# The Prevalence of 3?

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1. Apr 2, 2015

### Wolfman29

Hi everyone.

I'm almost done with my senior year of university (and will be going to Davis in the fall for the Ph.D. program), and something occurred to me yesterday: the number 3 is very prevalent in our universe. So my question is "Why?"

A few examples:
• As far as we can tell, there are only three families of quarks
• As above, we think there are only three species of leptons
• Three color charges
• Three spatial dimensions
• The fact that cross-products work (a consequence of three spatial dimensions - cross product is only defined in three dimensions)
So I'm sure there are more, but these were the only ones I could think of off the bat (at least, the ones that are interesting and not just numerology).

Now, I came to thinking about this because, it just happens that as a mathematical quick, when performing rotations in 3-space, to entirely parametrize the transformation, you need three rotation angles (Euler angles). This is simply because 3 choose 2 is 3 (a "quirk" of the mathematics) - it turns out that there are exactly three distinct planes in 3-space, which leads to 3 distinct rotation angles. In 2-space, there is only one plane, so we require only one rotation angle. In 4-space, there are six planes, so we require six rotation angles.

So, is it possible that the fact that there are exactly three color charges, families of quarks/leptons, etc. is simply a result of such a mathematical quark that is a consequence of combinatorics?

Maybe this is all a coincidence and I am thinking too much into it. Or maybe we can explain the prevalence of the number 3 by using the anthropic argument. I'd like to hear your thoughts on this.

2. Apr 2, 2015

### Staff: Mentor

You are thinking too much into a coincidence. 10% of all single digit numbers are 3. That's all it is.

3. Apr 2, 2015

Staff Emeritus
The number of quarks needs to be equal to the number of leptons for the theory to be consistent. There also need to be 8 or fewer generations for the theory to exhibit asymptotic freedom (i.e. match experiments), and at least 2 colors.

4. Apr 2, 2015

### Greg Trayling

It's true that we can only speak of an axis of rotation in three dimensions. In any higher-dimensional space, one simply refers to a plane of rotation, since there is no unique axis perpendicular to any plane.

It is possible to define a cross-product in seven dimensions using an involution called triality, but this is an obscure and singular exception. In general, any attempt to define a cross-product in a space of dimension higher than three can be rotated to where it is inconsistent with the original definition. (If you want more threes, triality is a unique discrete three-fold transformation which preserves the commutation relations of the rotational generators of the space, and completely preserves the generators of G2, but is not itself a rotation. Apply it three times and you're back to the initial space).

There are some loose connections arising from writing the Standard Model in Clifford-Algebraic notation, where the number of fundamental particles is restricted to the the available degrees of freedom in the spinor space of the algebra, but there has not been a really convincing demonstration of this that ties everything together.

I've lost track of how many sleepless nights I've had pondering why space is three-dimensional. It's ultimately a good thing, because you couldn't tie your shoes in anything higher.

Last edited: Apr 2, 2015
5. Apr 2, 2015

### Teichii492

It's simple really. There is no significance, you've just allowed yourself to be swayed by faults in your cognition and it's definitely not worth losing sleep over. The same as it's not worth losing sleep over why the dimensionless parameters take the values they do.

It's just apophenia with some of your cognitive biases thrown in

6. Apr 2, 2015

### SteamKing

Staff Emeritus
This is not true. The cross product can also be defined in seven dimensions.

http://en.wikipedia.org/wiki/Cross_product

The cross product in 3 dimensions can be uniquely defined.

7. Apr 2, 2015

### Wolfman29

I understand that it is likely just a coincidence (with apophenia and cognitive biases thrown in, as Teichii said), but (and I hate to say this...) that doesn't mean that there isn't something to it. In particular, the fact that the cross product can be defined by seven dimensions (you learn something new every day!) raises another question indicates that there is something mathematically unique about 3- and 7-dimensional Euclidean spaces. And the fact that it can only be expressed uniquely in 3-dimensional Euclidean spaces is interesting as well.

I know that there are some questions in mathematics that are answered by "that's just how it works out," like (in particular) questions about group structures, questions in number theory, etc. I guess I am more comfortable with that than I am with that sort of thing in physics. So can those sorts of things I am seeing in physics be boiled down to those sorts of "coincidences" in mathematics? I guess what I am asking is, can we explain all of these 3s by some elegant argument (from mathematics) that says it has to be 3? I presume this sort of thing would require a TOE, but is it even possible in principle?

8. Apr 2, 2015

### Teichii492

Why do we find triplets satisfying? why are triads so effective in music? Why do we prefer to count to three in preperation for something? Why is the speed of light almost exactly 3 x 108 m s-1?
How comes 23 =8 and 32 = 9 are the only non trivial consecutive powers of positive integers?
In regards to your comment about 3 and 7 dimensional spaces? Don't you find it odd that the first convergent of π is [3; 7] = 22/7 = 3.14 ?
Just look at all those 3's, you can do this all day, but it isn't unique to three, i was just following your mindset here.

The problem is you can do this with many arbitrary choices, you are assigning an undeserved uniqueness to one particular number. This is how the minds of deluded conspiracy theorists work, making connections and extrpolations beyond meaning, purpose and productivity from sets of unrelated data.
The connections you are making are highly trivial at best.

One thing i can't understand is why you have chosen three (this is a rhetorical question) because there are much less trivial mathematical connections out there that deserve far more attention. I suggest you distact yourself with those rather than threem, things like monstrous moonshine or the signifiance of mirror symmetry (how two seemingly different calabi yau spaces can produce equivalent field theories), The connection of M-theory to the five string theories given large limits of the coupling constant.

You seem unsatisfied by the answer that "that's just how things are", i suggest you read Martin Rees's Just six numbers, it discusses a similar question but in regards to the physical constants of nature but be aware this is a feeling you need to shake since it's counter productive and the logical steps required to grant ideas like this their significance leads to missteps in important critical thinking tools that are necessary to think clearly about science.

9. Apr 2, 2015

### Staff: Mentor

Closed for moderation

We will leave it closed. Numerology is not a topic for the forum.

Last edited: Apr 2, 2015