The rotation problem and Hamilton's discovery of quaternions III

The rotation problem and Hamilton's discovery of quaternions III | Famous Math Problems 13c

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Күн бұрын

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
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Пікірлер: 35

@maxwellchiu6859 10 жыл бұрын

Prof. Wildberger: I studied engineering in college. Recently read Thomas Pynchon's "Against the Day" and had to look up quaternions. Had a hard time getting useful information until I found your videos. I now look forward to seeing your other lectures. I do this PURELY FOR ENTERTAINMENT - that's how good these lectures are. Thanks.

@timelsen2236 2 жыл бұрын

Thanks to Norman for this absolutely fantastic post on a difficult topic. Your the best.

@khayathicham 4 жыл бұрын

Thank you for this excellent pedagogy, or may I say andragogy. You solved my quest of understanding quaternions after many months of non fructuous lectures. I am watching these in a row like a Netflix series. Highly recommended and again thank you. Highly informative, Clarity, precision, avoidance of pompous and misleading terms and concepts. I'm impressed by the style of this particular lecture.

@DcCornnnnn 7 жыл бұрын

Thank you for giving these lectures on quaternion! Learned so much from them.

@jathavedanmadambi2323 Жыл бұрын

Really inspiring . Especially when guaternions are finding applications in many fields.

@chasr1843 6 жыл бұрын

This lecture is a masterpeice

@ehypersonic 11 жыл бұрын

Make me so happy to watch, only a few clicks away, this kind of lectures. I find impressive the amount of knowledge that one can find in the internet but how few people really get into it. Keep the good work.

@njwildberger 11 жыл бұрын

Good point, will mention this next time!

@rwconn 11 жыл бұрын

Another great video! Thank you so much for spending the time making them. I hope that there are more Math Foundations videos coming because I have been on the edge of my seat for awhile now. Thanks again for the videos and I look forward to more in the future.

@RichardAlsenz 2 жыл бұрын

Remarkable!

@relike868p 11 жыл бұрын

That's a good way to visualize quaternions!

@shafrannilamdeen2856 Жыл бұрын

thank you sir! for this valuable information

@7steelrainbow 8 жыл бұрын

Thank you very much for this great lecture. Just wonderful! ( For those who've got some difficulty at the last slide of this video, like me, I recommend to keep watching the next video, the 'part-d'. There, the professor more deeply explains about the picture of 4d space shown at the last slide and tells how rotations are resulted by multiplying quaternions. )

@dsfgoppudfgihdsf 5 жыл бұрын

I think these videos are fantastic...i think all your videos are fantastic. ty.

@DrDanielHoward 9 жыл бұрын

Excellent, so clear and informative. Thank you.

@marzioforte9364 11 жыл бұрын

Just an enthusiastic thank you for providing lectures 13c and 13d. I appreciate the style and methodology you use to present information. It is great. Thanks again.

@aniliitb10 9 жыл бұрын

all your lectures are so awesome !!

@riadhpc 8 жыл бұрын

Thank you very much

@njwildberger 11 жыл бұрын

Excellent! Keep at it.

@mikedavid5071 2 жыл бұрын

Thank you Mr. Wildberger for doing this. The rational approach must be best. I mean it’s math right?!

@brendawilliams8062 3 жыл бұрын

Thankyou.

@jethomas5 4 жыл бұрын

You have explained very clearly how to make this work. It bothers me that to use quaternions for 3D rotations the usual way (I'm not yet comfortable doing it without angles and sines etc) we must first compute the sine and cosine of the half angle, and then do two quaternion multiplications. The second multiplication repeats half of the first one and cancels the other half. It's computationally simpler, starting with the unit vector A you want to rotate around, the vector B you want to rotate, and the sine and cosine of the angle theta to rotate. Split B into 3 orthogonal vectors, B1 parallel to A, B1=(A.B)A, B2 the part of B perpendicular to A, B2 = B-B1, and B3 perpendicular to both of them, B3=AxB. Then the rotated vector is B1+cos(theta)B2 +sin(theta)B3 There are lots of KZfaq videos by CS people explaining how to do quaternions, when they could get the same result in a little more than half the time. Maybe it's better to use quaternions to do 4D rotations. A single quaternion multiplication -- not two -- can be used to generate an elliptical orbit with the t dimension showing the deviation in time.

@peterhi503 11 жыл бұрын

Masterful. Best to you.

@alexmoroz3357 Жыл бұрын

Thank you!

@Smeak686 11 жыл бұрын

Thank you for this video.

@pickeyberry4060 2 жыл бұрын

Video Content 00:00 Introduction 03:19 Recall: Complex numbers 10:31 Laws 15:13 We connect quaternions and 2×2 complex matrices! 17:20 Conjugation 31:51 Main theorem 36:26 Inverses 40:20 The quaternion inner product 44:53 Picturing quaternions

@njwildberger 11 жыл бұрын

It is an inherent property of the sphere--at least if we assume that the multiplication is smooth in the obvious--if x and y move smoothly on the sphere, then so does xy. The usual proof connects with the fact that there is no non vanishing vector field on the sphere.

@mikedavid5071 2 жыл бұрын

L^2=j^2=k^2=ijk=-1 Is on the plaque on the bridge.