Should I email or text the professor as a apology if it seems that he is ignoring my email There is a professor that I work with in remote. We are in touch primarily via email about research collaboration. Recently, I'm trying to brainstorm a research idea for my master thesis. I'm new to research and I need a little bit hand-holding on finding a good idea. Over the summer, I mostly work on my own to fill in the background knowledge and try to keep the professor informed about my progress. 

The professor usually replies email quite quick during the academic time. But, in the summer, the speed of response can be 10 - 12 days. I totally understand because it is a summer time. Last time when I ask for some inputs on what references I should take a look. He replied with a pointer to some research group. I'm pretty excited to get the response. One of the reasons is that I have read some papers from the research group pointed out inside the email. I replied immediately later that night of the same day. This is the last contact we had so far. Around a week later, I sent out another email looking for doing some research work in some other project as a stepping stone. No reply for this as well.

Now, I notice he has updated his website in the past two days. I'm figuring he is back to work. But, I still haven't got any reply. Now, I'm in this hanging position and not know what to do next. Maybe because of my emails I send out earlier? What I should do in this case.

Thanks much!
I think you should apologize only if you have done something wrong. That doesn't seem too be the case here. He is likely just busy. It would be good not to flood him with things for a bit, but likely he is catching up on a backlog of work. 

It is even possible that he thinks you are working fine for the moment and don't need an intervention. 

Stay calm and carry on. If he is unhappy in any way you will likely hear.