The Islamists talk of restoring the glory of the caliphate, an earthly Islamic paradise that is to reach across the globe from Istanbul to Indonesia.
It will ''enjoin good and forbid evil and eliminate every vice'' under a single righteous ruler, according to the group Islamic Awakening.
Timbuktu still bitter over Islamists' sharia rule
Residents of Mali's ancient city of Timbuktu denounce the Islamist militants who controlled their city for 10 months.
But in case anyone is tempted by the vision, we should consider the reality.
Last week, the French military successfully routed the Islamic extremists who had taken control of the northern African nation of Mali, which had been a poor but stable democracy for 20 years until the past year or so.
Mali, and its fabled ancient city of Timbuktu, is a former French colony whose land area happens to be twice that of France.
So the liberated people of Mali are now free to tell the story of their experience when the Islamists swept into town, heavily armed in their battered utes and LandCruisers.
Here is how the self-declared agents of Allah set about delivering the glorious vision of the vice-free caliphate when they had the opportunity.
Women were systematically repressed and violently abused. Salaka Djikke, 25 years old, was targeted by four members of the religious police in Timbuktu for accepting an evening ride on the back of her boyfriend's motorbike.
''The four men began shouting at her in Arabic, a language few in the city speak, while one of them slapped her and another lashed her with a whip,'' reports a British newspaper, The Independent. ''I couldn't understand what they were shouting and why they were whipping me,'' she said.
Djikke was locked in an office of the Malian Solidarity Bank that the Islamists had converted into a makeshift station for the religious police. It was so crammed with scores of other women that she couldn't lie on the concrete floor and had to sit instead.
Their crimes ranged from failing to wear a veil to speaking with unmarried men. Djikke's punishment was standard - 95 lashes of the whip, administered to her in the public market.
Another Timbuktu woman, Fadimata Alainchar, a charity worker, told CNN that women who refused to cover their bodies as ordered by the militants' version of sharia were imprisoned or raped.
Far from an earthly paradise, it's a barbaric vision of the caliphate.
''When entering the city, the signboard which was: 'Welcome to Timbuktu the City of 333 Saints' is now 'Welcome to Timbuktu, the gate to the application of the Sharia','' she said.
Women wearing glasses reported that the Islamists took their spectacles from them.
Women were raped in a systematic way. The US-based Human Rights Watch made this report: ''Victims, witnesses, and family members of victims told Human Rights Watch about a wave of abductions of women and girls by armed groups. Witnesses described the abductions by rebels of at least 17 women and girls as young as 12.
''A 14-year-old girl told Human Rights Watch that six rebels held her captive in Gao and raped her over a period of four days. A Timbuktu resident told Human Rights Watch that he saw three Arab militiamen drag a girl of about 12 from her mother into an abandoned building, where she was gang-raped.''
Staff of the Office of the UN's Special Representative on Sexual Violence in Conflict said that rape ''was condoned by top commanders'' of the militants as ''a tactic to subjugate local populations'', Bloomberg reported. ''Each night a different district would be required to provide a number of women and girls to the rebels.''
The militants' version of sharia imposed summary amputations and death. In a case last July in a remote town called Aguelhok, CNN reported, Islamists forced a man and a woman into two holes and stoned them to death, alleging they had committed adultery.
In October in Timbuktu, a man accused of theft was strapped to a chair in the public square and screamed as Islamists hacked their way through his wrist with a saw, then held his ruined arm aloft and repeatedly shouted the greatness of Allah, a doctor, Ibrahim Maiga, told The New York Times.
A local Islamic preacher whose family had lived in Timbuktu for seven centuries, Mahalmoudou Tandina, tried to stop the summary brutality by ''going to the Islamic court with stacks of Islamic law books under his arm'', The New York Times reported.
He was ignored. ''Islam was whatever they said it was,'' the paper reported him as saying. ''They did not respect the holy book. They respected nothing but their own desires.''
Yet the Islamists lay claim to transcendent authenticity. One of the self-appointed Islamist commissioners of Mali, Aliou Tour, said last August: ''We don't have to answer to anyone over the application of sharia. This is the form of Islam practised for thousands of years.''
They pressed children to join them as killers. Hundreds, and probably thousands, of children were bought and enlisted as child soldiers. Amnesty International quoted a local 16-year-old who'd been sold into the Islamists' ranks by a relative of his teacher. The boy was apparently fed drugs to make him more tractable.
''They trained us to shoot aiming at the heart or feet. Before the fighting, we had to eat rice mixed with a white powder and a sauce with a red powder.
''We also had injections. I had three. After these injections and eating the rice mixed with powder, I would turn like a motor vehicle, I could do anything for my masters. I perceived our enemies like they were dogs and all that was in my mind was to shoot them.''
The Islamists destroyed ancient knowledge and ancient cultural sites. They attacked ancient shrines and tombs of the more tolerant Sufi form of Islam. They burnt the Ahmed Baba Institute, a modern building housing priceless documents dating back to the 13th century, destroying thousands of Islamic originals.
''It was one of the greatest libraries of Islamic manuscripts in the world,'' said Marie Rodet, an African history lecturer at London's School of Oriental and African Studies.
This is the vision of heaven on earth that the Islamists impose when they have a chance. The Taliban who took over a weak Afghanistan were not a one-off.
Wherever a state is vulnerable to the parasite of Islamist extremism, it will arrive and take over the host body to do what it did in Afghanistan, what it has tried to do in the southern Philippines, in Somalia, in Algeria, in Mali.
Far from an earthly paradise, it's a barbaric vision of the caliphate that the International Federation for Human Rights described in Mali as the ''descent into hell''.
The danger has not passed. It is just regrouping.
Peter Hartcher is the international editor.