Over the past several million years, the planet has had numerous, separate ice ages. Some of these, like the last one, were so extensive and had so much ice that world wide, sea levels dropped over 300 feet!
At those times, the straights at the exit of the Red Sea would have only been a few miles across, and the distant land could be easily seen from either side. I can image people, just like they are now, were very curious as to just what was on the other side. Logs float, and can be tied together to make rafts.
The rest, as they say, is prehistory!
Just what is the proof that humans began in Africa???? Maybe they earlier came TO Africa from somewhere else.....
Looks like the Red Sea was actually closed at the southern end, so theoretically people could have simply walked across from Africa into (present day Yemen) the Arabian peninsula.
I used to have a map of the same but 5 times larger on my old web server, but I cannot remember the link, and gave the server to someone else to administrate!
