In fact, according to the 20-year-old actor, "I owe my entire career, at least to this point, to 'Twilight,' and I'm very grateful for that."
In the movies, which are based on author Stephenie Meyer's best-selling fantasy romance novels, Gordon plays Embry Call, one of the friends, or "wingmen," of Jacob Black (Taylor Lautner).
Gordon, a newcomer at the time, got the much-coveted part due in part to a lucky break. He is a member of Meyer's LDS Church ward in Arizona, and "when she told me they were casting for the movie, I jumped at the opportunity."
And he said he has enjoyed the chance to play a "strong" Native American character in the film. Gordon is a member of Arizona's Hualapai tribe, which is probably best known for maintaining and operating the Grand Canyon's "Skywalk."
"I love being able to honor my roots and my heritage, especially in movies that are so hugely popular," he said by telephone from Los Angeles.
Gordon said his mother and manager, Camille Nighthorse, has been responsible for his cultural upbringing.
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And that will certainly be on Gordon's mind when he appears in Utah this weekend. He is one of the celebrity participants in the Native American Celebration Golf Tournament in Lehi July 23. And on July 24, Pioneer Day, he's appearing at the 16th annual Intertribal Competition Powwow in Liberty Park.
Gordon will be signing autographs in the afternoon and will be joined by other Native American entertainers, including his mother and actor/stuntman Patrick Shining Elk.
Gordon said he is "looking forward to enjoying the traditional music and entertainment."
In fact, he said he is a fan of many different music styles. He describes his band, Touche, as "progressive metal."
He and Touche got to perform at the red-carpet premiere of "The Twilight Saga: Eclipse" earlier this year.
"It was fun to play our music in front of that many people," he recalled, adding that "my co-stars were even dancing on stage while we played."
Gordon and Touche were originally set to be part of this year's Vans Warped Tour bill, but "unfortunately things got too crazy and we had to drop out."
Those "things" include Gordon's busy movie career. While he's here, Gordon is expected to sign on for a feature-film project with writer/director Eric Hendershot ("Down and Derby"), titled "Phantom Justice."
He's also about to start work on "Into the Darkness," a horror film that also features Mischa Barton and John Schneider.
Then, of course, there's "Breaking Dawn," the final part of Meyer's "Twilight Saga." The fourth book will be split into two movies, which will be made under the guidance of director Bill Condon ("Dreamgirls").
To save costs, Condon and officials at Summit Entertainment decided to shoot the films back to back.
"It's going to take us several months to shoot the whole thing. It's going to be absolutely crazy," Gordon said. "We'll take a break early next year and then have to dive right back in."
Gordon said he is hoping his character will get more to do in these two films, though he knows "obviously, people aren't just going to these movies to see me, unless they're my friends or family.
"It doesn't help that my character changes shape when he goes into action, so I don't really get to do as much physically as I would like."
Still, Gordon said he is looking forward to having a career after "Twilight."
"It will be nice to do something else besides these movies," he said. "But I am a little scared at the same time. Hopefully my newfound fans will follow me along to whatever else it is I do after that."
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As for the three Eclipse roles up for Twilight Saga grabs in the casting notice, here's the lowdown:
