This week, American history merges with Greek myth and Marvel comics in Culver City and a dance on the beach grapples with water and the politics that control it.

5. Mystery at the museum

Last year, at the Metropolitan Museum in New York, visitors had the chance to solve a mystery using a phone app — apparently Madame X, the red-headed subject of John Singer Sargent's famous 19th-century painting, had been murdered at a gala. Costumed performers (a detective, a reporter) helped solve the crime. Performers will appear intermittently when something similar happens at the Hammer this weekend. On arrival, museum visitors check out handheld radios, which will be triggered by transmitters around the museum, clueing visitors in to plot points in an unfolding mystery play, the exact nature of which has yet to be announced. It's organized by radio station in residence KChung. 10899 Wilshire Blvd., Wstwd.; Sat., Dec. 7. (310) 443-7000, hammer.ucla.edu.

4. Media meld

The group Parallax Beach, which includes a video artist, composer, saxophonist and performance artist, tries not to privilege any medium over another. This means in performances like the one they did in Long Beach this year, spinning discs might be projected on perpendicular screens while a muscular, shirtless figure in face paint flexes and turns slowly between them and a live saxophonist interrupts electronica. Nothing vies for more of your attention or takes a back seat to anything else. It's a club-ready vision of perfect cooperation. The group performs at Pehrspace. 325 Glendale Blvd., Echo Park; Mon., Dec. 9, 9:30 p.m., $10. (213) 483-7347, pehrspace.org.

3. Degrees of support

Artist John Burtle probably is less known for studio work than for the frequency with which he shows up for friends and does things with other people — the “gallery” he had on his arm with different proposed designs tattooed into his skin, his work with free-ranging radio station KChung, the time he's spent volunteering at artist-run spaces. So the fact that “Support Constructs,” his first show at Michael Benevento gallery, includes paintings and sculptures that hang on walls or sit on floors at first seems uncharacteristically conventional. But then you realize he's asking the same question he grapples with in other arenas: How does one thing support another in an interesting way? No canvas hangs on its frame in an obvious way. Sometimes Burtle applies paint to canvas piecemeal after peeling already-dry chips off a plastic sheet. Sometimes paint stands alone, like the thick, three-dimensional human heart made of acrylic, encased in a band of black and protruding from a back wall. 7578 Sunset Blvd., Hlywd.; through Dec. 21. (323) 874-6400, beneventolosangeles.com.

2. Water wars

Water and power, the political and financial kind, always have a tense relationship, which is what Beyond Control , the performance Caroline Caycedo organized with choreographer Rebecca Hernandez, grapples with. Caycedo has been thinking particularly about the violence surrounding Colombia's El Quimbo dam, which corporations support and many farmers oppose. She chose the beach, right where Pico Boulevard ends, for the performance's location, and dancers will appropriate paramilitary moves, police strategies of containment and the fluid motion of water. Audience members will start out surrounding the performers but likely later will find themselves surrounded. Where Pico terminates, Santa Monica Beach; Sat., Dec. 7, 2 p.m. (310) 453-3711, 18thstreet.org.

1. Hair-raising

Hair stands on end in Jim Shaw's current exhibition at Blum & Poe. It's thick, sandy blond and stiff, thanks to fabric hardener, and more than 5 feet long, which means the hair would reach at least someone's calves were it actually hanging from a head. But there's a see-through resin sculpture of a split-level house where a head should be. The hair sculpture stands in front of a big, operatic mural on which black-and-white superheroes, gods and goddesses tumble around in front of a faded painting of a Mark Twain–worthy steamboat. Shaw's entire, epic show is like this: hyper-reality meets domesticity meets Americana meets myth. 2754 S. La Cienega Blvd., Culver City; through Dec. 21. (310) 836-2062, blumandpoe.com.

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