While Alexander was sieging Tyre Darius send him a messager with a proposal: Alexander would get all the land west of the Eufrate, 10000 talents, and the hand of Darius' daughter, but in return Darius' family had to be reunited, and a permanent alliance between both empires should be signed. Alexander rejected the proposal as everything that Darius offered was in his reach anyway, and he could marry Darius' daughter with or without the approval of her father he said.
Tyre fell in the hands of Alexander and he continued his journey southwards. Before he could enter Egypt he had to deal with another coastal city called Gaza. Just like Tyre was this city extremely well defended, and just like Tyre it refused to let Alexander in the city. The city was placed on a high rock, and attacking it was almost equal to suicide, but for Alexander difficulties were only possibilities to prove his invincibility. He attacked and indeed took the city. All male citizens were killed while the women and children were made slaves. Again Alexander had shown that he was invincible, but also that he can be ruthless if needed.
The stories of Alexanders victories had of course reached Egypt, as well as the rest of the known world, and the Persian satrap in Egypt wisely decided that trying to stop Alexander would be useless. The Persian satrap simply did not have enough men to form an adequate army, so he did not resist the combined Greek-Macedonian invasion. The Egyptians even welcomed Alexander in their land as they finally would get rid of the hated Persians. Alexander was seen as a liberator and made pharaoh. The Egyptian pharaohs were always seen as Gods, so Alexander even more felt like a God. But what finally convinced him that he was a God was an oracle which called him the son of God.
With the conquest of Egypt Alexander secured the homeland from a Persian naval attack, and now it was time to move to the heart of the Persian empire. He moved through Syria, crossed the Eufrate and Tigris and met the Persian army at the other side of Mesopotamia: in the plains of Gaugamela near the city Arbela. For Darius it was now a matter of winning the battle, or his empire would be destroyed by Alexander.
Phase one: attacks of the Persian cavalry.
Darius had once again proved that he was an excellent organiser. The size and diversity of his army could only be compared to the army of Xerxes during his attack on the mainland of Greece. His men came from all the regions of his vast empire: Scyths, Bactrians, men from numerous small tribes, but also Greek mercenaries. Besides footsoldiers he had chariots, and even 15 war-elephants, but again the biggest part of his army consisted of cavalry. That is why he had ordered his troops to level the battlefield so he could use his cavalry and chariots optimally.
Alexander on his turn had also prepared himself for the battle. He had captured, and interrogated, several Persians and thanks to this information he was able to reconstruct the Persian army. On top of that he sent out several reconaissance missions, and he was even in charge of one of them, to get as much information as possible about the Persian army. Now he knew how Darius was planning to use his forces, and he placed his army in a sloped formation. A big part of the phalanx was positioned behind the first phalanx as reserves. As a result he did not have enough man to form a strong left wing. It was the job of the cavalry to secure this wing.
The Persian army used a completly different formation: the whole army formed one straight line, with in the center the chariots moved a little ahead of the rest, with archers to back them up. Darius wanted to attack the right wing of Alexanders army because there were the most experienced forces. If he could defeat them, the Macedonian-Greek army would surely fall apart and be an easy target he thought. He sent more and more cavalry to his left wing to increase his superior power in that area even more. Alexander saw the danger and knew that he had to do something. He could use his reserve phalanx, but he rather kept them ready to fill up gaps in his formation. Instead he moved his whole army towards the Persian left wing.
Both armies could have continued this for a while, but then they would leave the area which was leveled by Darius for his cavalry. The moment had come to attack and he ordered Bessos to attack with the left wing. Thousands and thousands horsemen charged forwards where they crashed into the right wing of Alexander which consisted of light and heavy cavalry, supported by light forces and archers. Alexanders right wing, under the command of Ariston, had a difficult time in the beginning and was not able to hold position, until a part of the reserve phalanx moved forwards and joined the battle.
Phase two: the Macedonian formation breaks.
The attack of Bessos on the Greek-Macedonian right wing had created a weak spot in the Persian formation. His cavalry had moved forward for the attack, and the Persian phalanx had to stretch out a bit to keep in contact with the charging cavalry. Alexander was lucky that he had placed his elite heavy cavalry just in front of this spot. He charged forward and ordered his phalanx to attack at the same time. Darius saw that Bessos is not making any progress, and ordered his complete right wing, under the command of Mazaeus, to attack in the hope of overwhelming Alexander with his numeric advantage.
A part of the Indian and Persian heavy cavalry spots this gap, and manages to move through it to the heart of the Greek-Macedonian army: the baggage. They kill the guards and liberate many Persian prisoners of war before the reserve phalanx stops their attack with the help of the light footsoldiers. This Persian attack is stopped, but the gap in the formation still exists and Parmenio is having difficulties keeping the left wing together. He sents a messager after Alexander who immedeatly stops his pursue of the Persians and turns around.
Alexander wants to move towards the center of the Persian army and attacks it in the back with his cavalry, but he crashes into the retreating Persian cavalry who had attacked the baggage of Alexanders army several minutes before. This way Alexander can not complete his manouevre, and Parmenio must wait a little longer before reinforcements will decrease the pressure on the left wing some. Then the chances suddenly change: Parmenio not only manages to stop the attack of the Persian cavalry, he even manages to push them back with his Thessalian cavalry! The Persians start to retreat in this area of the battlefield, and Alexander can move closer to the center. Darius feels threatened with Alexander and his elite forces so close to him, and decides to flee as fast as he can. The same situation as during the battle of Issus occurs: the remainder of the Persian army (this time the left wing) hears that Darius is leaving the battlefield, and decides to do the same.
The results of the battle.
At the moment of Darius' departure was the battle far from decided. Alexander might have forced an important break-through with his heavy cavalry, but just like at Issus left Darius way to early. His left wing and center were still intact, and both parties would have had most likely equal chances if he could close the gap in the left of the center, and assemble his right wing again. The results of the battles of Issus and Gaugamela might have been completly different if the Persians had a more determined high king.
Now the way to the heart of the Persian empire was open for Alexander and important cities like Susa, Babylon, and Perspepolis were conquered by him one after the other. Darius became a fugitive in his own country, and was eventually killed in one of the northern provinces by one of his own officers. Alexander declared himself high king of Persia, and executioned the murdered of Darius according to Persian laws. He had quickly conquered the central provinces, but the eastern provinces in the mountains were much harder. After three years of warfare he finally managed to subject them to his leadership.
The next years Alexanders problems were more of a political than a militaristic nature. As high king of Persia he had despotic power, and slowly he evolved into a despotic and tyranic person. He executioned the son of Parmenius as he suspected him of treachery, and he also killed Parmenius as he was afraid for revenge. On top of this he realised that he could not keep such an enormous empire together if he would not have a good relationship with its inhabitants. He started to dress and act more and more like a Persian in the hope of establishing this relationship, but the result was that he alienated himself from the Greeks and Macedonians.
Finally Alexander moved eastwards and invaded India in 326 BC. Maybe he could not stop his hunger for more conquests, or maybe he wanted to increase his prestige as political leader by more military victories. In India he gained many more victories, but when he had crossed through the Punjab his army revolted for the first and the last time: they refused to move any further to the east and wanted to go home. He turned around, and the last part of his expedition through India was more like an exploration. In 324 he arrived in Susa again, and started to make new plans. He wanted to conquer the areas dominated by Carthage, stop the growing power of Rome, and finally conquer the Arabian peninsula. If he would have achieved this he would have conquered the whole known world. Alexander died in 323 because of a fever.