Talk:Prime spiral (Ulam spiral)
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Iris 6/15 I made some changes according to pr.Maurer and Abram's comments. I'm still working on the Euler section. Other than the Euler section and pf. Maurer's comment about vertical and horizontal lines having constant 2nd differences (which I will be talking with him tomorrow), I made all the changes.
Smaurer1 11:03, 12 June 2010 (UTC)
I've gone through the latest version now.
The demonstration and movie showing that the 2nd differences are 8 is lovely. If you think about it, this demonstration shows that not only are diagonals quadratic in the Ulam Spiral, but so are horizontal and vertical lines -- in each case if you start far enough from the origin and keep going away from the origin. Basically, you have to go far enough out on the line so that to find the first difference of any consecutive terms by counting off along the spiral, you have to go once around a square. Think about it, you have a proof looking for a theorem, that is, a correct technique that merely needs the right restrictions to be applicable.
Throughout you say "line" when you mean "line segment". To a mathematician a line is infinite. Actually, all your results are correct for half lines - infinite in one direction. Your claim is not actually all that interesting for line segments: every finite sequence of numbers is polynomial. Also, as we discussed, it appears that the line y=x (through the origin) is the only line on which the Ulam numbers are polynomial in both directions with the same one polynomial.
Now that you've added the movie showing that , and explained how to get a quadratic from that, then the first part of your example wih 5, 19, 41, 71, 109 is redundant; we already know it will be quadratic and don't need the parabola. In fact, I think this whole example of determining the specific polynomial 2 ways (I am not sure I would call it deriving the polynomial) really belongs in the difference table helper page, not here. It's not really central to the Ulam Spiral discussion, even though it is very nice.
I have a number of detailed comments that we can discuss in person. I've also made a few individual edits in the Newton Interpolation subsection.
Individuals' statuses
Abram, 6/9: I notice that the page has been revision since our meeting on 6/8, but I haven't yet had a chance to review these revisions. My comments dated 6/8 are from a discussion on 6/8, but were not posted on the discussion page until the evening of 6/9. Clearly, some of them have already been addressed.
Active Comments
Things that are great about this page
- Great brief intro to the spiral. The context in which it was invented and the description of how we don't know formulas for prime numbers really help make the page seem relevant.
- Good use of examples in the quadratic polynomials section
- Sentence like, "It might seem strange that..." do a good job guiding the reader through the content
- Abram, 6/9
Reframe the description of quadratic polynomials along diagonals
All entries along a diagonal can be described with quadratics that have a leading coefficient of 4, not just the prime entries. Rephrase this section to make that more clear. (Abram, 6/8)
--I rephrased the section so that any diagonals can be described through quadratics. pf. Maurer and I talked, and it turns out that there are some exceptions, so I'll have to fix that point. (IRis, 6/11)
Also, it would be good to include a proof of this fact, rather than simply an example. A proof that this is true along the main diagonals (that go through entry number 1) won't be too ugly. We haven't yet figured out if doing a proof along other diagonals is or is not nasty. (Abram, 6/8)
--I provided the proof with the small animation. (Iris, 6/11)
- Can you write an explanation that walks through each step of the animation? (Anna 6/25)
Also, in this section, there's a lot of text in pretty big paragraphs. Can you break it up into smaller chunks? (Anna 6/25)
Elaborate on the Euler section
Does this section generalize in any way to work with numbers other than 41? Does the fact that there are 40 consecutive prime diagonal entries give a hint of some larger pattern, or is it just an isolated curiosity? In general, make the significance of this material (and whether or not it is even seen as significant) a little clearer. (Abram, 6/8)
Add A "Why It's Interesting" section
We have discussed that this spiral is not something mathematicians have studied seriously, but on the other that the patterns you describe are not just people's minds trying to find patterns, which indicates the spiral could be significant. Having a section about this could be interesting. (Abram, 6/8)
--I have added a "Why its interesting" section. I actually addressed the problem of people's mind trying to find patterns in the more mathematical section where I compare the Ulam spiral for prime numbers and random numbers. I'm not sure whether I should move this section (Iris, 6/11)