Hello everyone,
I have been busy with other things lately, but I
took some time last night to solve a simple problem
which occurred to me. It did not involve any advanced
reasoning (just high-school geometry), but I think
it might be of interest to the group.
I wanted to know what size cubes (in cubies per edge)
of all dimensions were theoretically constructable,
i.e. what cubes were not definitely inconstructable
without the pieces falling out.
It turns out that for dimensions higher than 3, two
different types of cubes are theoretically constructable -
those in which the pieces would not fall out regardless
of how one turns the faces, and those in which the pieces
would only fall out when the corners are sufficiently
far from the center of that face. All other cubes
would be inconstructable, in that the pieces would
definitely fall out no matter how you rotate the faces.
For the first type of cube, a d-dimensional cube with
n cubies per edge is theoretically constructable
if the following inequality holds:
(d-1)(n-2)^2 < n^2
This means that in three dimensions, up to 6x6 cubes
could be constructed (we are not considering cubes
that are not cube-shaped, so the 7x7 and higher V-Cubes
would not count), in four dimensions, up to 4x4 cubes, in
five through nine dimensions, up to 3x3 cubes, and in
dimensions higher than nine, only 2x2 cubes. For the
second type of cube, the matter of whether one could be
theoretically constructed reduces to the previous formula
using one lower dimension, or:
(d-2)(n-2)^2 < n^2
The reason for this is that the type of rotation which
keeps the pieces closest to the center of that face is
a 90 degree coordinate-axis aligned rotation.
Concerning the second type of cube, the closer the two
sides of the first inequality, the farther the corner
pieces could be from the center of that face without
falling out. An interesting thing to note is that
equality holds in the first inequality for a 3^10 cube, so
the corner pieces would only fall out when the corners
are the farthest distance possible from the center of the
face (or sufficiently close to that distance, depending
on how well the actual cube mechanism was designed).
In a 3^11 cube, equality holds in the second inequality,
which means the pieces would definitely fall out, but
only when they are the farthest possible distance from
the center of the face using a 90 degree coordinate-axis
aligned rotation (i.e. at 45 degrees).
I hope this simple yet interesting result was of value
to the group. Tomorrow, I plan to continue working
on the analogous permutation formulas for five-dimensional
cubes that I discovered for four-dimensional ones.
All the best,
David
I think you've just redefined the spirit of going to far. V-Hypercubes
would be your solution though.
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Hello everyone! First of all, I'd like to thank Melinda for inviting me to
this mailing list!
My name is Chris Locke, and I recently solved the 3^4, 4^4, and 5^4
puzzles. Roice's solution guide was indispensable for getting started.
After solving the 3^4 using Roice's guide (although I had to use my own
notation for moves), I was familiar enough with the Magic Cube 4D interface
to move on to the higher order puzzles. Macros were critical, as I don't
think I would've had the patience to go through the algorithms by hand every
time. The 4^4 cube presented a new challenge, as I had no guide to start
with. So the first thing I did was try out the algorithms I already knew
for the 3D case. It turned out, that by using conjugation and/or
commutators, I was able to build new algorithms to accomplish the tasks I
wanted. For instance, I had one algorithm that would swap two pairs of face
pieces. I would then take that algorithm, then after applying it move one
of the affected edges such that it would flip its orientation, then undo the
original algorithm to get an new algorithm for building edge pieces. As for
parity, the 3D algorithms were sufficient I found for solving face parity
situations, but I didn't know how to deal with edge parity cases. Luckily,
I never ran into edge parity problems! Once I moved to the 5^4 cube, I
realized that for moving center and face pieces around, I couldn't just use
the 4^4 algorithms, as they weren't general enough. However, I had never
solved a 3D 5^3 cube yet, so I had to look up some 3D algorithms. Then,
using the same ideas as before, I was able to discover ways of doing
3-cycles to center and face pieces. Although if I remember correctly, it
wasn't just one commutator any more; I think one of the moves ended up being
a quadruple nested commutator (of the form [[[[a,b],c],d],e]), although the
original 3D move was already a double commutator. Took a bit of work to
find moves that actually did what I wanted to, but eventually I succeeded.
Although I didn't macro the full algorithm, as inputting a 46 move algorithm
was a little bit much... so instead I macroed the 22 move triple commutator,
then simply did the middle move and inverse by hand :D. As for building
edges, I was able to just use my 4^4 move to accomplish this task. I found
that the biggest difficult with these larger cubes was patience... patience
to find the right pieces and being able to find out exactly where you want
them to go. There's a lot going on after all. Oh yeah, and I don't think
you'll be seeing me attempt the 5D cube any time soon... the main reason I
was able to cope with the 4D cube was because of the friendly interface.
Being able to navigate in 3D really helped me in solving it. But when I
look at the 5D cube, it is just a scary mess! And it seems difficult to
input moves based entirely and chosing which axis and whatnot. I wouldn't
be surprised if I tried sometime though, just not soon :). Oh yeah, and
that Magic Cell 120 thing is a monster... although the one redeeming feature
of it is that it is so large, it's practically 3D!
As for myself, I finished my undergrad this summer, majoring in physics and
mathematics. This fall, I moved to Tokyo, where I'm working at a
university, on scholarship as a research student. That basically means I do
research, but don't have to do any of the other things that grad students do
:D. After this research term is over, I will enter grad school, although in
exactly what disipline I haven't yet decided. Obviously, something
physics/math related though. It will probably involve programming too
(although most research does now anyway), as I have lots of experience now
in that field. My favorite past-time would have to be ice hockey. I still
closely follow the Vancouver Canucks from Tokyo, and also am playing ice
hockey here on a weekly basis. Outside of hockey, I like music, watching
TV, playing some video games, and just hanging out with friends. I only
started solving cubes and whatnot this summer. Although I hate memorizing
algorithms and don't want to "practice" it, so I'm not very fast at all!
Well, that's about it!
Chris Locke
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Hello everyone! First of all, I'd like to thank Melinda for inviting me to this mailing list!
My
name is Chris Locke, and I recently solved the 3^4, 4^4, and 5^4
puzzles. Roice's solution guide was indispensable for getting
started. After solving the 3^4 using Roice's guide (although I had to
use my own notation for moves), I was familiar enough with the Magic
Cube 4D interface to move on to the higher order puzzles. Macros were
critical, as I don't think I would've had the patience to go through
the algorithms by hand every time. The 4^4 cube presented a new
challenge, as I had no guide to start with. So the first thing I did
was try out the algorithms I already knew for the 3D case. It turned
out, that by using conjugation and/or commutators, I was able to build
new algorithms to accomplish the tasks I wanted. For instance, I had
one algorithm that would swap two pairs of face pieces. I would then
take that algorithm, then after applying it move one of the affected
edges such that it would flip its orientation, then undo the original
algorithm to get an new algorithm for building edge pieces. As for
parity, the 3D algorithms were sufficient I found for solving face
parity situations, but I didn't know how to deal with edge parity
cases. Luckily, I never ran into edge parity problems! Once I moved
to the 5^4 cube, I realized that for moving center and face pieces
around, I couldn't just use the 4^4 algorithms, as they weren't general
enough. However, I had never solved a 3D 5^3 cube yet, so I had to
look up some 3D algorithms. Then, using the same ideas as before, I
was able to discover ways of doing 3-cycles to center and face pieces.
Although if I remember correctly, it wasn't just one commutator any
more; I think one of the moves ended up being a quadruple nested
commutator (of the form [[[[a,b],c],d],e]), although the original 3D
move was already a double commutator. Took a bit of work to find moves
that actually did what I wanted to, but eventually I succeeded.
Although I didn't macro the full algorithm, as inputting a 46 move
algorithm was a little bit much... so instead I macroed the 22 move
triple commutator, then simply did the middle move and inverse by hand
:D. As for building edges, I was able to just use my 4^4 move to
accomplish this task. I found that the biggest difficult with these
larger cubes was patience... patience to find the right pieces and
being able to find out exactly where you want them to go. There's a
lot going on after all. Oh yeah, and I don't think you'll be seeing me
attempt the 5D cube any time soon... the main reason I was able to cope
with the 4D cube was because of the friendly interface. Being able to
navigate in 3D really helped me in solving it. But when I look at the
5D cube, it is just a scary mess! And it seems difficult to input
moves based entirely and chosing which axis and whatnot. I wouldn't be
surprised if I tried sometime though, just not soon :). Oh yeah, and
that Magic Cell 120 thing is a monster... although the one redeeming
feature of it is that it is so large, it's practically 3D!
As for myself, I finished my undergrad this summer, majoring in
physics and mathematics. This fall, I moved to Tokyo, where I'm
working at a university, on scholarship as a research student. That
basically means I do research, but don't have to do any of the other
things that grad students do :D. After this research term is over, I
will enter grad school, although in exactly what disipline I haven't
yet decided. Obviously, something physics/math related though. It
will probably involve programming too (although most research does now
anyway), as I have lots of experience now in that field. My favorite
past-time would have to be ice hockey. I still closely follow the
Vancouver Canucks from Tokyo, and also am playing ice hockey here on a
weekly basis. Outside of hockey, I like music, watching TV, playing
some video games, and just hanging out with friends. I only started
solving cubes and whatnot this summer. Although I hate memorizing
algorithms and don't want to "practice" it, so I'm not very fast at all!
Well, that's about it!
Chris Locke
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