And is his slightly blackened left eye all but winking at the wholesale violence that preceded the camera’s click? The attack on a meet-and-greet event with a congresswoman outside a supermarket; the killing of six people, including the chief federal judge in Arizona and a 9-year-old girl; the wounding of 13, including Representative Gabrielle Giffords, shot in the head.

Since last Saturday’s shooting frenzy in Tucson, investigators and the news media have spent the week frantically trying to assemble the Jared Loughner jigsaw puzzle in hopes that the pieces will fit, a clear picture will emerge and the answer to why will be found, providing the faint reassurance of a dark mystery solved.

Instead, the pattern of facts so far presents only a lack of one, a curlicue of contradictory moments open to broad interpretation. Here he is, a talented saxophonist with a prestigious high school jazz band, and there he is, a high school dropout. Here he is, a clean-cut employee for an Eddie Bauer store, and there he is, so unsettling a presence that tellers at a local bank would feel for the alarm button when he walked in.

Those who see premeditation in the acts Mr. Loughner is accused of committing can cite, for example, his pleading of the Fifth Amendment or the envelope the authorities found in his safe that bore the handwritten words “Giffords,” “My assassination” and “I planned ahead”  or how he bided his time in the supermarket, even using the men’s room. Those who suspect he is insane, and therefore a step removed from being responsible for his actions, can point to any of his online postings, including:

“If 987,123,478,961,876,341,234,671,234, 098,601,978,618 is the year in B.C.E then the previous year of 987,123,478,961,876, 341,234,671,234,098,601,978,618 B.C.E is 987,123,478,961,876,341,234,671,234,098, 601,978,619 B.C.E.”

What the cacophony of facts do suggest is that Mr. Loughner is struggling with a profound mental illness (most likely paranoid schizophrenia, many psychiatrists say); that his recent years have been marked by stinging rejection  from his country’s military, his community college, his girlfriends and, perhaps, his father; that he, in turn, rejected American society, including its government, its currency, its language, even its math. Mr. Loughner once declared to his professor that the number 6 could be called 18.

As he alienated himself from his small clutch of friends, grew contemptuous of women in positions of power and became increasingly oblivious to basic social mores, Mr. Loughner seemed to develop a dreamy alternate world, where the sky was sometimes orange, the grass sometimes blue and the Internet’s informational chaos provided refuge.

He became an echo chamber for stray ideas, amplifying, for example, certain grandiose tenets of a number of extremist right-wing groups  including the need for a new money system and the government’s mind-manipulation of the masses through language.