Geometric Structures: Session XII
Geometry, New Geometry, and What We Teach
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
Euclid's Elements are undoubtedly one of the great landmarks of intellectual history, not merely for geometry but for knowledge in general. Geometry has progressed a lot since Euclid, not only in extending and clarifying what he did but also in many new developments. These developments include analytical geometry, the development of projective geometry and the classical non-Euclidean geometries. However, big threads have been explored, such as: ideas about tilings and polyhedra (Euler's polyhedra formula); distance (taxicab distance, Hamming distance, and Levenshtein distance); and curvature. Beyond these there are remarkable "small" masterpieces with many implications that such results have in convexity geometry (Helly's Theorem), graph theory (Euler's traversability theorem, and its extension to the Chinese Postman problem), and discrete geometry (Sylvester-Gallai Theorem (if S is a set of points in the plane not all on a line, there is some pair of the points which have no other point of S on the line they determine)). One could easily go on virtually forever.
We currently live in a "standards"-based culture in mathematics. Vastly over simplifying, curriculum is defined in terms of "lists" of things that students are "expected" to know/master. I am not a big fan of the Standards movement in mathematics but I am a much bigger fan of the original NCTM Standards than the so-called Common Core State Standards. Here is my problem with Standards based mathematics. What happens to people to whom one teaches the mathematics one has deemed important and who, for whatever reason fail to master it all, or master it but when asked to use it or recall it at a future time are unable to do so? How accountable should teachers be for the inability of students to show mastery of Standards on standardized tests?
Consider things in this context. You recently worked on some problems in graph theory which dealt with graphs. There were exercises which dealt with graphs of polyhedra and with taking plane graphs and constructing from them medial graphs and line graphs. There was also practice on whether you could tell whether a graph was plane and 3-connected because by the amazing theorem of Steinitz, graphs which have both of these properties can be realized by convex 3-dimensional polyhedra. You also looked at issues having to do with spanning trees of polytopes which are important in looking for nets of polyhedra. Spanning trees come up in a wide variety of operations research problems. Currently, none of this material is directly part of the Standards-based geometry curriculum of the CCSS, though there are some parts of the NCTM Standards which would allow teachers to include some of these ideas in conjunction with the Standards specified.
However, for a moment imagine they were Standards. Students should be able to construct the dual of a plane graph, draw the medial graph of a plane 3-polytopal graph, draw the line graph of any graph, etc. Now what? As a teacher, one teaches this material. Some of the students learn this material and can answer any mechanical question you pose. A few have been inspired by this material to learn more about where these ideas than than the teacher was able to treat in class. However, other students did not do so well on the problems, and a few students did not have any idea where to get started.
Well, does this material rank as so important that if someone does not master it at all or fully they cannot learn other new geometry? Will a student who cannot do these problems be unable to become a lawyer, doctor, historian, policeman, nurse, truck driver, or Supreme Court justice? How does the importance of the skills involved here compare with adding fractions (2/15 + 4/21), factoring polynomials, adding polynomial fractions ((1/(x-2) + 1/(x+3)), solving equations such as x2 + x + 1 = 0 (which involves using complex numbers), and other skills often tested for in conjunction with college admissions in the United States? Will a student who cannot do these problems be unable to become a lawyer, doctor, historian, policeman, nurse, truck driver, or Supreme Court justice? If these were high school skills, would students who could not do them fail to be admitted to college?
I think it is important for each of you to think through where you stand on issues of this kind. Whereas issues about duals of graphs, etc. are not currently tested for, we could easily remedy that. Furthermore, knowing how to factor polynomials or solve quadratic equations is not in my opinion directly in the skill set of what would be prized for someone who was to sit on the Supreme Court!
Geometry is a very important subject for many careers. It is also important for daily life, and it has results of intrinsic excitement for some people. For some people, they can trace their reason for becoming a professional mathematician to the experience of geometry they learned in high school. Others hit a wall when they got to the traditional proof based geometry that constituted a traditional 10th grade geometry course of 50 years ago. They wrote off mathematics and science as future careers. Are some parts of mathematics being used as a hoop to have students jump through, which though they have limited value for many careers, prevent students from achieving their career goals because they have troubles with these mathematics skills?
These are all important issues for America and other countries.
Returning to some questions more directly related to mathematics, now that we have looked at a variety of ways that axiomatic geometry is tied to some of the geometric topics we have looked at previously, as well as to other parts of mathematics, let us consider some applications of Euler's polyhedral formula within mathematics. One of the nice examples of this is to graph theory and the theory of planarity.
Fact: K3,3 and K5 cannot be drawn in the plane without accidental crossings.
To see how one might prove that the complete graph on 5 vertices does not admit a plane drawing, compute the number of vertices and edges of this graph and use this to see how many faces a plane drawing of this graph would have to have.
Fact: There are five combinatorial regular 3-dimensional convex polyhedra, that is polyhedra all of whose faces are p-gons and where each vertex has the same valence.
This fact can be established using Euler's polyhedral formula. A bonus in using this approach is to use "analogous" thinking to look at what might be regular tilings of the plane. One can establish that there are three such tilings: 4-gons, 4 at a vertex, 3-gons, 6 at a vertex, and 6-gons, 3 at a vertex. Two of these tilings are fairly common; one sees them often used as flooring tiles. What about the third tiling that one almost never sees? Can you explain its scarcity?