Does anyone know any resources for Quaternions for truly understanding them? I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want to just memorize the formulas for using rotation quaternions, I want to truly understand them.
It's the most difficult thing I've ever tried to learn. Does anyone know any good resources for understanding quaternions? 
 A: One piece of quick advice before I recommend stuff on quaternions. Understand how complex numbers produce rotations in the complex plane first. Maybe you've already done that... if so, that'll be a helpful foothold.
Several questions on this site might be helpful:
How do quaternions represent rotations?
How can one intuitively think about quaternions?
How do you construct the quaternion and the multiplication rules, like Hamilton did?
Is there a geometric realization of Quaternion group?
Quaternions and Rotations
Then there is the wiki page devoted to this topic:
http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
If you have funds and patience there are a few books:
http://www.amazon.com/Quaternions-Octonions-John-Horton-Conway/dp/1568811349/ref=sr_1_5?ie=UTF8&qid=1363353978&sr=8-5&keywords=quaternions+and+rotations
http://www.amazon.com/Quaternions-Rotation-Sequences-Applications-Aerospace/dp/0691102988/ref=sr_1_1?ie=UTF8&qid=1363353978&sr=8-1&keywords=quaternions+and+rotations
http://www.amazon.com/Rotations-Quaternions-Double-Groups-Mathematics/dp/0486445186/ref=sr_1_2?ie=UTF8&qid=1363353978&sr=8-2&keywords=quaternions+and+rotations
One more thing: if you've only been studying it for a week, don't get discouraged! There is no reason to expect that you will get it all completely so quickly. I took up the same task that you are describing several months ago. I've had a lot of fun picking up the basic idea, and I'm still learning a lot about it all the time. Even after this time, I would not say I "truly understand them," but I definitely have a better grip on quaternions and their relationship to rotations.
As the old saying goes, "Don't worry about going slowly, worry about standing still."
A: I know that I am very late in this response, and that probably you found some paths to follow on quaternions, however, I thought that you might like to know these original resources on quaternions that I discovered when doing my own research to learn about quaternions myself:

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*An Elementary Treatise On Quaternions_Peter Guthrie Tait

*Introduction to Quaternions_Philip Kelland, Peter Guthrie Tait

*On Quaternions_William Rowan Hamilton

*Elements of Quaternions_William Rowan Hamilton
Also, to grasp Quaternions intuitively, I recommend this book that I recently found: Visualizing Quaternions by Andrew Hanson.
Best of luck in your learning ventures!
A: I may be a bit late to the party but having found this thread, I thought I'd share one good site I've recently stumbled across for anyone else who ends up seeing this thread:
Visualising Quaternions - Grant Sanderson and Ben Eater
This site allows you to physically play with some parameters while receiving the knowledge to understand what you are doing. For me, interacting with quaternions was definitely the best way to understand more about what I am actually doing in my head as the tool allows you to freely interact and see the result even after the lessons, a playground of sorts.
A: I found this website helpful for receiving a feeling for quaternions. The sites provide many references and describe in detail which systems have been used and the relationships to other representations of rotations such as Euler angles. 
However, the design will not please everybody - I have to admit that I had to get used to it too.
