As much as it brings on shuddering memories of ninth grade geometry lessons, any soccer field is as much an homage to Euclid as Pele or Mia Hamm. One large rectangle to begin with, the field is further subdivided into smaller and smaller rectangles -- a 6-yard box within an 18-yard box that rests within each half of the field. The goal frame adds its element of perpendicularity. A circle, with its painted diameter right out of a textbook, sits in the middle of the field.
And if the United States is to end a World Cup drought that stretches back more than 15 years to a sun-drenched day in the Rose Bowl, the triangle might be the shape that matters most. Its lines invisible as it shifts, sometimes obtuse, other times acute, it is the product of points occupied by Lauren Holiday, Carli Lloyd and Megan Rapinoe.
It might not be a right triangle, but it needs to be the correct one next summer.
"I think we're all soccer players -- we have a soccer player's brain, which works well," Lloyd said of the trio which started together the past six games, including the first three of the CONCACAF Women's Championship. "Lauren can possess the ball, can turn with pressure on her back and can find the gaps and seams. I think that's really kind of what I'm trying to do is find those gaps and seams where I can face up. And Pinoe is smart with her movement, as well.
"If we can form triangles all around the field, we're really going to pick teams apart."
That Holiday, Lloyd and Rapinoe are three of the most talented players in American women's soccer at the moment is a starting point safe from all but the most contrarian views. All are in their primes, Lloyd approaching her fifth major tournament, Holiday her fourth and Rapinoe her third. All are also midfielders, if not exclusively so throughout their careers in some cases. And United States coach Jill Ellis believes in a 4-3-3 system that utilizes three midfielders.
It isn't upper-level math to add all of that up if the objective is to get them on the field at the same time.
As things stand, they equal the midfield. The triangle in its entirety.
"We certainly want those players on the field," Ellis said. "And so we're looking at the functions of what we want out of them."
There is skepticism in some quarters, not unfairly so until evidence otherwise accumulates, as to how that will work. Unlike many World Cups and Olympics past, a lineup that includes those three lacks a traditional defensive midfielder in the mold of Shannon Boxx, a role Keelin Winters and Jen Buczkowski fill well in the National Women's Soccer League and Julie Johnston might fill for the national team in years to come. It means Holiday, the 2013 NWSL MVP and an unstoppable force as an attacking midfield presence during FC Kansas City's run to a championship this past season, inherits an unfamiliar role behind Rapinoe and Lloyd, closer to the back line than the forward line.
But if it's a puzzle to which the pieces seem imperfectly suited, Ellis is gambling that it's the people who will make it work.
Every triangle, after all, has three sides.
Getting at people. It is the phrase Ellis used in explaining why she wants Lloyd to play a box-to-box role, why she believes the 32-year-old who is coming off one of her best professional seasons is at her best when taking on opponents -- getting at them. There is nothing Lloyd likes more than someone who would stand in her way. They're the only ones you can push aside.
The oldest of the midfielders and the most experienced at the international level, Lloyd also came the closest to never making it to the senior team in the first place. A New Jersey product who played college soccer at Rutgers, a far cry from blue blood programs like North Carolina, Stanford, UCLA and Virginia that regularly stock the national team, she nearly walked away from the sport when her progress stalled with the youth national teams more than a decade ago.
Frustrated when no longer able to succeed on athleticism alone, she nearly gave up on herself before anyone else had a chance to do the same.
"Everything was always blaming the coach, blaming teammates," Lloyd said of her youth. "It was never looking inside of myself saying, 'You know what, I've got to make all these players around me better, so I have to look inside myself and see what I can do better.' I just had a totally different mindset. It's great to look back, but I do wish I had a coach that would have benched me when I was giving 80 percent, not reward me when I was giving 80 percent.
"My 80 was better than most people's 100, but you don't learn a great deal out of giving 80 percent."
Cut from the U-21 team by coach Chris Petrucelli, in retrospect deservedly so in her own estimation, Lloyd finally got that message. She committed mentally, and she found a trainer in James Galanis whom she felt provided the physical blueprint to make the revelation matter. By 2005, a new U-21 coach told Lloyd, a player whose college exploits she heard about but whom she had never seen, that she was one of the most gifted players the coach had encountered. That coach? Ellis, who subsequently took Lloyd, Rapinoe and Holiday (née Cheney) as part of a team that won that year's U-21 Nordic Cup in Sweden. That was the same summer Lloyd made her debut for the senior team.
Almost a decade later, Lloyd is closing in on 200 career appearances and has scored more goals as a midfielder than anyone for the United States, but she still carries the chip on her shoulder -- and readily admits to using it as motivation -- from those who doubted her place in the lineup or the upper echelon of the women's game. Like the Depression-era generation that remained painfully frugal decades after their own financial lots were secure, never willing to let those memories of lean times soften or fade, she seems always in search of a challenge and in search of fuel, lest that fire fade away.
"She smiles when she scores," Rapinoe said. "But not too many other times."
Any number of referees who have earned her wrath will attest that Rapinoe isn't always a bucket of chuckles on the field, either, but she is nevertheless a counterweight in both style and demeanor to Lloyd. Rarely the biggest, fastest or strongest player on the field growing up, Rapinoe's game grew through subtlety and subterfuge. The space nobody else wants or thinks to concern themselves with -- such as the space above a defender's head that a sand-wedge chip pass might arc through -- is the space in which she reigns.
The result of all that imagination was what she described as a tough relationship, at times and especially early on, with Pia Sundhage, the coach under whom she first gained a foothold with the national team after coming back from two ACL injuries that cut short her initial introduction under Greg Ryan. The Swedish coach wanted a wide player who would deliver crosses, a skill Rapinoe displayed for posterity when she set up Abby Wambach's equalizing header against Brazil in the 2011 World Cup. But only as time went on, first under Sundhage but more so still during Tom Sermanni's tenure and now under Ellis, did she gain full freedom to stray in search of invention.
The player most likely to find a parallel world at the back of a wardrobe, she needs that freedom. And the United States needs her to find it to diversify its attack.
"I like to take risks, and I think I've become better at calculating which ones to take and which ones not to take," Rapinoe said. "I like to enjoy myself on and off the field. I think there are times in my career when I've just gotten too serious, or tried to play too serious, and I don't think that suits me on or off the field. I think in ways I felt like, 'Oh, I have to try and be more serious and that's more professional,' but there's really nothing that proves that. It's different for every person.
"For me, I think when I'm enjoying myself and taking risks and doing things that other people aren't doing, aren't thinking about doing, that kind of sets me apart but also makes me happy."
All of which leaves Holiday, the player Ellis a decade ago recruited to come to UCLA as a target forward, in the almost Pirlo-esque conductor role behind the booming timpani and soaring violins at the top of the American lineup.
It isn't the role Holiday was born to play, but Ellis is banking on it being the role she has not only the skill but the demeanor to master.
"I think [Holiday's] just normally calm and very relaxed and has a cool head and has a sort of good ability to not get too high or too low," Rapinoe said.
If the story of the midfield is in some way the story of the twists and turns that led each player to the present moment, Holiday's begins when she nearly ended up on the other side of the country for college at North Carolina. How close was it? By her telling, about the width of a scrap of paper. At an impasse in choosing a school, she settled matters by literally pulling UCLA's name out of a hat. That it was actually the second name that came out of the hat, Virginia's having been pulled and promptly ignored, suggests it was less an exercise in chance than focusing her mind on what she really wanted all along.
Nevertheless, the Indiana teenager who wanted a change of scenery got that and more.
A player who had struggled to the point of discouragement to make the regional Olympic Development Program teams throughout much of her youth, Holiday emerged as one of the rising stars of the women's game at UCLA. She also met future husband Jrue Holiday, all of which suggests she -- or the hat -- made a wise decision. But dropped in a wholly new world in Los Angeles, she first had to tame what she described as her innate "confidence" and what the rest of us might call stubbornness. It was partly through conversation with Ellis that Holiday came around to the idea that, in the language of California, sometimes it's best to go with the flow and figure out the positives in a situation rather than fixate on the negatives.
"It did change my perspective," Holiday said. "Even though maybe I didn't agree with things, I could accept it for what was good. I think that opened my horizons and I was able to branch out from there."
They call that foreshadowing in Hollywood.
It is still strange to see Holiday so often removed from the play in front of an opponent's goal. The past two seasons with FC Kansas City, where coach Vlatko Andonovski entrusted her with running the show and reaped the benefits that came with that decision, showcased a player coming into her own as an attacking force, an Americanized version of the No. 10 role otherwise filled by the likes of Kim Little and Vero Boquete. She may share the sentiment, but she also sounds like someone looking for the positives.
"I love playing the 10," Holiday said. "I think that's where I'm probably most comfortable, and that's where I feel the most free -- not having to think. I think that will be my favorite position forever. I think that's me, that's where I love to play. But I think there are parts of the 6 I find very interesting and very intriguing, being able to hit a ball to spray it wide, and can I hit that person's feet every time? There are challenges there I think can only make me a better player."
That, in turn, will make Lloyd and Rapinoe better players by freeing them to be themselves. And that, in turn, will make the United States a better team.
That's the plan, at least.
"[Holiday] more or less wants to probably get higher up on the field and be more involved in our attack," Abby Wambach said of the learning curve. "But the reality is she's so good at sitting and holding the ball for us and keeping possession, so that our back five can build the ball up out of the back. [What separates her is] her ability to spray the ball wide -- not only get the ball wide but have it be precise and accurate on the touch line, whether it be to [Sydney Leroux] or Christen Press or Heather O'Reilly.
"If we can get a quick change of point through our No. 6, I don't know if there is a team in the world that can beat us."
It's all about making the geometry work.