Plotting with sagetex: implicit plots revisited


Earlier I had looked into Implicit Plots and found a way to do them which depended on whether they were connected or disconnected. Recently I found a way of determining the points that Sage is using to construct the plots. This eliminates the need for two cases BUT we have to take more care in plotting the graph. I've added information on plotting implicitly to the Plotting with Sagetex page. The key breakthrough was determining a way to determine the points that Sage used in plotting functions implicitly.  Sage is using a grid of points, which I called gridDim. The gridDim is the number of points on each side of the side of the grid. Since the plot is going from x=-3 to x=3 and y=-3 to y=3 there are (250)x(250) points that will be turned "on" by Sage if they are on the curve and "off" if they are not on the curve.Since an implicit plot is solving the equation f(x,y)=0, in order to have it plotted by the computer I chose to consider the point on the curve if it was "close" by making sure

if abs(C1.xy_data_array[i][j])<.02:

The value .02 was chosen from my experiments with plotting; change to .01 and the plot thins out too much for my liking while .03 starts to thicken the plots too much. You can find a more in depth discussion as well as 2 sample templates to experiment with, on the Plotting with Sagetex page.

Here are some stories which caught my eye recently:

  • hosts a piece on micoagressions.There is increasing pressure on college admin to think "correctly".From the article, "Early this year, the University of California’s president, Janet Napolitano, asked all deans and department chairs in the university’s ten campuses to undergo training in overcoming their “implicit biases” toward women and minorities. The department heads also needed training, according to the UC president, in how to avoid committingmicroaggressions, those acts of alleged racism that are invisible to the naked eye. A more insulting and mindless exercise would be hard to imagine. But Napolitano’s seminar possesses a larger significance: it demolishes any remaining hope that college administrators possess a firmer grip on reality than the narcissistic students over whom they preside."
  • The New York Times has an excellent piece on "The Singular Mind of Terry Tao". Terry Tao is considered to be one of the top mathematicians alive today. From the article, "But it turned out that the work of real mathematicians bears little resemblance to the manipulations and memorization of the math student. Even those who experience great success through their college years may turn out not to have what it takes. The ancient art of mathematics, Tao has discovered, does not reward speed so much as patience, cunning and, perhaps most surprising of all, the sort of gift for collaboration and improvisation that characterizes the best jazz musicians. Tao now believes that his younger self, the prodigy who wowed the math world, wasn’t truly doing math at all. ‘‘It’s as if your only experience with music were practicing scales or learning music theory,’’ he said, looking into light pouring from his window. ‘‘I didn’t learn the deeper meaning of the subject until much later.’’
  • has another piece on the LA public school system's attempts to get a star teacher fired."Esquith, who teaches low-income and minority fifth graders, made a joke about the difficulty he was having raising enough funds for the yearly Shakespeare production, which he finances through donations to his non-profit. As I previously noted, “While reading a passage from Huckleberry Finn in which ‘the king came prancing out on all fours, naked,’ Esquith remarked that if he couldn’t raise additional funds for his annual production, he supposed ‘the class would have to similarly perform naked.’” The joke was overheard by another teacher, who deemed it inappropriate and reported Esquith to the principal, Jonathan Paek....Paek made Esquith sign a humiliating letter of apology:...Esquith was suspended anyway, and now sits in one of the infamous “rubber rooms” while he waits for the district to decide his fate. Officials are doing their best to dig up dirt on him".
  • The reports on "The federal government is spending $125,000 to study adjectives that could be perceived as sexist or racist.". A good use of taxpayer money?
  • reports on "How 'Adjunct' Professors are Exploited". From the article, "According to data collected by the Chronicle of Higher Education, adjuncts at one college and two universities near my home in Southeast Florida earn between $1,380 and $3,000 to teach a fifteen-week, three-credit course. My own university’s published rates range from $1,500 to $3,000. A national survey found the average pay for a three-credit course to be $2,700...Given that the typical equation for calculating preparation and grading time for a three-credit course is three hours for every one hour of class time, it’s safe to assume that adjuncts put in a good 135 hours during a semester. That works out to just over $10 an hour for someone making the lowest rate and about $22 an hour for the higher rate based on the rates listed above...A report from the University of California at Berkeley found that nearly a quarter of all adjunct professors receive some form of public assistance, like food stamps or Medicaid. Many must, as grown educated adults with advanced degrees, live with their families, and struggle to afford basic food requirements. One adjunct professor reported, “I lived off of fried potatoes and onions for the semester. I actually lived better as a grad student than I do now.”.."

tkz-euclide: Snell's Law



Snell's Law is a formula in basic physics which follows easily from Calculus. This makes it worth deriving in your class. I've put together 2 diagrams (the PDFs can be downloaded from the Graphics page) and I've posted the approach to designing these graphics using just the basics of Altermundus' tkz-euclide package. That's posted here and the tex files are attached for you to modify/adjust the colors or notation.

Here are some stories that caught my eye this week:

  • livescience has an article "New Brain-Like Computer May Solve World's Most Complex Math Problems" which looks at a new type of computer design, "In contrast, Massimiliano Di Ventra, a theoretical physicist at the University of California, San Diego, and his colleagues are building "memcomputers," made up of "memprocessors," that both process and store data. This setup mimics the neurons that make up the human brain, with each neuron serving as both the processor and the memory...Now, Di Ventra and his colleagues have built a prototype memcomputer they say can efficiently solve one type of notoriously difficult computational problem. Moreover, they built their memcomputer from standard microelectronics...The new memcomputer solves the NP-complete version of what is called the subset sum problem. In this problem, one is given a set of integers — whole numbers such as 1 and negative 1, but not fractions such as 1/2 — ."
  • The Journal Gazette reports, "A 15-year-old high school student visiting Boston's Museum of Science has uncovered a math error in the golden ratio at a 34-year-old exhibit....Joseph noticed minus signs in the equation where there should have been plus signs. He left a message at the desk and later received a letter from the museum's exhibit content developer, Alana Parkes, informing him the equation would be corrected. Parkes wrote the mistake had been there for a "very long time" without being noticed."
  • The Daily Beast notes "Only one in eight Common Core-aligned textbooks actually meet Common Core standards—and none by textbook giants Pearson or Houghton Mifflin Harcourt—but they were repackaged and sold to public schools anyway, at taxpayers’ expense."
  • The Statesman Journal tells the story of a letter sent by a school to parents "The third paragraph of the letter begins: "Children must be picked up on time. If they are not picked up on time we will call DHS and you will then have to pick them up at court the next day." The letter also talks about dropping children off in the morning. "We will be serving breakfast at 7:45 a.m. at no cost," the letter said. Cafeteria will close at 8:15 a.m. for breakfast. If your child is running late, please feed them at home before sending them to school. Please do not drop your children off before this time. There will not be supervision. If children are dropped before 7:40 the staff will call the authorities."". Parents are upset and a district spokesman says the letter was sent in error. So----they said it but didn't mean it.
  • RT reports on the politicization of history in school. "In addition, the books will teach that the Civil War was caused by “sectionalism, states’ rights and slavery" ‒ in that particular order so as to emphasize the idea that slavery was not the main driver of war, and that, in the context of the 1860s, states' rights and slavery were not intertwined. Slavery was "a side issue to the Civil War," conservative state education board member Pat Hardysaid in 2010 when the board was considering new standards. “There would be those who would say the reason for the Civil War was over slavery. No. It was over states’ rights.""
  • has the article, "'Free' Community College Already Costs 50 Percent More than Originally Estimated". From the article, "For those who missed the whole thing, here are the basics: President Barack Obama and several leading Democrats want to make community college free for students who would then transfer to four-year colleges for bachelor's degrees and for those who are in training programs for occupations that the state determines to be part of an "in-demand industry sector." The federal government will pay two-thirds of the cost, but states will have to chip in one-third to participate...None of these plans seem to seriously deal with the fact that its government subsidies to education that have driven up these costs. That massive debt middle class students are racking up isn't paying for instructors. It's paying for boatloads after boatloads of administrators making plum salaries for managing these reform programs that the government insists will make colleges better. The government isn't making college cheaper. It's making it even more expensive.". Quantity of education doesn't equal quality so this seems like an even bigger waste of money. Remember this post? where you'll find the quote from a Michael Snyder article, "In Japan, Finland and the Netherlands, young adults with only a high school degree scored on par with American Millennials holding four-year college degrees, the report said". The article also says "It also mandates community college systems provide plans for "institutional reforms and innovative practices to improve student outcomes." I suppose writing on a slip of paper "Grade inflation!" would not qualify."  If they could deliver a quality public school education that resulted in literate students with good math skills there'd be no need for this. But they can't. So what's the quality of education that students will get in a school that needs to "improve student outcomes". No question mark there because it's more of the same low quality education. Excellence in education is found mainly in private schools and public schools in affluent areas where the parents help to assure the quality.
  • Solve my maths has a post Concentric Circles Problem that would be useful for a lot of math teachers. The Humor section of the site has some nice content too.
  • notes, "A bill introduced in the New Jersey State Senate last month seeks to put a three-year moratorium on the growth of charter schools in the state.In a blistering op-ed this week in NJ Spotlight, Dale Caldwell, the head of school at the Village Charter School in Trenton, argues that the New Jersey Education Association (NJEA), the state's largest teachers union, is behind the legislation. The union's goal: to preserve its monopoly over public education in the state.". The posted video is worth your time. The "blistering op-ed" piece is here.

Resource: Internet Archive


Who doesn't like quality, free resources? The Internet Archive explains on their About page: "The Internet Archive is a 501(c)(3) non-profit that was founded to build an Internet library. Its purposes include offering permanent access for researchers, historians, scholars, people with disabilities, and the general public to historical collections that exist in digital format.

Founded in 1996 and located in San Francisco, the Archive has been receiving data donations from Alexa Internet and others. In late 1999, the organization started to grow to include more well-rounded collections. Now the Internet Archive includes: texts, audio, moving images, and software as well as archived web pages in our collections, and provides specialized services for adaptive reading and information access for the blind and other persons with disabilities.".

What separates the Internet Archive from other resources is threefold: the quality of their resources, the range of resources (books, software, audio, video) and the ability to read through many of the books by flipping through the text (as well as downloading it in a variety of formats). Take a look at Kotov's "Grandmaster at Work"


Notice the 2 red boxes? The box at the bottom displays a variety of formats that you can download the book in. The box near the top right surrounds the "Full screen" button. Press that button to go into full screen mode.


Clicking on the right hand page flips the book forward, while the left hand page flips you back. So you can browse the resource online. Notice that the screenshot above has two more red boxes. The one in the top right hand corner will activate the voice reading of the book. The red square in the bottom left hand corner is a slider that can quickly get you to deep inside the book without flipping each page.


My only complaint is that the Search feature wasn't as helpful in finding the resources. I failed to find some books through searching "mathematics" which turned up in other unrelated searches.Here are some links to get you started:

Spivak: Calculus book, Supplement for the book, Dugopolski: Precalculus, Beginning and Intermediate Algebra,lots of CK-12 series books CK 12 AlgebraCK 12 Algebra II with Trigonometry, MOOCulus Sequence and Series Textbook, Advanced Math 2Python Programming, Soltis: What it takes to become a chess master, Botvinnik: Half a Century of Chess, Alburt: Test and Improve Your Chess, Kosikov: Elements of Chess Strategy, lots of old Schaum's books, and so much more! I've added the link to the Internet Archive to the sidebar.

Here are some stories which caught my eye the last week:

  • The Intercept looks at "NO CHILD LEFT UN-MINED? STUDENT PRIVACY AT RISK IN THE AGE OF BIG DATA". From the article, "“What if potential employers can buy the data about you growing up and in school?” asks mathematician Cathy O’Neil, who’s finishing a book on big data and blogs at In some of the educational tracking systems, which literally log a child’s progress on software keystroke by keystroke, “We’re giving a persistence score as young as age 7 — that is, how easily do you give up or do you keep trying? Once you track this and attach this to [a child’s] name, the persistence score will be there somewhere.” O’Neil worries that just as credit scores are now being used in hiring decisions, predictive analytics based on educational metrics may be applied in unintended ways. Such worries came to the fore last week when educational services giant Pearson announced that it was selling the company PowerSchool, which tracks student performance, to a private equity firm for $350 million. The company was started independently; sold to Apple; then to Pearson; and now to Vista Equity Partners. Each owner in turn has to decide how to manage the records of some 15 million students across the globe, according to Pearson."
  • reports on famous author Judy Blume warning about censorship in today's world.
  • Huffington Post has a piece on Dr. John Urschel, professional football player, on why more kids don't like math.
  • Microagrressions, which I mentioned in this post, are back again. Although I learned "America is a melting pot", that's now a color blindess microaggression because it denies a person of color's racial/ethnic experience. The College Fix can help get you up-to-date on on the latest witch hunt. From the article, "University of Wisconsin-Stevens Point officials have advised faculty that the term “America is a melting pot” is a racial microaggression. The common phrase was among a list of examples of so-called racial microaggressions used “as a discussion item for some new faculty and staff training over the past few years,” a campus official told The College Fix in an email. Other phrases on the list included: “You are a credit to your race,” “where are you from,” “there is only one race, the human race,” “I believe the most qualified person should get the job” and “everyone can succeed in this society if they work hard enough.”. Take some time and look at the lists from University of Wisconsin and University of California. Lots of the examples on the list are ambiguous in the sense that it presumes you know WHY a comment was made. So the example, to a woman of color about "I would never have guessed you were a scientist." is considered a microaggression. because it's assumed you said it because she's a woman which means it could be perceived as insulting her intelligence. Apparently it's okay to to say it to a white male, though, because it wouldn't be an attack on his intelligence....wait, what? Or if you've mistaken a faculty of color mistaken for a service worker then it assumes you've done it because they are of color and not because of how they were dressed, where they, or who they looked like. Heck, I've been mistaken for someone working in a store that I was shopping in for who knows what reason.  Should I have been insulted? HOLDING AN OPINION that, "Affirmative action is racist" IS FORBIDDEN because it makes it seem like one group gets extra privileges. And the common practice of empathy, such as a someone saying, "As a woman, I know what you go through as a racial minority" is enough to cause a problem. How can a woman possibly know what racial discrimination is like. Your AMBIGUOUS ACTIONS are now under assault. "A person asks a woman her age and, upon hearing she is 31, looks quickly at her ring finger" is a problem because the reason WHY you did that was you thought "Women should be married during child-bearing ages because that is their primary purpose.". If faculty are being taught that examples like they've listed are transgressions then you get an indication of how today's young are looking at the world. It's a less tolerant, "he/she said this which made me feel ___, therefore they must pay the price". Imagine spending money to get an education and coming out less educated and less tolerant. ZeroHedge has a piece on how "hate speech" is used to destroy "freedom of speech".
  • There's an annoying piece that's getting a lot of play. From the Western Morning News we hear "Myth that men are naturally better at maths than women debunked". Now you'd think that such a conclusion would be based on some test scores which would show that women scored just as well (or better) than men. No such case. From the article, "US psychologist Dr Shane Bench, from Washington State University, who led a study that involved assessing the ability of men and women to predict their performance in maths tests, said: "Gender gaps in the science, technology, engineering and maths fields are not necessarily the result of women's underestimating their abilities, but rather may be due to men's overestimating their abilities." His team conducted two studies of 300 undergraduates who were asked to have their maths skill tested before guessing how well they had fared. In the first study, participants received feedback about their real performance before they were again asked to take a test and predict their scores. For the second study, the students only sat one test without receiving any feedback, and were questioned about any plans to pursue maths-related courses Across both studies, men were consistently found to overestimate the number of problems they solved correctly while women's appraisal of their own abilities was more accurate. After receiving feedback about how well they did in the first study, men were then better at estimating their scores in the second test.". Got that? With no information on actual math scores, what does this mean?? Suppose, for example, women scored 75% on the test and then estimated they scored about 75% whereas men scored 80% and estimated they scored 85%. Then the women are more accurate at gauging their performance, but since their performance is worse, how would that debunk the claim that men are better than women at math. And to make things worse, the research said "After receiving feedback about how well they did in the first study, men were then better at estimating their scores in the second test.". I'm not sure how this "research" proves anything. It only seems to show that, without feedback on performance, women are better at appraising their performance than men. But with feedback on performance (which is what happens in the real world as students get feedback on each test throughout the semester) men are better at appraising their performance. But none of this has to do with mathematical expertise.
  • The PC climate claims another high school teacher. has the report, "An Illinois high school teacher was fired after stepping on the American flag to prove a point about free speech. The teacher, Jordan Parmenter, had been using a flag as a pointer during class on May 15. At least one student accused him of being disrespectful toward the national symbol, so Parmenter dropped the flag on the ground and stomped on it, according to Word quickly spread, and soon enough, demonstrators appeared outside Martinsville Junior-Senior High School. Parmenter wrote a letter of apology, but the school board voted 6-0 to fire him....The school board had a golden opportunity to show kids that honoring the values the flag represents is more important than honoring the flag itself. Instead, they imparted a different lesson: that no act of defiance goes unpunished by the government. Perhaps that’s an important lesson as well.".  Beware the angry mob.
  • The PC climate claims a college teacher as well. The Advocate has the story of an LSU professor, Teresa Buchanan, fired for using salty language. The teacher is fighting back with a lawsuit. From the article, "She said the university is trying to dictate how she teaches and in the process is impinging on her academic freedom. “The occasional use of profanity is not sexual harassment,” Buchanan said. “Nor is the occasional frank discussion of issues related to sexuality, particularly when done in the context of teaching specific issues related to sexuality.” LSU spokesman Ernie Ballard declined comment Friday on Buchanan’s dismissal, saying it’s a personnel matter and involves possible litigation. Buchanan was fired even though a committee of five faculty members that presided over an 11-hour dismissal review hearing held on March 9 recommended that she keep her job. While the committee found that her adult language and humor violated university policies that protect students and employees from sexual harassment, it found no evidence Buchanan’s comments were “systematically directed at any individual.” The committee recommended she be censured and agree to quit using “potentially offensive language and jokes” that some found offensive."