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 RIO DE JANEIRO — The Olympics are as much about what warms the heart or initiates debate as who wins the medals. But what warms the heart or initiates debate can be a matter of perspective in a sprawling event that brings together more than 200 nations and territories. 
 
 There was little division on Tuesday in the Olympic Stadium, when two runners — Abbey D’Agostino of the United States and Nikki Hamblin of New Zealand — offered a helping hand to each other after a collision in the 5,000 meters and then urged each other on to the finish. 
 
 “I mean, that girl is the Olympic spirit right there,” Hamblin said afterward, speaking to reporters about D’Agostino. 
 
 But on Sunday in the women’s marathon, when the German twins and training partners Anna and Lisa Hahner decided — spontaneously, they insist — to join hands as they crossed the finish line deep in the pack and far from the medals, they quickly drew sharp criticism. 
 
 German track and field officials accused them of publicity seeking and treating the Olympic marathon “like a fun run.” 
 
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 This is not just a German point of view, of course, but it does reflect the range of expectations as athletes navigate the cultural norms that relate to competition and sportsmanship. 
 
 The Egyptian judoka Islam El Shehaby, who refused to shake the hand of his Israeli opponent Or Sasson, was following his nation’s standards — one that Olympic officials expressly reject. He was ejected from the Games. 
 
 There was no outcry, only amusement, when David Katoatau, an exuberant weight lifter from Kiribati, danced off stage on Monday with the goal of heightening awareness of the threat posed by climate change to his tiny nation in the central Pacific Ocean. 
 
 The Hahner twins had no political agenda when they joined hands to end their race, but their move, to their surprise, was still polarizing. 
 
 “Victory and medals are not the only goal,” Thomas Kurschilgen, sports director of the German Athletics Federation, said in an email on Tuesday. “Still, every athlete in the Olympic competitions should be motivated to demonstrate his or her best performance and aim for the best possible result.” 
 
 That approach, according to Kurschilgen, is what divides elite sport from mass-participation sport and what he thinks the Hahners failed to grasp. 
 
 “Their main aim was to generate media attention,” he said. “That is what we criticize.” 
 
 The twins, who placed 81st and 82nd in the marathon, do not see it that way. Contacted on Monday, they initially declined to speak, but Anna Hahner later sent an email in which she said they had not planned on finishing hand-in-hand and had done their best individually. 
 
 “In all the marathons we ran together before, there was a point in the race we had to split up,” Anna said. “This was also the case in the Olympic marathon.” 
 
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 Anna said she started faster, and then Lisa’s group caught up with her at around the 17-kilometer mark, at which point Anna said they ran about three kilometers together. 
 
 “But then I realized I couldn’t run this pace, and I had to let them go,” Anna said. “Lisa was always not far from me. After 40 kilometers, there was a turning point, and I knew, ‘Okay Anna, two kilometers to go to close the gap to Lisa.’ I invested all I had and 300 meters before the finish line, I was next to Lisa. It was a magical moment that we could finish this marathon together. We did not think about what we were doing.” 
 
 But Anna said the symbolism was not lost on either twin. “We trained the last four years to participate in this marathon,” she said. “Neither the time nor the position was what made us happy but to know that we did the best that was possible that day.” 
 
 And yet the sisters were well short of their best marathon performances. Anna’s personal best in the marathon is 2 hours 26 minutes 44 seconds. Lisa’s is 2:28.39. In Rio, Anna finished in 2:45.32, and Lisa in 2:45.33 despite coming early to the Olympic city to acclimatize. 
 
 Clearly, the large time gap between their previous performances and their Olympic performances made Kurschilgen and others only more convinced that they were not seeing the twins’ best effort. 
 
 There is a school of thought at the Olympics and elsewhere that it can be better for an athlete’s profile and even their bottom line to lose memorably — however unintentionally — than to win routinely. D’Agostino and Hamblin would not have become global talking points on Tuesday if they had simply collided and soldiered on without interacting. 
 
 Nor would the British runner Derek Redmond have become part of an “Olympic moment” if he had simply won his 400 semifinal at the 1992 Olympics instead of tearing his hamstring and insisting on continuing the race. His father came onto the track to assist him as he hobbled to the finish with the Barcelona crowd cheering him on. Even though Redmond was disqualified for receiving outside aid, the image and the memory were and remain powerful. 
 
 It is the unexpected and uncalculated gesture that so often moves the masses most, no matter what the culture. 
 
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 “This is Olympism,” said Mary Wittenberg, former director of the New York City Marathon who has attended many Olympics. “Yes, athletes go as hard as they can to win the shiniest medals they can most of the time. We love that, and we want to see those medal dreams come true. That said, there are moments when it’s not all about that and most often, as in the case of the Hahners, it’s also because that’s not in the cards at a given event. So an athlete makes a conscious decision to make the most of the moment in a different way.” 
 
 It is worth noting that the Hahners were not the only twins to finish side by side on Sunday. Kim Hye-song and Kim Hye-gyong of North Korea finished 10th and 11th on Sunday in an identical time of 2:28.36. 
 
 No word from North Korea whether this was perceived officially or unofficially as an uncompetitive act. ||||| Jamaica's Usain Bolt greets fans as he celebrates winning gold in the men's 4 x 100-meter relay final, during the athletics competitions of the 2016 Summer Olympics at the Olympic stadium in Rio de Janeiro,... (Associated Press) 
 
 Jamaica's Usain Bolt greets fans as he celebrates winning gold in the men's 4 x 100-meter relay final, during the athletics competitions of the 2016 Summer Olympics at the Olympic stadium in Rio de Janeiro, Brazil, Friday, Aug. 19, 2016. (AP Photo/David Goldman) (Associated Press) 
 
 RIO DE JANEIRO (AP) — Michael Phelps won more medals than anyone else, again. And then he said farewell, also again. 
 
 Usain Bolt kissed the finish line goodbye after enhancing his Olympic legacy. Simone Biles and Katie Ledecky delivered under the burden of enormous expectations. Two strangers went from rivals to forever-linked by a display of kindness, an entire island seemed to celebrate a tennis match, and a gold medalist scampered home to avoid legal issues after a robbery story unraveled. 
 
 And then the home team won soccer gold. 
 
 The Rio de Janeiro Olympics were not perfect. 
 
 But there were moments — some great, some dreadful and some downright ugly — that will not, and should not, be forgotten. 
 
 Here's a look: 
 
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 MEDAL LEADERS 
 
 The U.S. dominated the medal count, a rare romp in a non-boycotted games. There's still more medals to win Sunday, but the margin of overall victory could be the most in a fully attended games in 68 years. Phelps won six medals, five of them gold, to increase his career haul to 23 Olympic titles and 28 medals overall. Ledecky and Biles each won five medals; Biles will carry the U.S. flag into the closing ceremony. And U.S. shooter Kim Rhode has now won a medal in six consecutive Olympics — and she might not be done. 
 
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 BEST FINISH 
 
 Bolt kissing the finish line after his ninth and final Olympic gold — in nine final races — was a perfect ending. He ran the anchor leg of the 4x100-meter relay for Jamaica, won emphatically to become the third athlete ever with nine golds in track and field, and insists that this will be the end of his Olympic career. 
 
 "Nothing left to prove," Bolt said. 
 
 He's right. 
 
 "I am the greatest," he added. 
 
 Right again. 
 
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 BEST ACT OF SPORTSMANSHIP 
 
 An easy pick. 
 
 In the women's 5,000-meter heat, Abbey D'Agostino of the U.S. and New Zealand's Nikki Hamblin — strangers before that day — were involved in a tumble. D'Agostino helped Hamblin up, encouraging her to finish the race. D'Agostino tore a knee ligament in the fall and obviously couldn't finish. Hamblin wound up finishing last of 17 in the 5,000 final, so neither left with a medal. 
 
 Instead, they got so much more. 
 
 "That girl is the Olympic spirit right there," Hamblin said of D'Agostino. "I've never met her before. Like I never met this girl before. And isn't that just so amazing?" 
 
 Yes, it is. 
 
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 BIGGEST EMBARRASSMENT 
 
 Another easy pick, even in a games where an Egyptian was sent home after failing to shake an Israeli judo opponent's hand. 
 
 Ryan Lochte is a 12-time Olympic medalist, and the odds that he'll have a chance to ever swim for a 13th are as murky as some of the pools were in these Rio Games. Lochte's story that a robber put a gun to his head quickly unraveled, his three teammates who were companions that night all were left to answer legal questions after he scurried home, and more repercussions from the U.S. Olympic Committee are likely coming. 
 
 "It's traumatic to be out late with your friends in a foreign country — with a language barrier," read part of the apology that Lochte released on social media Friday. 
 
 Maybe so, but it bears noting that language barrier or no language barrier, the incident occurred after Lochte left a night out at ... Club France. So it would seem like not all foreign atmospheres struck him as traumatic before now. 
 
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 BREAKOUT STAR 
 
 We asked. You voted. 
 
 Narrowing the list of potential candidates to four — Simone Manuel (four swimming medals), Joseph Schooling (beat Phelps for a gold in the pool), Monica Puig (whose tennis gold was Puerto Rico's first Olympic title in any sport) and Ibtihaj Muhammad (the barrier-breaking fencing medalist) — AP asked its Twitter followers to have a say in deciding this one. 
 
 The response was overwhelming, and one-sided: Puig is the breakout star of these games. 
 
 Reduced to tears many times after draping the Puerto Rican flag across her body after the gold-medal match, Puig beat two of the world's best five players to win the title. 
 
 "To do this ... is everything," Puig said. 
 
 The Olympic ideal, personified. 
 
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 BEST COACH 
 
 Due respect to Mike Krzyzewski, Geno Auriemma, Martha Karolyi and the Japanese wrestling coach who found himself getting body-slammed twice in celebration by women's gold medalist Risako Kawai, this one goes to Ans Botha — the 74-year-old great-grandmother who guided South Africa's Wayde Van Niekerk to Olympic gold and a world record in the men's 400 sprint. 
 
 Her style is simple. When the smile goes away and the voice gets stern, it's time to get serious. Sort of like most great-grandmothers. 
 
 "She's an amazing woman," Van Niekerk said. "Her work ... I think it speaks for itself." 
 
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 BEST COUPLE 
 
 Decathlon champion Ashton Eaton of the U.S. and his wife, heptathlon bronze medalist Brianne Thiesen-Eaton of Canada, would be a very easy selection here. 
 
 But what Kate Richardson-Walsh and Helen Richardson-Walsh did was even more rare. 
 
 They're married, and played for the same gold-medal winning British women's field hockey team. In a games that had more openly gay athletes than ever before, theirs was a moment that surely resonated with many around the world who don't know the first thing about field hockey. 
 
 "To win an Olympic medal is special. To win an Olympic medal with your wife standing next to you ... we will cherish this for the rest of our lives," Kate Richardson-Walsh said. 
 
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 BEST BACK-AND-FORTH 
 
 When the U.S. women's soccer team was ousted by Sweden (and former U.S. coach Pia Sundhage), American goalkeeper Hope Solo raised eyebrows with her assessment of the match. 
 
 But her former coach had the perfect rebuttal. 
 
 "We lost to a bunch of cowards," Solo said. 
 
 "It's OK to be cowards if you win," Sundhage countered. 
 
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 BIGGEST OFF-FIELD STAR 
 
 A tough category, especially after Leslie Jones of "Saturday Night Live" parlayed hilarious Twitter commentary into an invite to Rio. Matthew McConaughey hung out with everyone from the U.S. women's rugby team to Phelps, and Gisele Bundchen's catwalk in the opening ceremony was one not to forget. 
 
 But the pick here is Zac Efron, who flew to Rio just to surprise the U.S. women's gymnastics team — particularly Biles, who doesn't hide her enormous crush on the actor. 
 
 He also saw Bolt and the U.S. men's basketball team, but let's face it, no one is going to ignore invites to hang with Biles and the U.S. gymnasts right now. 
 
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 BEST SPIN 
 
 No one might have been better at their job during the Olympics than Mario Andrada, the spokesman for the Rio Games organizing committee. 
 
 When something went wrong it was his job to explain it — and do so with the world watching and listening. He and International Olympic Committee spokesman Mark Adams held daily hourlong briefings with reporters, and Andrada offered up some quote gems. So we'll go ahead and award his head-scratching statements their own set of medals. 
 
 The bronze: "Numbers mislead," he said, as part of an answer about ticket sales and poor crowds. 
 
 The silver: "Let's give these kids a break," he said of Lochte and his U.S. swimming teammates, after their incident. 
 
 The gold: "Chemistry is not an exact science," he said, talking about why the water at a diving pool went from blue to green. 
 
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 BEST BRAZILIAN MOMENT ... 
 
 Stand up and cheer, City of God. 
 
 One of your own is now an Olympic champion. 
 
 Slums, or favelas, are everywhere in Rio — none more infamous than the City of God, which rose to fame through the 2002 movie of the same name that depicted life in the slum. Rafaela Silva grew up there, and she won gold in judo at these Rio Games. 
 
 Now that's a made-for-Hollywood story. 
 
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 ... THAT IS, UNTIL THIS HAPPENED 
 
 How's this for a storybook ending for Brazil? 
 
 Soccer gold — and in most dramatic style, too. 
 
 Brazil vs. Germany, men's soccer, gold-medal matchup. Neymar gives the hosts a 1-0 lead, but the Germans tie it and ultimately it goes to penalty kicks to decide the Olympic champion. 
 
 Germany makes its first four. 
 
 So does Brazil. 
 
 Then Germany is stopped on its fifth, and the stage is set for Neymar — who else? — to set off a national celebration. His shot sails into the back of the net, Brazil wins and the roar could be heard all over Rio as Neymar collapsed on the field in a quivering mass of joyous tears. 
 
 Forget all the problems these games had. Forget how many Brazilians opposed them in the first place. 
 
 They got their perfect finish to an imperfect Olympics.
– The 2016 Olympics came to an end when Rio officially handed things over to 2020 host Tokyo Sunday night in a jubilant closing ceremony that featured, among other things, Japan's Prime Minister Shinzo Abe rising from a pipe dressed as Super Mario. A look back at some of the best moments from Rio: Usain Bolt's triple-triple. "I am the greatest," the Jamaican runner declared after winning gold in a third race at a third consecutive Olympics. Few would quibble with that statement. Michael Phelps' swim into history. He took another six golds in Rio, giving the "once in 10 generations" swimmer a staggering 23 golds over his career—14 more than any other athlete has ever scored. Team Refugee. For the first time, the Olympics featured a team made up of refugees from the world's conflict zones. "You have inspired us with your talent and human spirit," IOC President Thomas Bach said in his closing address. Simone Biles' amazing performance. She won a record-tying fourth gold in one of the greatest Olympics a gymnast ever had, displaying what Penelope Blackmore at the Guardian calls "jaw-dropping power, laser precision, and pure dynamism." Almaz Ayana's record-breaking run. Some 27 world records were broken in Rio. The first to fall inside the Olympic Stadium was the women's 10,000-meter record, which the Ethiopian runner beat by more than 14 seconds. Liina, Lily, and Leila Luik. The Estonian women set a different kind of Olympic record by becoming the first triplets ever to compete in the marathon. Helping hands. America's Abbey D'Agostino and New Zealand's Nikki Hamblin showed true Olympic spirit—and won Fair Play Awards—by helping each other after a tumble in the women's 5,000 meters. The Brits who are being called the best couple. Kate and Helen Richardson-Walsh not only became the first same-sex married couple to compete in the Olympics, they won the field hockey gold. Changing of the guard. Phelps had an Olympic swimming career that might never be matched unless dolphins start competing—though he was beaten in the 100-meter butterfly by 21-year-old Singaporean Joseph Schooling, who grew up idolizing Phelps. Soccer gold. Brazilians breathe soccer, and they got what the AP calls a "perfect finish to an imperfect Olympics" when their team beat Germany to take the gold.