## 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:

• The Washington Post (Matt Schudel, July 15): "Her work was deeply theoretical, but other mathematicians considered it boldly original and of untold future importance." And a bit of mathematics: "She [said] that she became interested in mathematics after her older brother told her about a shorthand way to add all the numbers from 1 to 100. The trick, devised in the 18th century by Carl Friedrich Gauss, is to add the outermost pairs of numbers: 1 plus 100, 2 plus 99, 3 plus 98, and so on. Each time, the sum is 101. There are 50 pairs of numbers. Multiplying 50 by 101 yields the answer: 5,050." Schudel links to Erica Klarreich's Quanta article (reprinted in Wired) about Mirzakhani, written at the time of the Fields Medal award, which in turn links to an excellent video about her, made by the Simons Foundation and the International Mathematical Union.
• The New York Times (Kenneth Chang, July 16): Chang quotes Peter Sarnak (Princeton): "She was in the midst of doing fantastic work. Not only did she solve many problems; in solving problems, she developed tools that are now the bread and butter of people working in the field. ... [S]he was a person who thought deeply from the ground up. That's always the mark of someone who makes a permanent contribution." Chang sketches the setting of Mirzakhani's work on the trajectories of balls on variously shaped billiards tables; he quotes Amie Wilkinson (Chicago): "When these trajectories unwind, they reveal deep properties about numbers and geometry." He also shows a picture of a newsstand in Iran on Sunday: Mirzakhani's picture is on the front page of every paper. "Some news outlets took the unusual step of running a picture of her without a head covering."
• Le Monde broke the news on July 15 and published a full obituary by Nathaniel Herzberg on July 17. In between came a report on the Iranian press coverage of Mirzakhani's death. "The Iranian press salutes the mathematician Mirzakhani after her death in the United States" (lemonde.fr Agence France-Presse, Juny 16). They describe how several newspapers published on their front page photographs of Mirzakhani without a hair covering, and how unusual this is. "This happened with the conservative daily Hamshahri, with the headline 'The genius of mathematics has given way before the power of death,' and with the business daily Donaye Eghtessad which announced 'the final departure of the queen of mathematics.' [Iranian] President Hassan Rohani had published on Saturday [July 15] on his Instagram account a picture of the mathematician without a veil, mourning her 'sad disappearance.'"
• The New Yorker online has a posting, "Maryam Mirzakhani's Pioneering Mathematical Legacy," by Siobhan Roberts (July 17). Roberts relays conversations with several mathematicians who knew Mirzakhani, and recounts an amusing anecdote with a mathematical twist. Manjul Bhargava told her that at the Fields Medal ceremony, the actual gold medals got mixed up: "I received Martin [Hairer]'s, who received Maryam's, who received Artur [Avila]'s, who received mine. ... After the ceremony, it was very busy, and there was little chance for all four of us, or even say three of us, to be in the same place simultaneously. Also, due to constant photo shoots, we each needed a medal with us at all times so that we could fulfill our duties and pose with one when asked." He ran into Mirzakhani and they worked out a solution. "Maryam and I exchanged our medals; then Maryam waited to run into Martin to exchange medals with him, while I waited to run into Artur to exchange medals with him." As Bhargava explained to Roberts: "A four-cycle cannot be expressed as the composition of fewer than three transpositions, or 'swaps' ... moreover, those last two swaps could now be carried out in parallel, making it better than any other possible solution. We had this amusing mathematical conversation very quickly, exchanged medals, and then ran off to our next obligations."
• Quanta magazine posted on July 24 "The Beautiful Mathematical Explorations of Maryam Mirzakhani" by my Stony Brook colleague Moira Chas. "She was not the star of her lectures; the only stars were the mathematical ideas. She spoke calmly and clearly and radiated deep enjoyment of the process." Moira goes on to explain several aspects of Mirzakhani's research, and concludes: "It is heartbreaking not to have Maryam among us any longer. It is also hard to believe: The intensity of her mind made me feel that she would be shielded from death."

## 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:

## 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 $CE=2.4$.

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