In two dimensions, a lattice polygon is a polygon in a Cartesian coordinate plane such that the two coordinates of each vertex are integers. In three dimensions, a lattice polyhedron is a polyhedron such that the three coordinates of each vertex are integers.
(a) Prove that a lattice triangle cannot be an equilateral triangle.
(b) Is it possible for a lattice tetrahedron to be a regular tetrahedron?
Click here to see the solution.
John Donne and Mathematics
I recently found myself thinking about the play Wit by Margaret Edson. I saw a production a couple of years ago, and was extremely moved by it. The play is about a professor of literature who is in the hospital dying of ovarian cancer. The play has been made into a movie, which I haven't seen.
The literature professor is an expert in the poetry of John Donne, and a major motif in the play is a teacher's insistence on the correct punctuation in one of Donne's sonnets. She complains about an edition in which a semicolon has been replaced by a comma. I suppose this could have been an excuse for a put-down of pedantry, but on the contrary the playwright made me believe that the correct punctuation was important, even vital.
In the same way, most students must regard the distinctions that mathematicians make as mere pedantry. Why make a big deal over the difference between rational and irrational numbers? According to the calculator, sqrt(2) = 1.414213562, and if you use that value for any practical application it won't matter that it is not exact. But it does matter. I wish I had the skill of Ms. Edson to make my students understand that this is not an unimportant detail, but rather is the heart of mathematics itself.
The literature professor is an expert in the poetry of John Donne, and a major motif in the play is a teacher's insistence on the correct punctuation in one of Donne's sonnets. She complains about an edition in which a semicolon has been replaced by a comma. I suppose this could have been an excuse for a put-down of pedantry, but on the contrary the playwright made me believe that the correct punctuation was important, even vital.
In the same way, most students must regard the distinctions that mathematicians make as mere pedantry. Why make a big deal over the difference between rational and irrational numbers? According to the calculator, sqrt(2) = 1.414213562, and if you use that value for any practical application it won't matter that it is not exact. But it does matter. I wish I had the skill of Ms. Edson to make my students understand that this is not an unimportant detail, but rather is the heart of mathematics itself.
You Haul 19 Pounds
(Title with apologies to Merle Travis.)
A couple of days ago I requested an examination copy of Single Variable Calculus by John Ragowski from W. H. Freeman. Today a 19-pound package arrived at my door, containing 5 books: The book I requested in hardback plus Volume II of the paperback version, plus both paperback volumes of the Early Transcendentals version, plus a 1425-page Instructor's Solution Manual (Early Transcendentals). In addition, there was an Instructor Resources CD, and a nice canvas bag with the publisher logo and the slogan "No Teacher Left Behind". To top it all off, the fulfillment service mistakenly slipped in a packet of signage for a Bruegger's Bagels franchise. (I wonder if Bruegger's Bagels got a packet of calculus materials, and if so what they made of it.)
While I appreciate W. H. Freeman sending this so me so promptly, I doubt that all this is necessary. One book would have been enough for me to make an adoption decision. Sending out all these books seems to be a very non-sustainable practice. Even if I adopt the book, I have at least 3 books that I will never use. I have to ask how much this practice contributes to deforestation, the burning of fossil fuels, and the high price of textbooks.
So, are the books any good? I don't know yet, but it looks pretty much like a dozen other calculus textbooks.
A couple of days ago I requested an examination copy of Single Variable Calculus by John Ragowski from W. H. Freeman. Today a 19-pound package arrived at my door, containing 5 books: The book I requested in hardback plus Volume II of the paperback version, plus both paperback volumes of the Early Transcendentals version, plus a 1425-page Instructor's Solution Manual (Early Transcendentals). In addition, there was an Instructor Resources CD, and a nice canvas bag with the publisher logo and the slogan "No Teacher Left Behind". To top it all off, the fulfillment service mistakenly slipped in a packet of signage for a Bruegger's Bagels franchise. (I wonder if Bruegger's Bagels got a packet of calculus materials, and if so what they made of it.)
While I appreciate W. H. Freeman sending this so me so promptly, I doubt that all this is necessary. One book would have been enough for me to make an adoption decision. Sending out all these books seems to be a very non-sustainable practice. Even if I adopt the book, I have at least 3 books that I will never use. I have to ask how much this practice contributes to deforestation, the burning of fossil fuels, and the high price of textbooks.
So, are the books any good? I don't know yet, but it looks pretty much like a dozen other calculus textbooks.
Lockhart's Lament
Paul Lockhart, a research mathematician and K-12 math teacher, has written a scathing critique of the way that mathematics is taught in our schools. His point of view is that mathematics is an art and needs to be taught as something that is as inherently enjoyable as music or painting, rather than as a subject that must be endured so that the students can pass their exams and the country can become more competitive. Unfortunately, most teachers have never done any real mathematics and have never learned to appreciate mathematics as an art.
For a copy of this paper, go to Keith Devlin's MAA Column
where you can read a short appreciation of Lockhart and link directly to the paper. It is impassioned, funny, and as as over-the-top as a good polemic should be.
I suggest this paper to my mathematics education students at Cambridge College just to shake things up a bit.
For one book in the spirit of Lockhart's ideas, see Trimathalon: A Workout Beyond the School Curriculum by Judith and Paul Sally.
For a copy of this paper, go to Keith Devlin's MAA Column
where you can read a short appreciation of Lockhart and link directly to the paper. It is impassioned, funny, and as as over-the-top as a good polemic should be.
I suggest this paper to my mathematics education students at Cambridge College just to shake things up a bit.
For one book in the spirit of Lockhart's ideas, see Trimathalon: A Workout Beyond the School Curriculum by Judith and Paul Sally.
Why do students have such trouble with quantifers?
During a course I taught this summer in Non-Euclidean Geometry for middle school and high school teachers I used Joel Castellanos' excellent NonEuclid program. I assigned as homework a few problems from the list of activities that accompanies the program. Only one of the 13 students in the class answered the following question correctly:
The correct answer is that the statement is not true. Counterexamples are easy to come by. For example, consider a regular hexagon, whose center coincides with the center of the Poincare Disk. If the vertices of the hexagon lie on a sufficiently large circle, as in Figure 1, a little experimentation should convince the student that it will be impossible to enclose the hexagon in a triangle. A simple proof (not required for the homework) is based on the fact that with proper normalization the area of a hyperbolic triangle is equal to the defect = (pi - sum of the angles). Thus, no triangle can have an area greater than pi. However, the hexagon can be decomposed into six triangles, each of which has defect = 2*pi/3 - eps, where eps > 0 can be made as small as desired by increasing the radius of the circle. Thus the area of the hexagon = 4*pi - 6*eps can be made significantly larger than pi. Since the part cannot be greater than the whole, no triangle can enclose such a hexagon.
Twelve of 13 students showed, in effect, that given a triangle, they could find a regular hexagon inside the triangle. One student even wrote that her initial attempt didn't work because her hexagon was too big, so she had to use a smaller hexagon. Clearly, her problem was in the interpretation of the question. I am convinced that that was the problem of the other students as well, since almost all of them had previously constructed "large" regular hexagons. See Figure 2 for a typical student production.
Figure 1:
Figure 2:
This can be viewed as a problem with understanding the difference between existential and universal quantifiers that bedevils college students, as anyone who has taught beginning calculus knows. I think that there is a psychological component as well. Most students originally go into mathematics because they are good at following directions. For example, they are asked to multiply two polynomials, and they are rewarded when they can do so. The idea of discovering that something is impossible rubs the wrong way. It is satisfying to be able to create a regular triangle inside a given triangle. It is disturbing to have to conclude that there is a hexagon which cannot be enclosed in any triangle.
If our teachers think that solving a problem in mathematics consists of following some procedure to produce a positive result, how are students going to view mathematics as a search for truth, whether the result be positive or negative?
In Euclidean geometry, any polygon can be completely enclosed in some sufficiently large triangle. This is so obvious a statement that I have never even seen it written as a theorem. In, hyperbolic geometry, this is not an obvious statement. Is it a true statement?.
The correct answer is that the statement is not true. Counterexamples are easy to come by. For example, consider a regular hexagon, whose center coincides with the center of the Poincare Disk. If the vertices of the hexagon lie on a sufficiently large circle, as in Figure 1, a little experimentation should convince the student that it will be impossible to enclose the hexagon in a triangle. A simple proof (not required for the homework) is based on the fact that with proper normalization the area of a hyperbolic triangle is equal to the defect = (pi - sum of the angles). Thus, no triangle can have an area greater than pi. However, the hexagon can be decomposed into six triangles, each of which has defect = 2*pi/3 - eps, where eps > 0 can be made as small as desired by increasing the radius of the circle. Thus the area of the hexagon = 4*pi - 6*eps can be made significantly larger than pi. Since the part cannot be greater than the whole, no triangle can enclose such a hexagon.
Twelve of 13 students showed, in effect, that given a triangle, they could find a regular hexagon inside the triangle. One student even wrote that her initial attempt didn't work because her hexagon was too big, so she had to use a smaller hexagon. Clearly, her problem was in the interpretation of the question. I am convinced that that was the problem of the other students as well, since almost all of them had previously constructed "large" regular hexagons. See Figure 2 for a typical student production.
Figure 1:
Figure 2:
This can be viewed as a problem with understanding the difference between existential and universal quantifiers that bedevils college students, as anyone who has taught beginning calculus knows. I think that there is a psychological component as well. Most students originally go into mathematics because they are good at following directions. For example, they are asked to multiply two polynomials, and they are rewarded when they can do so. The idea of discovering that something is impossible rubs the wrong way. It is satisfying to be able to create a regular triangle inside a given triangle. It is disturbing to have to conclude that there is a hexagon which cannot be enclosed in any triangle.
If our teachers think that solving a problem in mathematics consists of following some procedure to produce a positive result, how are students going to view mathematics as a search for truth, whether the result be positive or negative?
Update on Quadrilateral Paper
Great news! Our paper, "Constructing a Quadrilateral Inside Another One" was accepted without further changes by The Mathematical Gazette, and should appear in the November 2009 Issue. This is the same version that I have posted on scribd.
NES/MAA Meeting
Last weekend I attended the annual meeting of the Northeastern Section of the Mathematical Association of America (NES/MAA) held this year at St. Michael's College, near Burlington Vermont.
While on-campus housing was available, Leslie and I chose to stay at a Days Inn across the street from the campus, which worked fine, except for their WiFi, which unusably slow. Weekend weather was a mix of rain, clouds, and sun. The locals said that the rain was needed, and it was easy to put up with. The meeting was Friday afternoon through early Saturday afternoon, and we stayed on through Sunday morning, spending Saturday night doing the tourist thing in Burlington. Burlington is a beautiful small city, with lots of nightlife, particularly since there was a Jazz Fest going on. We had a tasty meal outdoors at the Irish Pub. Just after we ordered a downpour began, our waiter came out and reduced the height of our cafe umbrella, and we had a great time eating as the temperature cooled down and rain poured down inches from us.
I gave a 15-minute talk on my paper (described elsewhere on this blog), "Constructing a Quadrilateral Inside Another One". The talk was one of seven "Contributed Papers", which were scheduled in three rooms between 8 AM and 9 AM on Saturday morning. My talk was the first one, and not surprisingly the audience was small, but the talk was well received. My presentation used both PowerPoint and Geometer's Sketchpad and (since the local PC did not have Sketchpad) I had hook up my laptop. Fortunately, everything worked fine.
The talks were quite interesting, and the organizing topic seemed to be mathematical modeling in biology and environmental science. Even though Leslie is very much a non-mathematician, she is quite interested in the application areas, and she attended and enjoyed a couple of the talks. The most interesting talks for me were those by George Pinder, Christopher Danforth, and Charles Hadlock.
George Pinder of the University Vermont described a simulation of alcohol-assisted bioremediation of superfund sites.
Chis Danforth, also of UVM, gave the after-dinner Battles lecture entitled "Chaos and the Mathematics of Prediction: Hurricane Katrina, Harry Potter, and Happiness." The reference to Hurricane Katrina had to do with the difficulty of predicting weather, the Harry Potter reference is about the difficulty of predicting which children's book out of hundreds published annually might be the next blockbuster hit, and Happiness refers to trying to determine the emotional well-being of large populations over time by an analysis of the numbers of positive and negative words published online or in song lyrics.
Charles Hadlock of Bentley College spoke on Agent-Based Modeling in Teaching and Research. Agents are individuals who are part of a large population and whose behavior is directed (either probabilistically or deterministically) by nearby individuals. Think of Conway's Game of Life. Agent-based models are easily programmed, making them a good choice for student learning, and resulting animations, shown by Charles, have a mysterious beauty that appears very much like natural phenomena.
I found Mohammed Salmassi's short talk on using Spherical Easel for geometry education to be the most immediately useful, since he convinced me to use that software as part of my course on non-Euclidean geometry that I am teaching this summer.
While on-campus housing was available, Leslie and I chose to stay at a Days Inn across the street from the campus, which worked fine, except for their WiFi, which unusably slow. Weekend weather was a mix of rain, clouds, and sun. The locals said that the rain was needed, and it was easy to put up with. The meeting was Friday afternoon through early Saturday afternoon, and we stayed on through Sunday morning, spending Saturday night doing the tourist thing in Burlington. Burlington is a beautiful small city, with lots of nightlife, particularly since there was a Jazz Fest going on. We had a tasty meal outdoors at the Irish Pub. Just after we ordered a downpour began, our waiter came out and reduced the height of our cafe umbrella, and we had a great time eating as the temperature cooled down and rain poured down inches from us.
I gave a 15-minute talk on my paper (described elsewhere on this blog), "Constructing a Quadrilateral Inside Another One". The talk was one of seven "Contributed Papers", which were scheduled in three rooms between 8 AM and 9 AM on Saturday morning. My talk was the first one, and not surprisingly the audience was small, but the talk was well received. My presentation used both PowerPoint and Geometer's Sketchpad and (since the local PC did not have Sketchpad) I had hook up my laptop. Fortunately, everything worked fine.
The talks were quite interesting, and the organizing topic seemed to be mathematical modeling in biology and environmental science. Even though Leslie is very much a non-mathematician, she is quite interested in the application areas, and she attended and enjoyed a couple of the talks. The most interesting talks for me were those by George Pinder, Christopher Danforth, and Charles Hadlock.
George Pinder of the University Vermont described a simulation of alcohol-assisted bioremediation of superfund sites.
Chis Danforth, also of UVM, gave the after-dinner Battles lecture entitled "Chaos and the Mathematics of Prediction: Hurricane Katrina, Harry Potter, and Happiness." The reference to Hurricane Katrina had to do with the difficulty of predicting weather, the Harry Potter reference is about the difficulty of predicting which children's book out of hundreds published annually might be the next blockbuster hit, and Happiness refers to trying to determine the emotional well-being of large populations over time by an analysis of the numbers of positive and negative words published online or in song lyrics.
Charles Hadlock of Bentley College spoke on Agent-Based Modeling in Teaching and Research. Agents are individuals who are part of a large population and whose behavior is directed (either probabilistically or deterministically) by nearby individuals. Think of Conway's Game of Life. Agent-based models are easily programmed, making them a good choice for student learning, and resulting animations, shown by Charles, have a mysterious beauty that appears very much like natural phenomena.
I found Mohammed Salmassi's short talk on using Spherical Easel for geometry education to be the most immediately useful, since he convinced me to use that software as part of my course on non-Euclidean geometry that I am teaching this summer.
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