Message #3696

From: damienturtle@hotmail.co.uk
Subject: Re: Physical 4D puzzle V2
Date: Fri, 28 Apr 2017 20:50:24 +0000

Hi Melinda,

The V2 looks great! It is such a nice coincidence that it has the colour scheme I use in MC4D too, I want one even more now!

It is nice that you have a reasonably short way to do 4D reorientations, I still wonder if there’s still some better way to do it though since it takes a few steps. Now for your questions. First of all, it has already been correctly pointed out that there are 6 orbits, a factor of 3 from orientation and a factor of 2 from permutation. Secondly, I believe that it is in some sense the 2^4. While it can do things that are not legal twists of the 2^4, it can do everything the 2^4 can do. Therefore, if we are careful and correctly restrict how we manipulate it, we get a puzzle that is mathematically identical to the 2^4 as we usually understand it, and that’s good enough for me. Finally, I do not think it is an issue that opposite faces touch, it seems it is just a quirk of the implementation rather than suggesting that they are actually adjacent. In fact, I believe this can be avoided with the pieces you already have!

How do we avoid opposite faces touching? Turn it inside-out. It seems a little strange at first maybe, but bear with me. Instead of having, say, pink and purple facing outwards on each cube, they could point towards the centre of the cubes and would therefore not be visible from the outside. The correct way to do this seems to result in the two cubes looking like mirror images to each other, with the colours forming windmill patterns on each face. This might be a confusing description, so I made a very crude image in paint to demonstrate, which should be attached to this post (or uploaded to my folder at the very least, I’m not sure how to work this stuff correctly). This might not be the best way to play with it since ‘stickers’ are not visible, but it does prove a point. I think that the same reorientation technique shown in the video should also work for this with essentially the same effect. I wonder if there is a shorter way to perform a 4D rotation which starts in the sort of configuration you have shown, and ends with the cubes inside-out like this? Right now, I have no idea.

Lastly, a honey-based mechanism seems … messy. It is very simple and cheap though, so maybe we missed a trick there :).

Matt