But Egypt is also committed to a peace treaty and security cooperation with Israel, whose goal of blocking the success of the radical Islamic Hamas in Gaza is quietly shared by Mr. Mubarak. Egypt receives over $2 billion a year in aid from Washington, which is also committed to stopping the rise of Hamas, largely because the group refuses to recognize Israel's right to exist and opposes American-sponsored peace talks with Israel.
Hamas, by creating a border crisis after Israel extended its import restrictions to Gaza in another failed effort to stop rocket and mortar fire into Israel, managed to divert criticism of its management of Gaza.
As important, Hamas earned a degree of respect for action even from its opponents in Gaza, further consolidating its control. Now Hamas is trying to force Cairo to acknowledge that control and to deal directly with Hamas to solve the border crisis.
Abdel Moneim Said, director of the Ahram Center for Political and Strategic Studies in Cairo, blamed Hamas for recklessness. "Hamas wanted to pull Egypt in through many things: the tunnels, pushing people to Egypt, and now they are trying to blame Arabs for not rescuing the people in Gaza," he said. "Hamas is managing the crisis by distributing the blame, although it is the one responsible, or partially responsible at least, for what is happening."
But it will be very difficult now for Mr. Mubarak to reseal the border completely. More likely, he will work something out with Mr. Abbas and Israel to allow a regulated border crossing. But even that will resound as a Hamas victory, because both Egypt and Israel will have been forced to a concession that they could have negotiated freely with Mr. Abbas any time in the last six months.
Mr. Abbas will meet with Mr. Olmert on Sunday, with the news, as usual -- this time, the border crisis -- overshadowing their efforts to push along peace talks. In a speech on Saturday in Ramallah, Mr. Abbas made it clear that he would press for his plan for the Palestinian Authority and his own presidential guard to once again take over the Gazan side of the crossings from Israel into Gaza, and from Gaza into Egypt, despite the Hamas takeover of Gaza last June.
Mr. Abbas repeated that he would not talk directly with Hamas until it apologized for its "coup" in Gaza and handed over power there to the Palestinian Authority. In saying that, he essentially rejected an Egyptian invitation for talks with Hamas in Cairo that the exiled Hamas political leader, Khaled Meshal, had accepted on Friday.
Israel has previously rejected Mr. Abbas's proposal because it would reopen the crossings and take pressure off Hamas, as well as putting the crossings effectively under the control of Hamas.
But Israel's recent effort to further intensify the closing of Gaza, by cutting off nearly all supplies and forcing the extension of rolling power cuts to more than 12 hours a day, clearly backfired, giving Hamas a kind of moral pretext in the Arab and Palestinian world to break through the Egyptian border. As the daily newspaper Haaretz said in an editorial on Friday, "The siege of Gaza has failed."
Defense Minister Ehud Barak of Israel, with no adequate answer to stopping rocket and mortar fire from Gaza and reluctant to have a major military incursion that wouldn't stop them either, intensified the closing in what his aides called "an experiment."
"The experiment blew up in their faces," said Shlomo Avineri, a political scientist at Hebrew University. "The whole theory of putting pressure on a population to put pressure on their government doesn't work. It didn't work in Lebanon in 2006, and it didn't work now."
The policy was denounced by the European Union and the United Nations as "collective punishment" and as illegal under international law. "Whether it's against international law or not, the fact is that the policy was ineffective," Mr. Avineri said. "Barak made a mistake in thinking it would turn the population against Hamas; it did the reverse."
Israel's larger error, after pulling out of Gaza in 2005, was to view it almost entirely as a security problem, with a main focus on smuggling of weapons and rockets from Egypt into Gaza. "The whole relationship with Egypt became subsumed under questions about smuggling," Mr. Avineri said. But the relationship of Gaza to Egypt has major strategic and political implications, he said. "Why should the border be sealed between two Arab populations?" he asked. "Israel should support some regulated border regime."