There is obviously the Hagia Sophia, but along with Greek Fire, these are the only monumental achievement that I can think of that were invented by the Byzantines.
The Byzantine Empire existed for one thousand years. Constantinople was the centre for learning, it was the most powerful empire in Europe for a very long time. How come we hear so little about its technological advancements? Why don't we hear anything about Byzantine philosophers?
Not to stifle any further discussion, but I would refer you to these two [excellent] (http://www.reddit.com/r/AskHistorians/comments/25a93l/the_byzantine_empire_had_access_to_all_the/chfkcek) [responses] (http://www.reddit.com/r/AskHistorians/comments/25a93l/the_byzantine_empire_had_access_to_all_the/chfkclv) by /u/Ambarenya on this very topic.