How many angles it's needed to define 2-plane in 4d?

I mean the angles between arbitary 2-plane and euclidean orthonormal 2-planes which common origin lies at that 2-plane ⊂ R⁴. I think the orthonormality of 2-planes (bivectors) is unambiguous in normal cartesian 4d system, isn't it?

I got a solution that three could be enough; angles by ruled parity in relation to e.g. xy, yz, zw -planes (set 4d as x, y, z, w coordinates). The unknown 2-plane goes through the origin, you remember.

I tried to check all the degrees of freedom. Can anyone study a solution?

• You first need to say how you define the angle between two $2$-planes. Are you using the definition in this answer or something else? Oct 20, 2020 at 16:51
• Yes, just that.
– Eusa
Oct 20, 2020 at 19:01
• The space $Gr(2, \Bbb R^4)$ of $2$-planes in $\Bbb R^4$ is a $4$-dimensional space, so roughly speaking one needs four parameters to specify an arbitrary $2$-plane. See en.wikipedia.org/wiki/Grassmannian for details. Oct 24, 2020 at 22:06
• Ok. Right now I found John C. Baez on twitter commenting: twitter.com/johncarlosbaez/status/1290328126761836545 - there are 4 parameters; 2 for self-dual and 2 for anti-self-dual (non-isoclinic). Just imaging if either could be isoclinic via serials - but is it then inevitably discrete? Not necessary...
– Eusa
Oct 24, 2020 at 22:53

Let yz be the 2-plane under operations.

Now, we can define isoclinic rotations α (xy, zw), β (yz, wx) and γ (xz, yw).

My statement: In order α -> β -> γ with appropriate parameters we can rotate yz-plane to any 2-plane through the origin of euclidean R⁴.

How to proof?

• An analogue: in 3d you can rotate straight line in any direction by two plane of rotation e.g. xy, yz - zx is not necessary. Why not the same principle for 2-plane were ok in 4d? Double isoclinic rotation by xy&zw, yz&wx, xz&yw - antisymmetric self-dual or anti-self-dual wouldn't be necessary? I know that even-dimensionality makes a difference but i think it has been taken account by the chosen mechanism; angles can be as small or as large as needed - several rounds by other parameter can finetune the effect of smaller... At least i want to know how to study the setup...
– Eusa
Oct 28, 2020 at 10:31
• So it's seems to cover better when rotating simultaneously by infinitesimal increments 0...(α, β, γ) and angles can be up to infinity. Of course one of them need not to be greater than pi.
– Eusa
Oct 30, 2020 at 15:03