If you are in a Euclidean vector space V of dimension n and you have two totally orthogonal subspaces S and T of dimensions k and n-k, then a rotation s of S and a rotation t of T can be continued to a rotation v of V such that the restriction of v to S is s and the restriction of v to T is t.

As i1 = i, ii = -1, ij = k, ik = -j, multiplication from the left with i means rotation of the (1,i)-flat (as with the complex numbers) and an analogous rotation if the (j,k)-flat. These rotations can be continued to a rotation of four-space around the origin.

Analogous for j and k.

I heard that quaternions are used in geometric optics, but I am not a physicist.

Dear Johann,

        I agree with you.

But if you represent  i  as the left multiplication by  i  in the space of quaternions that contains  i  itself, it seems to me that this is not an explanation of  i  in terms previously known geometrical concepts.

So I cannot be completely satisfied of your correct solution.

Dear Giancarlo,

if you identify x := 1x_0+ix_1+jx_2+kx_3 with the vector (x0,x1,x2,x3)^T, written not as a line but as a column, then the matrix M =

0   1   0   0

-1  0   0   0

0   0   0   1

0   0  -1   0

is a rotation matrix, and Mx corresponds to ix. The matrix M does not contain i.

Dear Johann

,        now I agree completely with you.

Maybe the algebra of quaternions is isomorphic with the rotohomoteties of the real 4-dimensional euclidean space, or someting like this.

But I was really looking for an exact interpretation of quaternions in the  3-dimensional space following, I believe, what Hamilton did.

Do you have any idea?

Dear Giancarlo,

there is a vector-field interpretation of quaternions, in which the generators of the algebra of linear vector fields in the 4-dimensional Euclidean space, are tangent to the 3-dimensional sphere.

Dear Igor,

thank you for your example.

I'm also interested for more simple examples

to be used in teaching quaternions to young people.

Dear Christian,

please see my answers

to Johann and Igor.

I think, you are looking for a geometric interpretation of quaternions via rotations of a 3d space. One possible answer is discussed in my article "On a Quaternionic Analogue of the Cross-Ratio" just before Lemma 1. Hope this helps.

Dear Matvei,

I dont see what I look for.

Can you give me further information about your paper?

I'm looking for an easy exact geometrical interpretation

of the qualernions  i  j  k  in the usual 3-dimensional real space,

to be used to teach quaternions in a geometrical way to young students.

the paper

"On a Quaternionic Analogue of the Cross-Ratio" by E. Gwynne and M. Libine

can be downloaded from arxiv.org or my researchgate account.

See the discussion just before Lemma 1 there.

Dear Matvei,

I have seen your paper

and the discussion before Lemma 1.

It is an interesting representation

of quaternions

in term of nice complex 2x2 matrices.

But I search a geometrical real representation

in the ordinary 3 dimensional space.

This is not what I meant. I wanted to show you the map

H^x/R^x  ->  SU(2)/{+-Id}  ->  SO(3)

Dear Matvei,

can you give me the geometrical meaning

of  i  j  k  in simple words?

Thank you.

Any unitary quaternion can be interpreted as a rotation of 3-space. Namely, if you take q with norm one, it can be written as q=cos\theta + \sin\theta w, where w is a quaternion without real part , that is, w=xi+yj+zk, and norm one. Then q can be interpreted as the rotation of axis w=(x,y,z) and rotation angle \theta/2. The exact assignment is the map S^3 \to SO(3) sending q into the conjugation map x \maps qxq^{-1}, see for instance Chevalley's book on Lie groups, or many other sources. Your question refers to w=i,j,k, then they are rotations of angle \pi/2 around the three coordinate axis.

Dear Enrique,

It seems to me that the angle of the rotation is 2*\theta and not \theta/2.

In every case it seems to me that the map that to each unitary quaternion associate a rotation is not a bijection. Opposite quaternions have the same rotation.

Finally the rotations associated to  i  j  k  are of  \pi radiants not  \pi/2 , I believe.

you are right,  of course,it is 2*\theta, sorry. and i,j,k are \pi rotations, indeed.

yes, it is not a bijection, this is called a two-fold covering map, in fact both S^3 (unit quaternions)  and SO(3) (rotations) are Lie groups, and the kernel of the map is the center \pm id, this is why changing the sign of q does not affect the rotation.

the interpretation of quaternions as rotations was a great controverse betwen Hamilton (who was wrong) and Olinde Rodrigues (who was right),

Please see the attachment.

Dear Wiwat,

in your attachment I only see that

H can also be represented

by 2 × 2 matrices with complex entries.

I'm interested in real 3-dimensional

representations.

Dear Prof. Giancarlo Meloni,

Please see the attachment in the following useful link:

Matt Gravelle, Quaterions and their Applications to Rotation in 3D Space.

http://facultypages.morris.umn.edu/math/Ma4901/gravelle.pdf

I have found an (maybe the) exact interpretation

of unitary quaternions in the real 3-dimentional space.

In general a unitary quaternion is the usual associated rotation

endowed with the information, if  α  is the angle of the rotation,

if we rotate from 0 to α  clockwise or anticlockwise

Dear Giancarlo,

I see that you have repeated a number of times your plea for a simple verbal description of the geometrical meaning of the quaternion units i, j and k.

They each correspond to a rotation by pi in the right-hand sense.

I've found no simpler illustration of the workings of the unit quaternions than Chapter 11 of Roger Penrose's book, The Road to Reality.

If you can find Section 11.3 "Geometry of quaternions" in Chapter 11 "Hypercomplex numbers" (p. 204 in my editions - I have two copies), I'm sure you'll find it all perfectly clear, and you'll be initiated into the mysteries of spinors at the same time.

You might even find there's a PDF of the book online...

Hoping this helps - Paul

https://www.amazon.co.uk/Road-Reality-Complete-Guide-Universe-ebook/dp/B01BS7NTA6/ref=sr_1_1?s=books&ie=UTF8&qid=1481647967&sr=1-1&keywords=penrose+reality