Talk:Prime spiral (Ulam spiral)
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Iris6/28
I made all the changes to pf Maurer, Abram, and Anna's comments. I am ready for more comments
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
Other numbers and patterns section
- The triangle number section is really clear and concise. Is there any reason why they form that patter? You don't have to answer that question, but it might be interesting.
- In the prime numbers in lines section, you've got some layout kinks. You might want to center the big images here. The animation/picture in my browser ends up cutting off the equation numbering in a really weird way--the number jumps down to after the word "or" but above the next equation. Other than that, this section is quite clear and well written (Anna 7/7)
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)
- Nice job with the math content. See the separate discussion thread about making the description of "rings" and "diagonal half-lines" clearer. (Abram, 6/28).
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)
- Iris I added an explanation (6/28)
- Nice job with this. There are a couple of wording details that can be dealt with in a final pass through. The only substantive problem right now is that if the reader starts looking at the animation at the wrong moment, they won't see the "innermost light blue boxes" that you refer to. If we can't add a "play" button to the animation, you might want to tell the reader to wait until they can see three rings of blue boxes, and then watch the animation all the way through. (Abram, 6/28)
- Iris I added a sentence (6/29)
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)
- Iris I tried breaking them down. (6/28_=)
Clarify the description of "rings" and "diagonal half-lines"
It seems like we've decided to super-impose the blue spiral from Image 1 onto Image 3 so that we can point out examples of how the red lines give you numbers that are in the same "ring".
We also decided to change things so that either we come up with a new term for diagonal half-line or in some other way rephrase this, because a) many things that one would intuitively think of as a diagonal half-line are not diagonal half-lines according to this definition (e.g. the red lines in image 3), and b) many things that one would not think of as diagonal at all satisfy this definition (e.g. many horizontal and vertical lines). It's not that there's anything mathematically *wrong* with this. It's just a really confusing choice of terminology. (Abram, 6/28)
- [Iris] (6/30) I clarified the terms involving half-lines. I superimposed the blue spiral, and I added a couple sentences to describe what we meant by ring, but I'm not sure if this is clear enough.
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)
As a small thing in this section, make sure to point to Image 8 when you are talking about it in the text (Anna 6/25)
- Iris] I made this change (to both Abram and Anna's comment_ (6/28)
- Aha, I think I figured out what was confusing me. Look at all the material in the Euler section that starts off hidden (all the text after "To learn more about how to find Euler's polynomial, click show more". First, instead of describing this section as a way of "finding" Euler's polynomial, would it be accurate to describe this as a way of deriving why the Ulam spiral that starts at 41 generates the same outputs as Euler's polynomial.
- Second, you might want to point out that because the "central" diagonal line follows the rule about never staying in the same ring in both directions, you can actually plug in negative numbers to generate the numbers that are "downhill" from the center. As it stands right now, the x= -19 through 20 seem to come out of nowhere.
- Third, when you point out that x = -19 to x = 20 into 4x^2 - 2x + 41 generates prime numbers, this is actually a little bit confusing. The reason is that you are in the middle of showing how you get *to* the conclusion that this polynomial is in some sense equivalent to the Euler polynomial (once you do an appropriate transformation), but the fact that plugging x = -19 to x = 20 into this polynomial gives prime numbers actually comes *from* the fact that the these polynomials are essentially equivalent. It's not that you said anything wrong. It's just that pointing out an implication of statement X while you are in the middle of proving statement X is a bit disorienting.
- There are a few more small changes that could help clarify this section, but this will make a big difference. (Abram, 7/1)
I really like the way this section reads now. You've done a great job of flushing it out and explaining your pictures better. (Anna 7/7)
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)
- I think moving this section about how the patterns aren't random to the Why It's Interesting section is a really good idea. (Abram, 6/28)
- Iris I moved this section (6/29)