George Hammond wrote:

"Tesseract" is the name given to a 4D cube (a hypercube).
The Tesseract is particularly well known as the unit cube
in the 4 dimensional spacetime of Relativity.

(*)

While it is well known that an ordinary 3D cube has
13 (rotational) symmetry axes, namely-

The 3 normals of the cube.
The 4 body diagonals of the cube.
The 6 face diagonals of the cube.

Whence, 3+4+6=3D13 rotational symmetry
axes of the 3D cube.

The QUESTION at hand is: How many rotational
symmetry axes does a 4D cube have?

If anyone knows the answer to this question, or how to
compute the answer, or where to find the answer
I would very much appreciate the help.

PS: Interestingly, there is a simple formula for the total
number of rotations of an n-dimensional cube, namely
(n!)(2)^(n-1) .... see:
http://gregegan.customer.netspace.net.au/APPLETS/29/HypercubeNotes.html
This formula yields 24 rotations for a 3D cube
(which we can manually confirm to be correct
for the above 13 axes....and it gives 192 rotations
for the 4D cube....but it doesn't tell us the number
of individual rotational AXES in the 4D cube.... which
is what I'm trying to find. Any suggestions?
[Hammond]

Consult the Ph.D. thesis of Salomon Levi van Oss:

Record from online catalog of Delft University of Technology Library

http://library.tudelft.nl/ws/index.htm>Catalogue >Access to the catalogue

Title =20
Die Bewegungsgruppen der regelm=E4ssigen Gebilde von vier Dimensionen
Author Oss, Salomon L. van
Publisher Utrecht : z. uitg., 1894
Descr. 38 S. M. Taf., 8 o
Gen. note Giessen, Dr.-Ing.-Diss. 1894
Classific. BTA / Meetkunde (algemeen) / Geometry (general)
Shelf number CB 06164502

I made an excerpt, which I do not have at hand at this moment. Will post
it to the newsgroup when I will have found it among my paperwork.

An approach different from Van Oss's uses the several different
homomorphisms and isomorphisms of the 4D rotation group SO(4), the unit
quaternion group S^3 and its direct product with itself, and the 3D
rotation group SO(3) and its direct product with itself.
I did not do this exercise. Presumably it will amount to squaring the
order (24) of the rotational congruence group of the 3D cube, and
dividing this number (576) by two each time a 2:1 group homomorphism
turns up. Or something similar to this...

(*) BTW, the tesseract does not play any role in the geometry of Special
Relativity. It is an object from Euclidean 4D space with signature ++++,
whereas the signature of Einstein-Minkowski-Lorentz space is -+++ (or
+---, depending on the convention on signs).

Cheers: Johan E. Mebius

George Hammond wrote:

"Tesseract" is the name given to a 4D cube (a hypercube).
The Tesseract is particularly well known as the unit cube
in the 4 dimensional spacetime of Relativity.

(*)

While it is well known that an ordinary 3D cube has
13 (rotational) symmetry axes, namely-

The 3 normals of the cube.
The 4 body diagonals of the cube.
The 6 face diagonals of the cube.

Whence, 3+4+6=3D13 rotational symmetry
axes of the 3D cube.

The QUESTION at hand is: How many rotational
symmetry axes does a 4D cube have?

If anyone knows the answer to this question, or how to
compute the answer, or where to find the answer
I would very much appreciate the help.

PS: Interestingly, there is a simple formula for the total
number of rotations of an n-dimensional cube, namely
(n!)(2)^(n-1) .... see:
http://gregegan.customer.netspace.net.au/APPLETS/29/HypercubeNotes.html
This formula yields 24 rotations for a 3D cube
(which we can manually confirm to be correct
for the above 13 axes....and it gives 192 rotations
for the 4D cube....but it doesn't tell us the number
of individual rotational AXES in the 4D cube.... which
is what I'm trying to find. Any suggestions?
[Hammond]

Consult the Ph.D. thesis of Salomon Levi van Oss:

Record from online catalog of Delft University of Technology Library

http://library.tudelft.nl/ws/index.htm>Catalogue >Access to the catalogue

Title
Die Bewegungsgruppen der regelm=E4ssigen Gebilde von vier Dimensionen
Author Oss, Salomon L. van
Publisher Utrecht : z. uitg., 1894
Descr. 38 S. M. Taf., 8 o
Gen. note Giessen, Dr.-Ing.-Diss. 1894
Classific. BTA / Meetkunde (algemeen) / Geometry (general)
Shelf number CB 06164502

I made an excerpt, which I do not have at hand at this moment. Will post
it to the newsgroup when I will have found it among my paperwork.

An approach different from Van Oss's uses the several different
homomorphisms and isomorphisms of the 4D rotation group SO(4), the unit
quaternion group S^3 and its direct product with itself, and the 3D
rotation group SO(3) and its direct product with itself.
I did not do this exercise. Presumably it will amount to squaring the
order (24) of the rotational congruence group of the 3D cube, and
dividing this number (576) by two each time a 2:1 group homomorphism
turns up. Or something similar to this...

(*) BTW, the tesseract does not play any role in the geometry of Special
Relativity. It is an object from Euclidean 4D space with signature ++++,
whereas the signature of Einstein-Minkowski-Lorentz space is -+++

+---, depending on the convention on signs).

On 26 Mrz., 09:45, "George Hammond" <Nowhe...@notspam.org> wrote:

"Tesseract" is the name given to a 4D cube (a hypercube).
The Tesseract is particularly well known as the unit cube
in the 4 dimensional spacetime of Relativity.

While it is well known that an ordinary 3D cube has
13 (rotational) symmetry axes, namely-

The 3 normals of the cube.
The 4 body diagonals of the cube.
The 6 face diagonals of the cube.

Whence, 3+4+6=13 rotational symmetry
axes of the 3D cube.

The QUESTION at hand is: How many rotational
symmetry axes does a 4D cube have?

If anyone knows the answer to this question, or how to
compute the answer, or where to find the answer
I would very much appreciate the help.

PS: Interestingly, there is a simple formula for the total
number of rotations of an n-dimensional cube, namely
(n!)(2)^(n-1) .... see:http://gregegan.customer.netspace.net.au/APPLETS/29/HypercubeNotes.html
This formula yields 24 rotations for a 3D cube
(which we can manually confirm to be correct
for the above 13 axes....and it gives 192 rotations
for the 4D cube....but it doesn't tell us the number
of individual rotational AXES in the 4D cube.... which
is what I'm trying to find. Any suggestions?
[Hammond]

First you need to specify what you call rotational axis.
If you want only orientation preserving mappings (i.e. no
reflections),

then you also have to accept rotations around plains (e.g. keeping the
forth dimension fix and rotating the 3D projection in a standard
manner).

On Mar 26, 3:45 pm, "George Hammond" <Nowhe...@notspam.org> wrote:

The QUESTION at hand is: How many rotational
symmetry axes does a 4D cube have?

Rotations in 4 dimensions don't have axes

(i.e. 1-dimensional
invariant subspaces), they have invariant subspaces with either 2
dimensions (i.e. planes) or 0 dimensions (i.e. only the 0 vector is
unchanged by the rotation).

For example, a 4-d rotation might rotate vectors in the x-y plane by
some angle theta, and leave the z and w components of vectors
completely untouched. Then the whole z-w plane is an invariant
subspace of that rotation (just as the z-axis alone would be the
invariant subspace in the 3-d case).

But you can also have a 4-d rotation that rotates in *both* the x-y
plane and the z-w plane, and has *no* (non-zero) invariant vectors!

OK, so what about the rotational symmetries of the 4-d hypercube?

Here's a partial answer. Suppose you pick one of the 4 dimensions as
the direction from the origin to the centre of a cubic hyperface of
your tesseract, and call that vector e_i. Then there will be 13 axes
of rotation from all the 3d rotations of a cube that we can map into
the remaining three dimensions (those that are not e_i), and the 13
distinct planes we get that are spanned by those 13 directions along
with e_i will be preserved by those rotations that preserve our chosen
cubic hyperface. But we can choose e_i four different ways, so that
gives us 52 distinct planes that are preserved by various rotations.

That's not the full answer, though, because we've only accounted for 4
x 23 = 92 of the 191 non-trivial rotations (i.e. the 23 non-trivial
rotations of the cube, for each choice of e_i). What about the other
99 rotations? Do they all have invariant planes? Are some of these
planes ones that we've already counted? I don't know the answers to
these questions off the top of my head ...

On Mar 26, 3:45 pm, "George Hammond" <Nowhe...@notspam.org> wrote:

The QUESTION at hand is: How many rotational
symmetry axes does a 4D cube have?

[W]e can choose e_i four different ways, so that
gives us 52 distinct planes that are preserved by various rotations.

That's not the full answer, though, because we've only accounted for 4
x 23 = 92 of the 191 non-trivial rotations (i.e. the 23 non-trivial
rotations of the cube, for each choice of e_i).