h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

What symmetries does a tesseract have? A point does not have any symmetries, in a line reflection, in a plane reflection and rotation ect.
Is dilation a symmetry in 4-space, in the sense that the hypercube maps onto itself?
You seem to be the best person to ask.

This article:
http://www.jucs.org/jucs_6_1/the_automorphism_group_of
seems to be answering your question. I haven't looked at it yet, I don't know how clearly things are explained.

After a quick glance it doesn't look too frightening, will have to learn graph theory but it looks OK.
Thanks.

The advantage of graph theory is that it is very visual. And following the paper's explanations can be done, in principle, by specialising to the square and the cube.

Good luck

Well, the basic idea, I think, is that a symmetry of the tesseract is the same as a symmetry of its graph of vertices. In other words, if two symmetries of a tesseract permute the vertices in the same fashion, then the symmetries are actually equal.

Now, what is the "graph of vertices" of the tesseract? You just have big dots representing the vertices, and you draw a line between two dots if there is an edge between the two corresponding vertices of your tesseract. For a square, the graph of vertices could be just the same square. For a cube, you can make a 3D wire model of the cube. For a tesseract, you can also make a 3D wire model of this graph, in the way described in the Entry for example.

The advantage of the graph of vertices is that now you don't have to bother about the edges of the graph having the same length or being at a right angle to each other. The only thing which counts is, whether there is an edge at all or not. For example, you could arrange the "graph of vertices" of a square to be a rectangle, a rhombus or even just any 4-sided convex shape.

And to finish the description, we need to know what is a "symmetry of a graph". It is just a permutation of the vertices which respects adjacency:

- permutation of vertices: if they are labelled, say, 1,2,3,... it just means you re-label them, possibly in a different order;

- preserving adjacency: if there is an edge between two vertices, then there should be again an edge between their images under the aforementioned permutation. That is, if you have an edge i-j before relabelling, you must have an edge i-j after relabelling as well.

Hope this helps.



Incidentally, this is another example where defining a concept for an object with extra structure has to respect the structure. As for "ordered fields", where the order has to be compatible with addition and multiplication, here the "symmetry of a graph" is a permutation of vertices which respects the edges.

You are right, but visualising 4D is difficult, the extra symmetry is not dilation as we see it in 3D, but extrapolating from the perspective image, the outer box could map to one of the pyramids,that pyramid to the inner box ect; a permutation.
Its visualisation is hard, the mathematics in this instance is easier.
Not all symmetries look like rotations or reflections on the perspective image of the tesseract, but I find it difficult to visualise 4D.
Thanks

Everybody finds it hard to visualise 4D

That's why it is easier to just imagine these symmetries as a fancy way to permute your 3D wire model (vertices graph). Because you lost information with this 3D representation, you can indeed map the inner cube to one of the pyramids, or even (Shock! Horror!) to the "outer cube".

But, these are only *representations* of the symmetries - not the *actual* symmetries of the tesseract. These actual symmetries should preserve distances, right angles, volumes, etc, just as they do with squares or cubes.