happening in mathematics and the ways that it can be applied, but very few of these new developments are making their way into classrooms. For example, many new discoveries in geometry and discrete mathematics can be presented even in elementary school. Some of the materials that I have written or been involved with can be found here.

Particular interests of mine related to education are:

(a) Public perceptions of mathematics and mathematicians

Mathematics is much more a part of people's lives than they are aware. To demonstrate this I often call attention to articles in magazines and newspapers that I refer to as "mathematics in the news." These items often do not explicitly mention mathematics yet none the less some thought will show the connection.

(b) When to call a mathematician!

(c) Techniques versus themes for organizing the content of mathematics

(d) Training "ambassadors" for mathematics

(e) Providing a non-standard look at geometry for children.

(f) Mathematics in the News. Unfortunately, I have not found time to keep this going, but the American Mathematical Society (AMS) has been sponsoring a digest of mathematics in the media that is edited by Tony Phillips (SUNY-Stony Brook)

(g) CUNY is involved with planning for opening a new Community College. I have been part of the group planning what the mathematics program at the new school would look like. In conjunction with this, I have prepared some materials that might have a broader interest.

(h) Geometry gems

Extraordinary elementary results in geometry.

I have prepared some questions to assist undergraduates who want to enter graduate programs with the selection of a suitable school. I also have a page where one can learn about some of the distinguished women contributors to mathematics.

I have become involved with a group of mathematicians, mathematics educators, and other interested parties to help change the climate for Mathematics Education. The group calls itself Math Is More. In conjunction with this here are some comments I posted to the Math is More web page. I post comments about mathematics and mathematics education to the Math is More Discussion Formum on a regular basis. Here is a direct link to that discussion board.

Also in conjuction partly with Math is More and other educational efforts of mine, I have prepared some "stand alone" essays (or other materials) about Mathematics Education. These essays are available here.

York College is part of the Teacher Academy program at CUNY (now sadly confined only to students who have already been admitted) which offers free tuition (and other benefits) in exchange for teaching Mathematics or Science in the New York City public schools. I have been involved with the Teacher Academy at York, where I organized a Mathematics Education Seminar and developed some curriculum materials for the benefit of Teacher Academy Students (and other interested parties). A list of talks and abstracts for the talks given at the York College Mathematics Education Seminar will give you an idea of the range of topics our community has learned about and is interested in. Unfortunately, CUNY no longer supports the Teacher Academy, but the web pages above still are available.

Geometry Seminar:

I am one of the founding members, along with Jacob Eli Goodman, Erwin Lutwak, and Richard Pollack, of the Courant Institute Geometry Seminar. The offical seminar page is available here.

Course Materials and Talks:

Some materials I have developed for the Mathematics (and other) courses I taught at York College or the CUNY Graduate Center are available here

I have developed a course called Geometric Structures which I have taught at both the graduate and undergraduate level. Some materials which supplement and are variants of those available via the link above can be accessed below.

Geometric Strutures curriculum materials.

I will be developing some materials for a Game Theory course aimed primarily at future teachers which will deal with games in the sense that term is used by economists and political scientists, although I will do a bit with combinatorial games such as nim, etc. A syllabus is available for the material and "notes" will be added from time to time.

Game Theory curriculum materials.

Some materials related to the teaching mathematical modeling can be found via the link below.

Mathematical modeling curriculum materials

Here are some slides that I used for a talk about the combinatorics on words (Fibonacci word and Thue-Morse sequence) that I gave to York College's Math/CS Club

The slides below (pdf files) were designed for two talks to the York College Mathematics/CS Club related to the fact that NYC had introduced ranked voting ballots for use in elections and primaries. Due to scheduling problmes I have only presented presented the second set of these slides to the Club. The two files are very similar.

A mathematical view of voting and elections

A mathematical view of voting and elections - updated version of the file above

Another talk I gave (Oct. 2018) to the York College Math/CS Club dealing with partition problems in geometry (naming convex n-gons using partition pairs; eccentricity degree sequences for graphs) is available here:

Geometric Problems Involving Partitions

Another talk I gave (Oct. 2022) to the York College Math/CS Club dealing with an introducction to graph theory, planar graphs, and how graphs which are planar and 4-valent can be used to represent knots is available here:

Graph Theory and Knots

Various publications, slides from prentations (both in mathematics and mathematics education) and papers - many of these are related to my keen interest in 3-dimensional polyhedra and planar graphs

When Robert Bumcrot retired from Hofstra University I wrote a paper for his retirement festshrift which appeared on line, but the diagrams for the versions of the paper that one can find on the web "disappeared." So I am making a version of this paper available here with the diagrams.

Pancakes, Graphs and the Genome of Plants

The paper below (pdf file) was prepared to supplement an oral presentation on this topic, knots and polyominoes, at the MOVES III Conference sponsored by the Mathematics Museum in New York. My written presentation was not accepted for publication but perhaps some people might find it of interest. Comments are welcomed.

Polyomino Knots

Here are slides for a talk I gave at the Courant Geometry Seminar (October 2018) that deals with realizing plane triangulations with 3-polytopes that have isosceles triangle and/or equilateral triangle faces.

Isosceles and Equilateral Triangle Polyhedra

It is common to draw representations of convex 3-dimesional polyhedra in the plane. This expositiory essay compares and contrasts drawings that emphasize the 3-D nature of the polyhedron as compared with drawings that make sure the combinatoral type of the polyhedron is clear (plane, 3-connected graphs).

Drawings representing convex 3-dimensional polyhedra in the plane

Here are slides from a presentation I made at an MAA Session on Discrete Mathematics held at the 2019 Joint Mathematics Meetings in Baltimore. It deals with "lobbying" to encourage all mathematics departments to require a course titled Discrete Mathematics as a required course for the Mathematics Major, just as Calculus I is required as a Mathematics Major requirement.

A Required Discrete Mathematics Course is No Less Important for Mathematics Majors than Calculus I

The following article dealing with a partition of 6 approach to classifying tetrahedra in 3-dimensional space shows that there are 25 different partition types. It also deals with when one can tell if a collection 6 positive integers form the edges of a tetrahedron that can be built physically. Perhaps surprisingly even though the 6 numbers determine 4 faces that each satisfy the strict triangle inequality it still may not be possible to make a tetrahedron with the 6 numbers. This is a preprint version of the article that appeared in COMAP's newsletter, Consortium. Title: The Transition from Two Dimensions to Three Dimensions - Some Geometry of the Tetrahedron, Consortium 105, Fall/Winter, 2013, p. 1-7 (Authors J. Malkevitch and D. Mussa.)

Tetrahedra: The transition from two dimensions to three dimensions

Other Activities:

York College is a member of the Long Island Consortium for Interconnected Learning (Licil), housed at SUNY at Stony Brook (Alan Tucker, program director). For this program I have taught two Humanities Courses at York: Humanities 230, Fairness and Equity and Humanities 235 (Digital Revolution).

The Fairness and Equity Course has as its goal to show the value of developing analytical (mathematical) approaches to equity and fairness (elections, cost sharing, bankruptcy, fair division, organ transplants, etc.). I have developed a course syllabus, showing the range of topics that can be covered in such a course (but not realistic for a one semester course). I have also collected some instructional and resource materials for this course. A glossary of terminology related to fairness and equity questions (and game theory) is available here.

The Digital Revolution Course has as its goal to show how mathematical ideas, particularly ideas about different kinds of codes, support new technologies such as fax, wireless telephony, baggage and mail tracking systems web based technologies, etc. The course also is designed to treat the many civil liberties issues being raised by digital technologies. I have developed a course syllabus, showing the range of topics that can be covered in such a course (but not realistic for a one semester course.) I have collected some instructional and resource materials for this course.

Open Problems Suitable for Undergraduates (Also see Student Research section and Microworlds Research section below)

Recently COMAP has started a column in its newsletter Consortium where research problems for high school students and above appear. You can find very brief descriptions, title and abstracts so to speak, here. With permission I have posted these problems below.

Folding Clusters of Equilateral Triangles

Drawing of Number Sequences

Middle of a Graph


Convex Isosceles Triangles

From Trees to polyhedra

Arrangements of Curves

Folding Polyominoes to Boxes

Spanning Trees of n-Dimensional Cubes

3-Dimensional triangle faced polyhedra with only two edge-lengths

Graphs from Rectangles

glossary of terms that may of use in understanding some of the material discussed in the problem sets above and below.

Problem Set I

Problem Set II

Problem Set III

Here is short collection of open problems related to graphs and their realization as polyhedra in 3-dimensional space. The problems involve questions about folding and unfolding, realization with isosceles triangles, and polyhedra that have faces that are the union of regular polygons. Some of these are similar to problems in other notes I have written.

Problems related to realizing graphs as polyhedra and related problems about 3-dimensional polytopes

Here is a brief essay and some graph theory situations that probes how new mathematical problems arise. Some of the questions here have already been answered but I have listed no references in order to encourage students to explore these questions on their own.

Research Frontier

High School Student Research (also of interest for undergraduate and graduate students):

Students who discover an interest in Mathematics or Computer Science in high school can have the exciting experience of trying to do research. I have created a collection of unusual research problems (requiring almost no prior background in Mathematics or Computer Science). The areas these problems cover range from problems about polyhedra, tilings, nets, number theory, fairness, elections, etc. My York colleague T.C. Wu has kindly improved the version of these research problems which were developed with partial support from the NSF to the Long Island Consortium. One feature of the problems I have prepared is that there is a glossary of terms that might be unfamiliar to students that accompanies the problems. This glossary may be especially helpful for students who are just learning about graph theory. Also see the Microworlds section below on this page where there are links to more recent research problems which "beginning researchers" might work on.

I have also collected a primer of resources for students (high school level or above) who want to do research in Mathematics and Computer Science.

Students who might be interested in doing research on a collection of geometry problems related to the Art Gallery Theorems can find some problems and resources for getting started on that topic here

Here is a short collection of problems (pdf file) that is designed to encourgage students to think about problems related to the Gale-Shapley "marriage problem." What is involved is taking the preferences of n men for n women and these n women for the n men and pairing the men and women in a "stable" way. There are many applications to this circle of ideas, including the pairing of hospitals and medical residents. Alvin Roth and Lloyd Shapley recently won the Nobel Memorial Prize in Economics for work related to these ideas.

I have prepared a web page with a brief list of web sites where there are interesting and quick starting research projects suitable for high school students and undergraduates.

Microworlds of Research

Here are a series of research environments where with relatively little background one can have the experience of trying to solve problems which are both old and new. It is exciting to work on a problem on your own even if some of the questions have already been solved. Typically, new questions that have not been solved arise from this process, plus there is great enjoyment to be had in exploring problems that are new to you on your own. In particular, high school students should be able to make progress on questions of this kind.

Microworld 1

This set of questions involves the shortest tour to visit a collection of lines in the Euclidean plane.

Visiting Lines

Microworld 2

This set of questions involves the triangulations of simple polygons, and their visbiility properties.

Visibility Properties of Polygons

Microworld 3

(a) Two questions about the existence of tetrahedra with six distinct edge lengths.

Tetrahedra with six distinct edge lengths

(b) Some additional questions about tetrahedra in a partitions framework. The questions involve having right angles in the faces of the tetrahedra.

Tetrahedra with Right Angle Faces

Microworld 4

This research investigation involves using partition type as a way of keeping track of what kinds of tetrahedra can fill (tile) 3-space without gaps or overlaps.

Tiling Space With Tetrahedra

Microworld 6

This investigation looks at the eccentricies and degrees of the vertices of a connected graph from a partitions point of view.

Partitions and Eccentricities and Degrees of Vertices in a Graph

Microworld 8

This investigation looks at when plane polygons can have different numbers of sides yet have the same area and same perimeter.

Equal Area and Perimeter Polygons

Microworld 10

This investigation looks at distances that are produced by points in polyiamonds, polyominoes, and polyhexes.

Distances in polyiamonds, polyominoes, and polyhexes

Microworld 11

This investigation involves the suggestion that ideas about partitions and compositions might offer a new window to looking a problems involving tiling the plane with convex pentagons. A new type of pentagon was recently (2015) discovered.

Tiling the plane with convex pentagons

Microworld 12

This investigation calls for classifying the "partition types" of tetrahedra that can be inscribed in a sphere. A tetrahedron has 6 edges. The partition of 6, {4,2}, would mean that the tetrahedron has 4 equal length edges of one length and 2 equal edges of another length.

Classifying tetrahedra inscribed in a sphere


I enjoy preparing bibliographies that may be useful to students and researchers on various topics within (and related to) mathematics. Some of these bibliographies are very general and others are very specific. I welcome any corrections and suggestions for additional entries.

Code Bibliographies

Geometry Bibliographies

Fairness and Equity Bibliographies

Graph Theory and Combinatorics Bibliographies

Journals with Expository and elementary articles about Mathematics

Journals Dealing with Discrete Mathematics, Graph Theory, Combinatorics, and Related Geometry

Martin Gardner's Books

Mathematical Biology Bibliographies

Modeling and Applications of Mathematics Bibliographies

Operations Research Bibliographies

Biographies of Mathematical People

Popular Books About Mathematics

Mathematical Classics

History of Mathematics and Mathematics Curriculum:


It is interesting to consider who has served as a president of one of the major professional socieites in Mathematics (MAA, AMS, SIAM, NCTM)

Presidents of MAA, AMS, and SIAM

Links (These are some very impressive and useful sites I continually find valuable.)

Here is a list of the major professional societies that are directly or indirectly related to Mathematics and Computer Science:

American Association for the Advancement of Science (AAAS)

American Mathematical Society (AMS)

American Statistical Association (ASA)

Association for Computing Machinery (ACM)

Association for Women in Mathematics (AWM)

Conference Board of the Mathematical Sciences (CBMS)

Consortium for Mathematics and Its Applications (COMAP)

DIMACS: Discrete Mathematics and Theoretical Computer Science (Rutgers University)

Institute for Electrical and Electronic Engineers (IEEE)

Institute for Mathematics and Its Applications (UK)

Institue for Operations Research and Management Science (INFORMS)

Mathematical Association of America (MAA)

National Council of Teachers of Mathematics (NCTM)

Society for Industrial and Applied Mathematics (SIAM)

Sigma Xi (Sigma Xi)

Society of Actuaries (SOA)



Last updated 1/15/2021