The Ancient Era
Settle in place, and push out a settler to grab the moo spot in the southeast. Research a normal C3C path to hit the Republic slingshot in 1000bc. Barbs are no great problem, huts give me loads of axes and workers, plus one tech, and I meet the other AI on Alpha fairly quickly. I am able to trade sufficient techs from them to keep my wonder-building programme running, but towards the end of the era the Mongols start getting belligerent, and hit me with an army sufficient to seriously threaten Mecca, and keep the entire rest of my empire busy desperatley pumping out archers. I hit the medieval in 390bc when Cleo researches Construction, and I buy it with Republic.
The Artistry Bee-line
Gifting up Osman gets him Feudalism, which eventually becomes the only tech that I learn from the Great Library; when I finally get a galley across the eastern ocean in 190ad (after about 10 fails), I find the French have that tech too. I meet the remaining civs over the next 150 years. Meanwhile, I research up the top branch of the tech tree, as the only sensible medieval tech path in 20k is to head straight for Shakespeare, with a possible detour for Astronomy if the shields are stacking up fast enough that Copernicus won't delay Shakespeare (but might make some time back by speeding up research through Democracy). In this case, Copernicus is well worth stopping off for, as lacking any coastal wonders, I need it to start my golden age. I reach Astronomy in 370ad, get Copernicus in 440ad, and follow it with Shakespeare in 550ad.
The Expansion Begins
While Mecca has been dedicating itself to wonder-hogging, the rest of the empire has concentrated on building up my military - support costs paid by tech sales to the AI, who have been able to come up with fairly convincing gpt. The army isn't great, due to the desperate resource situation down at this end of Alpha. Not even any saltpeter on my side of the Egyptian border! So, I have a couple of spears covering plenty of catapults and longbows. It isn't very glamorous, but it should suffice. I dow Egypt in 610ad, and begin a plodding war, fighting over some of the most dreary desert imaginable. There is one big upside though; I get three leaders very quickly, after just 3 elite wins, and then another 5, and another 1. The first forms an army of course (longbows, and one musket upgraded from a spear after I take the Egyptian saltpeter), and the second rushes Epic.
The Industrial Era
As my 1-move units slog inexorably through the Egyptian deserts, my GA-fuelled research makes short work of the bottom branch of the medieval tech tree; nothing after Democracy takes more than four turns, so I hit Gravity in 750ad, and enter the industrial era. There are a lot of medieval wonders outstanding at this point, and big AI builds still in progress, so I am reluctant to bring Osman into the new era. Even if I did, as the only Sci civ, he probably wouldn't be keen on selling me his free tech,
and there's only one tech of interest he could get anyway, Steam. Why do I normally agonise over this?!? So I set research to Steam and leave the AI languishing half way through the medieval. I'll bring Osman up when it safe, and hope he gets Nationalism or Sanitation.
Finally Gaining Strategic Resources
With the completion of Steam in 820ad, I find with relief that I have coal; start on rails. But after rushing to Industry for a factory in Mecca, I realise that I have forgotten I still don't even have iron! Luckily, the Egyptian campaign finally progresses beyond Thebes in 870ad, and I can colony the nearby iron. Mecca's factory is rushed 910ad, and followed by a coal plant. By 1070ad, I have reached the end of the easily accesible Egyptian cities, although there are still some beyond Ottoman lands. But I'm not about to bother with ROPs just to reach those. I give Cleo peace and dow Osman instead (after bringing him forward - he got Steam, meh). A couple of Ottoman towns later, and I can finally claim his horse source for my own. I had previously traded for Ottoman horses, so had some ansars, and with Joan's research of Mil Trad in 1120ad, I get to upgrade to cavs. War should go smoothly from here on then.
War Going Smoothly
In 1260ad, I reach the penultimate Ottoman town - I need to keep Osman alive for his modern free tech, of course. He gets peace, and Cleo gets dowed again. Osman has reduced her to living on a galley moored off the north coast of Alpha. A tough galley though, it sinks a couple of my frigates / privateers before I finally send it to the bottom of the sea. I dow the Mongols as well: also victims of Ottoman aggression, they have only a single town, which I immediately claim. Alpha is virtually under my control, so I start massing the troops to head east from the northern end of Alpha, to hit the Japanese, but am distracted by a not-very-sneaky sneak attack by the Spanish. I quickly take the equatorial island from them, then jump from there into their territory on Beta. Spain is small and weak, so by 1380ad I am done, leaving them in a single town on the island next to Alpha's northwest peninsula, pumping war happiness into my empire.
Mopping Up
I dow Toku as soon as I have finished the Spanish campaign, and move smoothly around the Japanese peninsula, wiping them out in 1465ad. By this time, Joan has unwisely sneak attacked me too. Aggressive lot, these AIs...

Access to France is rather blocked by China, but I ship some units over from my core, and raze their northwestern cities. Joan buys Mao onto her side, and then secures 20 turns of peace for herself by researching Espionage. No such luck for the Chinese; caught in a pincer between my forces from the French and Japanese campaigns, Mao is eliminated in 1545ad. When the French 20-turn peace expires in 1605ad, it takes only two turns to complete that job, despite Joan's attempt to settle the huge tracts of empty land on Beta. Just Bella and I survive to witness Mecca hitting 20,000 culture in 1665ad.
The Culture
4000bc: Found Mecca
4000bc: Palace (622c)
3550bc: temple (1206c)
1525bc: Oracle (1932c)
1000bc: Revolt to Republic
0775bc: Pyramids (1672c)
0670bc: library (1212c)
0430bc: Mausoleum (736c)
0390bc: Ancient > Medieval
0150bc: Gardens (1304c)
0170ad: Great Library (1668c)
0210ad: cathedral (816c)
0270ad: universtity (1040c)
0280ad: colosseum (514c)
0440ad: Copernicus (836c)
0450ad: Golden Age begins
0550ad: Shakespeare (1408c)
0640ad: Golden age ends
0700ad: Michelangelo (828c)
0750ad: Medieval > Industrial
0820ad: Newton (756c)
0830ad: Epic (500c)
0990ad: Bach (654c)
1070ad: Suffrage (404c)
1150ad: Darwin (279c)
1220ad: Hoover (172c)
1255ad: Academy (82c)
1275ad: Pentagon (78c)
1300ad: Industrial > Modern
1300ad: Wall Street (146c)
1305ad: Internet + lab (378c)
1395ad: UN (216c)
1400ad: Apollo (106c)
1445ad: SETI (132c)
1470ad: Smith (117c)
1500ad: The Cure (99c)
1505ad: CIA (32c)
1535ad: Longevity (78c)
1540ad: MASH (25c)
1565ad: The Bomb (40c)
1590ad: Star Wars (15c)
1665ad: win with 20103 culture; 5125 Firaxis points, and 7852 Jasons