Tony Phillips

Tony Phillips' Take on Math in the Media
A monthly survey of math news

This month's topics:

Maryam Mirzakhani

Maryam Mirzakhani's death at age 40 on July 14, 2017 was prominently reported in newspapers all over the world. The combination of her youth, her gender, her mathematical talent and its recognition (the first woman and the first Iranian to win a Fields Medal, 2014) became a story with universal appeal. Some samples:

Rohani instagram acct screen-shot
Image (larger image) downloaded from President Hassan Rohani's Instagram account.
Le Monde also published a screen-shot of part of Mirzakhani's last Facebook posting, the day before she died.

Maryam Mirzakhani's facebook page
Image of Maryam Mirzakhani's last posting, downloaded from her Facebook account. (Larger image).

 

No more pentagonal tilings

Quanta magazine posted, on July 11, 2017, "Pentagon Tiling Proof Solves Century-Old Math Problem" by Natalie Wolchover. "One of the oldest problems in geometry asks which shapes tile the plane, locking together with copies of themselves to cover a flat area in an endless pattern called a tessellation. ... Now, a new proof by Michaël Rao, a 37-year-old mathematician at CNRS (France's national center for scientific research) and the École Normale Supérieure de Lyon, finally completes the classification of convex polygons that tile the plane by conquering the last holdouts: pentagons, which have resisted sorting for 99 years." What Rao actually did was to show that the 15 pentagonal tesselations (the last one was discovered in 2015) represent all the possibilities. There can be no more. (The Quanta article starts with an illustration of all 15). As Wolchover explains, "He used simple geometric conservation laws to impose restrictions on how a pentagon's corners -labeled 1 to 5- can possibly meet at the vertices in a tiling. These conditions include the fact that the sum of angles 1 to 5 must equal 540 degrees -the total for any pentagon- and that all five have to participate in a tiling equally, since they're all part of every pentagonal tile. Morever, the sums of the angles at a given vertex must always equal either 360 degrees, if the corners of the adjacent pentagons all meet there, or 180 degrees, if some corners meet along another pentagon's edge." This eliminated all but 317 sets of angles. Then a computer algorithm tested each of the 317 by laying down tiles one at a time, backtracking when an overlap was forced, until all possible ways forward were exhausted (which happened for all except the 15 known angle combinations). Wolchover's article links to a video showing the process in action. A couple of similar images:

bad choice of pentagon
Here the computer has arrived at a point where it obviously has to back-track.

tesselation 15
In this case the angle combination gives a tesselation. This one is in fact the last one discovered, in 2015 (larger image). Images courtesy of Michaël Rao.

 

Geometry problems on Long Island

Geometry was big news at Newsday the week of July 17, 2017. On Monday "GEOMETRY EXAM DOESN'T MAKE GRADE" was the topic-of-the-day headline on the front page, modified by "More LI Educators Saying" and "Call to lower passing Regents score as students continue to fail." Background: New York State has a centralized standard high school curriculum leading to a Regent's Diploma. Since 2013 the State has been implementing the Common Core standards, which has led to resistance. As the Newsday reporter John Hildebrand tells us, "In May, more than 90,000 students in grades three through eight on the Island alone refused to take state math tests - maintaining the region's position as the epicenter of [the] boycott movement." The Regent's geometry exam has become a special sore point: "only 64 percent of [the more than 135,000] students taking the exam statewide passed in June 2016."

What made matters worse this June is that two of the 36 questions on the geometry exam had more than one correct answer. That was acknowledged by the State. But an additional question turns out to have had no correct answer at all, and this the State so far has refused to admit. Details of this part of the story emerged on Wednesday, July 19, in a story ("Angling to fix Regents error") by Rachel Uda. Newsday printed out the question in full:

24. In the diagram below, $AC=7.2$ and $CE=2.4$.

Q24

Which statement is not sufficient to prove $\triangle ABC \sim \triangle EDC$?

  1. $\overline{AB}\parallel \overline{ED}$
  2. $DE = 2.7$ and $AB = 8.1$
  3. $CD = 3.6$ and $BC = 10.8$
  4. $DE = 3.0 ,~ AB = 9.0,~ CD = 2.9$ and $BC = 8.7$

 

Uda tells us that Ben Catalfo, a junior at Ward Melville High School, had discovered the error in early July while using the exam and the answer key to tutor geometry students. "According to the test's answer key, answer 2 was the correct choice. But Catalfo, who passed the Regents exam in the seventh grade, found that none of the answers was correct." He consulted William Bernhard, one of his teachers, who alerted the state Education Department. "Officials told Bernhard they were aware of the situation and would not change how the question would be scored." Catalfo has created an online petition "demanding the question be marked correct for all students." Uda quotes him: "The question is unfair because there are no correct answers. There are some kids who failed by a very small margin and there are some kids who might be in summer school because of this."

 

FLASH! (July 26) The State has agreed to regrade Question 24. An email came this morning from Benjamin Catalfo with the news. He wrote: "Today, mathematics wins."

 

Tony Phillips
Stony Brook University
tony at math.sunysb.edu