Wikipedia:Reference desk/Archives/Mathematics/2023 June 2

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June 2

Angles in 4-dimensional space

Three-dimensional space has 2 kinds of angles:

1. Dihedral angles are related to edges.
2. Polyhedral angles are related to vertices.

What kinds of angles does 4-dimensional space have?? What is each kind of angle related to?? Georgia guy (talk) 14:19, 2 June 2023 (UTC)

First I would consider dihedral and polyhedral angles as two different names or uses for the same thing. You can get a polyhedral angle from a dihedral angle between two planes by e.g. taking normal vectors of the two planes then considering the angle between those vectors. In fact as angles are generally defined as between two vectors this is the normal approach.

So what's the equivalent in 4D? You can measure the angle between two vectors using the dot product – that would be your polyhedral angle. The dihedral angle corresponds to the angle between two 3-spaces, such as the angle between the solids that bound a 4-polytope. Each 3-space has a vector normal in 4D, which given two 3-spaces gives two vectors between which you can calculate the angle. You could use this e.g. for the angles between solids of a 4-polytope.

It gets tricky though for angles between 2-spaces; in their most general form rotations in 4D can have two angles of rotation, each acting in two of the four dimensions – a "double rotation". You need other tools than just single angles to describe such rotations. 4D matrices e.g. or rotors. 92.40.5.34 (talk) 15:43, 2 June 2023 (UTC)

(edit conflict) In general, in an $n$-dimensional flat (Euclidean) space, $n\geq 2,$ two intersecting $(n{-}1)$-dimensional hyperplanes intersect at angles, which equal the angles between their normals. (I use the plural because if hyperplanes intersect at some angle, they also intersect at that angle's supplement.) If more than two hyperplanes meet at a vertex – in general you need at least $n$ intersecting hyperplanes to determine a vertex – the "polytopic angles" they form can be characterized by their pairwise angles.

You might find the article Angles between flats useful. This is about the principal angles which includes 0 for the dimensions in common. NadVolum (talk) 18:44, 2 June 2023 (UTC)

The article looks a bit confusing, but the relevant subsection is "Algebraic example" where they talk about 2 planes in 4 space. From that it's fairly clear that you need at least two angles to describe the relationship between the planes. --RDBury (talk) 20:19, 2 June 2023 (UTC)

Yep, the planes might only intersect at a single point in 4D rather than along a line as in 3D. NadVolum (talk) 20:38, 2 June 2023 (UTC)

The planes might not intersect at all, just like two lines in 3D need not intersect. For example, the plane given by $\{(\kappa ,\lambda ,0,0)~|~\kappa ,\lambda \in \mathbb {R} \}$ does not intersect the plane $\{(\mu ,0,\nu ,1)~|~\mu ,\nu \in \mathbb {R} \}.$  --Lambiam 10:42, 4 June 2023 (UTC)

That's true okay. However the article defines the angles as where the planes are translated to intersect so they'd make the lines in 3D intersect and then give the angle between them. NadVolum (talk) 11:23, 4 June 2023 (UTC)

There's more to it than that. Given two lines in 3D, even lines that don't intersect, it's still possible, straightforward even, to find an angle between them by considering just their directions, taking the dot products of their direction vectors to get the angle. The effect is the same as if they were both translated to pass through the origin.

Two planes through the origin in 4D intersect at a point, the origin. But that's not enough to calculate an angle between them, except in exceptional cases. It's hard to prove this negative. One way of thinking of it is planes in 4D are six-dimensional objects, as they have up to six independent components: ij, ik, il, jk, jl and kl. It's not possible to generate a 4D "normal vector" from such, as you can from a plane in 3D, a solid in 4D. You can treat it as a 6D vector but in doing so you discard the geometry of it. The approach at Angles between flats is one way of doing it but there are other ways of approaching it.92.40.5.34 (talk) 11:27, 4 June 2023 (UTC)