A Toy Model For Q8
By Doug Sweetser | March 20th 2012 12:11 AM | 14 comments | Print | E-mail | Track Comments

Trying to be a semi-pro amateur physicist (yes I accept special relativity is right!). I _had_ my own effort to unify gravity with other forces in...

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The idea for this blog was dead simple. In my second blog at Science 2.0, "Quaternions for a Third Grader",  I showed of my clay and pipe cleaner model of quaternion multiplication. A few months ago, David Halliday and I started talking about the finite group Q8, which over the real numbers becomes the quaternions. David pointed out my tetrahedron was half the cube it needed to be to represent Q8. That was the idea for the blog: make a cube to represent that finite group.

It was quite the challenge to make the tetrahedron out of clay an pipe cleaners. The structure is rather fragile. A few preliminary sketches of the cube indicated that each of the eight vertices would need 14 holes drilled. Clay was not up to the task. The more precise I could be, the better chance the result would look like a cube. After wandering the aisle at several craft stores, I saw ¾" wood cubes. That could work if I could drill precisely. How does one drill at an angle? Buy an angle clamp at Sears.

When doing an upgrade to a piece, it is not enough to improve materials. There also needs to be a filling out of the concept. What has been most interesting in the discussion are the relationships between Q8 and the groups Z2, Z4, and Z2xZ2. I decided to make those too, and include them so one could see the connections between them all.

This one image required I take a vacation day from work, but I do have something new the wall:

[Correction: The initial images had Z4 with only 2 vertices instead of 4.][This is now version 3, where I was able to find the errors in arrows found in Q8. The system was direct: do one simple arrow, and the rest follow because they must flow around a plane defined for a given color. Consider the two planes defined by Kelly green pipe cleaners:

The positive k's go one direction, the negative k's go the other. That was an easy data quality system to implement. I made mistakes on 6/48, or 1/8.
I could release a version 4 since I spotted another error. I was wondering what other kinds of checks I could do. One is to count arrow heads going into a cube. Three should be white, three should be black at each of the eight cubes. A 2/4 split indicates an error. That is not a test people on the web can do. Here is another test: for any face, one set of the the two arrow heads on parallel lines must point together, the other two must point apart. On the top, all point together, so there is an error. The yellow (+j) needs to be flipped.]

It became important to me to include the trivial group. Why? When a physicist says: "I am studying foo interacting with bar." What that statement does is define the trivial group. They will start with foobar and end with foobar. Another good reason to keep the trivial group is that it can be seen in all the other groups. Every group has the 1x1=1. I call that the "slacker" operation, the one that keeps going by doing nothing. It is vital to acknowledge the slacker that is mostly a good description of systems.

Another big insight I obtained was to spot the relationship between quaternions and complex numbers. Sure I have known that quaternions have three complex numbers as subgroups. This time I can see the cube getting squished and flattened until Q8 makes a transition to Z4. This could happen if only one spatial parameter matters in describing a system. Sure, the other two dimensions are there, but they don't matter in the description of the system.

There is another type of squish, going from Z4 to Z2. That represents a move from a directional graph to an undirectional graph.

[Correction: Zcan be seen in Z4 by deleting two fo Z4's elements. Not much of a squish.]

Every representation is flawed. The biggest issue with this is the image is static. I will update this blog Tuesday night with dynamic versions of these graphs.

[animation update]
Do the black pipe cleaners first, those that go from -1 to +1, -i to +i, etc.

All the points are center stage.

The blue pipe cleaners diversify at least in the tx plane:

A similar pattern holds for yellow and green and their corresponding planes:

And of course, these can all be combined to make an animated version of Q8:

Like the pipe cleaner static model, it is not easy to see what all is going on here since there is much overlap. There are three "blink" times: -1, 0, and +1.
[end update]

Doug

Snarky puzzle: What is required so that a system undergoing change can be described just as accurately by the group Z4 as by Q8?

Quaternion Jam session: Saturday, 11AM on Saturday, Google+ Hangout.

Next Monday/Tuesday: Quaternions, THE numbers of the Universe[title for an Ignite Boston talk I will give on Thursday, March 29]

Doug:

I haven't double checked Q8, but it is obvious (to me) that your "Z4" is not Z4.

Sorry.  :(

David

At least an issue is with a small one that is MUCH easier to regenerate :-)
From _http://groupprops.subwiki.org/wiki/Cyclic_group:Z4

The multiplicative subgroup of the nonzero complex numbers under multiplication, generated by  (a squareroot of ).

So the one in white is +1, and the one in blue is +i. This is my "too positive" error. I need a black cube for -1 and a -i too. Will try and correct tonight.
The correction has been made. One difference in my upgrade effort was not to use letters, relying instead on color. White is +, with all white being +1. The blue with a white dot is +i. One nice accident is that having black be minus on a black canvas makes it disappear. Maybe that is not good, will have to see how I feel about it. If the canvas was gray, it would be easier to see the black.

I don't think there is enough photographic info in the blog to check all the directional edges on Q8.

Doug:

The Z4 is correct, now, though you should still correct the prose:

There is another type of squish, going from Z4 to Z2. That represents a move from a directional graph to an undirectional graph.

I also think that a slight "twist" in your Z4 representation may help its relationship to subsets of Q8 become more obvious.  Basically, exchange the positions of the -1 and -i blocks, so the blue connections form a square, and the -1 connections form the central cross, just as they do in your Q8 representation.  This also helps illustrate the cyclic character of Z4.

Just a suggestion.

David

I did a sketch and like the idea. I need to buy new 3/4" wood blocks, and find a time to drill some holes. Writing at night is OK, but not high speed drilling as that can wake the little one.
Doug

Doug:

I think I agree that "I don't think there is enough photographic info in the blog to check all the directional edges on Q8."  However, there is enough to tell that at least some of your directional arrows are reversed (black when they should be white, and visa versa).

For instance, going from k, with a -k, takes you to 1, which you correctly have, but not to -1, as you simultaneously incorrectly have.  So I suggest you look for such conflicts.

Once you make the recommended "twist" in your representation of Z4, you will be able to use that to double check the three representations of Z4 within Q8.  (The cyclic nature of Z4 should "shine through" and help you here.)

The other directional arrows will be a little more difficult because you will have to be very consistent with what order you multiply the block with the arrow.

David

I did focus on getting the thing built, and should have given an hour to check the math. Here is the error on the top of the cube and how to read it. One takes a vertex, one of the cubes, multiplies it by the edge, and sees if the result is consistent with the multiplication table for Q8. The Kelly green cube with a white dot is +k. The Kelly green pipe cleaner is a k. If it has a white bead and a white arrow, then it is +k. The others are Indigo for i, and yellow (Jaundice) for j. Focus only on the Kelly green cross on the top of the cube, so the second term in the products will always be a k:
$\\ (-1) \times (+k) = -k \\ \\ (+k) \times (-k) = 1 \\ \\ (-i) \times (+k) = j \\ \\ (-j) \times (-k) = i$

By my tally I got...none of those right. Arg. This is going to take a number of hours to fix because it is confusing. One neat thing I have noticed were patterns along the surface. For example, I just noticed squares at 45 degree angles that are of one color. I bet when the arrows are done right (tricky), there will be a simple pattern. It should be a fun few hours doing this kind of error correction.

The main blog is now up to version 3 of the big images. As you suggested:
The cyclic nature of Z4 should "shine through" and help you here.
That makes error checking easy for Q8. My error rate was 6 out of a possible 48. Thing is, the top of the cube had three negative vertices, and I think 4 of the 6 errors. Those silly extra minus signs.

A fun conceptual insight was to see the similarity between complex numbers and cross products. One green pipe cleaner plane was about plus/minus one and k and looked just like Z4. The other had i, j, and k, but also looked just like Z4

<I forget my own tag to indicate I feel less confident about this part of the comment...>
So there are three complex numbers as subgroups in a quaternion, but also three of these cross product groups. Because the complex numbers have the same graphs as the cross product groups, they have the same multiplication table, and they will be closed. It is the closure that feels shaky. I know i2 is not in the plane :-)  The i, j, and k are just labels.

Yet there is a distinction. The complex plane uses only 2 symbols: 1 and k. The cross product one uses three: i, j, and k.  I think I have to go with the abstraction, without labels.

In the craft project, I could have included labels, but thought the pipe cleaner representation was too busy already to include letter beads. The lack of labels made it easier to make mistakes. Now that I look for cycles of arrows, it was easy to weed out the mistakes.
Mr Sweetser : "very late" is better than "never".
So a big thank You from Italy ....

the physical constants still are scalars or they became vectors,i times epsilon zero ,j*mu0 ,k*c^2 ?

mobber:

Right now, there is only a single physical constant per field coupling.  These are typically called "charges":  Electromagnetic (U(1) group), Weak Nuclear (SU(2) group), and Strong Nuclear (color, SU(3) group).  Finally add in the couplings with the Higgs field, that, together with the Higgs field "vacuum", yield the masses.

Of course, this is after getting rid of unit conversion constants like c, and charges in Coulombs, etc.  All the remaining constants are unitless (except for that pesky coupling in General Relativity ;) )

What we would like is some theory that reduces the number of physical constants.  Trying to group them into some "vector" or "tensor" form doesn't seem useful, especially since these physical constants only transform like scalars.

David

david:in every day life we neglect the theory of relativity,is' implications,because 1/c^2 is very little,in ours usual intervals-little space,much time.so we prefer to consider 1/c^2=0.what doug tries to do,as i understand it,is to modify the formalism as a way to expand the understanding about known paradigms,to facilitate new insights.today the "speed of light" is given BY DEFINITION:c=299792458m/s.i like a better definition:c=i*299792458m/s.this works so much better,Pythagora's still hold.yes,an imaginary value for this constant.....only a change in the formalism...we can continue neglecting a very small imaginary number,:).

error:the sign of a very small number,if we are barely 100 years old considering it's magnitude.....

mobber:

Your desire to set "c=i*299792458m/s" is in similar company with those that use ict for the "time" dimension (though your choice would simplify this back to ct, since the imaginary, i, becomes hidden).  You would be in rather good company, since Hawking likes the "imaginary" "time" approach.

Now, I used to be vehemently against the "imaginary" "time" approach, especially when in combination with Quantum Mechanics, and its own use of an "imaginary".  I always thought the two "imaginaries" would have to be different "imaginaries".

However, while working on my dissertation, I started some work on a complex form of General Relativity (with Hermitian [pseudo-]metric, but one that involved a complex spacetime manifold that could be describable as a real manifold [since our spacetime is obviously describable as a real manifold]).

In the course of that work, I found that I could properly ask the question of whether one could appropriately use "imaginary" "time", without having to introduce a new "imaginary".  I expected to prove that such simply was not consistent.  However, to my complete surprise, I proved that one could, indeed, transform, quite consistently, to an "imaginary" "time" description.

Unfortunately, my dissertation was getting too long as it was, so I left this work in my dissertation file, but commented it out.  Adding injury to insult, I later lost my entire dissertation file when trying to restore from a backup!

Someday, I hope to reproduce this work.

David