Talk:Prime spiral (Ulam spiral)

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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 \Delta^2 a_n = 8, 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.

Xingda's Comments

  • All images should be labeled correctly and explained correctly. For example, the part explains the hockey stick does not match the picture given by Pascal.
  • Certain explanation can be more easily grasped by putting a picture besides it. For example, the first part of the MME can be accompanied by a picture.
  • label the sequence in the Sierpinski Triangle and explain the picture exactly as it is drawn.
  • Explain why the hockey stick's head has to point in certain direction but not others.
  • put the first three patterns in the other patterns and properties in the basic description. They are very simple and should not come as the last part of MME.
  • make some of the obvious connection more obvious for example, put the relationship of a and b at the very beginning of the MME next to the conclusion of the Binomial Coefficients so that the connection is obvious and your whole argument comes full circle.

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

Layout of the diagonals section

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_=)
It looks much better now (7/14)
Use of subheadings and/or at least one picture to go with each idea might also help. For instance, sub-headings on "Definition of half-lines" and "Examples and equations for half-lines", or having Image 3 and Image 5 break up the text instead of being on the side. I don't really know, but it's something you could play with.
You may want to look for other places in the page where images that are on the side could look more appealing and be physically closer to the accompnay text if they were in-line as well. (Abram, 7/8)
Iris(7/12) I fixed this
There's a huge amount of white space around Image 3 now. Is there any good way to deal with this? (Abram, 7/14)
I'd be okay with the whitespace if you centered the image (Anna 7/14)
Iris(7/15) I fixed this
You could center image 8, or not. You could also put image 16 and 17 under the text instead of to the side, or not. (Abram, 7/16)
Iris(7/19) I tried putting the image 16, 17 under the text, but it doesn't work properly. I centered image 8
Image 8 still doesn't look centered on my browser, but that can wait for another person at another time. (Abram, 7/19)

A couple mathematical details

  • I have no idea what the last sentence in the Archimedean Spiral definition means. (Abram, 7/14)
Removing that last sentence took care of the problem. This page still doesn't identify exactly how you space out the dots on the spiral, but you could add that as a future suggestion or something. (Abram, 7/16)
Iris(7/18) I put up in future suggestion that we need a Archimedean spiral helper page
My guess is that an Archimedean Spiral helper page wouldn't actually address this placement of dots, because dots are not an inherent part of the Archimedean Spiral -- they're just added for the sake of the Sack's Spiral. But again, this can wait for another person at another time. (Abram, 7/19)

(Abram, 7/8)

Archived Commetns

A couple mathematical details

  • Diagonal lines either have a slope of +/- 1 or they move upward or downward at a 45 degree angle. But they don't have a slope of 45 degrees.
  • The beginning of the "Prime numbers in lines" section includes the sentence "The nonprime numbers that appear on prime-concentrated columns are all multiples of prime numbers." Of course that's true -- all numbers are multiples of prime numbers! I'm guessing there is something else you are trying to say here.
  • Add mouse-over definitions or external links for Gilbreath's conjecture and Goldbach's conjecture

(Abram, 7/8)

Iris I fixed all of these (7/8)
The Goldbach's conjecture mouse-over wasn't working on my browser. Otherwise, looks good. (Abram, 7/14)
Fixed (as described above). (Abram, 7/16)
  • In the triangular number section, the first sentence should read, "The nth triangular number, T_n, is given by the formula", otherwise you technically never define the symbol T_n (even though it's really easy to tell what you mean by it). (Abram, 7/14)
Iris(7/14) I dealt with this problem
Looks good. (Abram, 7/16)

Edits for the sake of explanation

  • There's still a sentence that reads "To learn more about the relation between the diagonal lines and quadratic polynomials, click show more."
Looks good. (Abram, 7/14)
  • The sentence "Indeed, as Image 4 illustrates, regardless of the exact number of the blue boxes, there are 8 more boxes in the outer ring than in the inner ring" is confusing. What is "the" outer ring and "the" inner ring? Seems like maybe it should say something like, "There are 8 more boxes in any given ring than there are in the ring that is one layer inwards from it" or something. (Abram, 7/14)
Iris(7/15) I fixed this
Looks good. (Abram, 7/16)

We could always do more

Iris, this page is looking quite good. We could always do more. For instance, there are a few slightly confusing sentences here and there. Also, we haven't subjected the "prime numbers in lines" section to nearly the same scrutiny that we have with the rest of the page. It's a recent addition, and it's a really nice section, but there are certainly ways to clean it up. However, you have already done a lot of really good work for this page, and you should feel free to say you feel done with it. (Abram, 7/14).

I actually feel like the "prime numbers in a line" section is very straight forward, and I can't come up with anything that would really help the section. So, like Abram, I think you can leave it as is. (Anna 7/14)

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.
Iris(7/8) professor Maurer and I have been trying to find an explanation, but we couldn't until now.
I'm happy to let this go for the time being. Anna? (Abram, 7/14)
I am too. I was just curious! (Anna 7/14)

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)
Looks good. (Abram, 7/8)

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.
Really nice clarification of half-lines. It might be nice for the definition of a ring to avoid use of variables (e.g. you could describe them as "concentric rings centered around the 1 at the center of the spiral" and give a couple examples of pairs of numbers that are on the same ring). I don't know if my suggestion was actually correct, and maybe this is impossible. It would just be nice. (Abram, 7/7)
Iris(7/8) I fixed a little bit.
Really nicely done. (Abram, 7/14)

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)
Agreed. Nicely done. (Abram, 7/8)

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)
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)
Nicely done. (Abram, 7/8)

Nit Picky Details (Also known as "doing more"--see Abram's comments below) Each of these comments should take you very little time to address. (Anna 7/14)

  • In the "Why is it interesting" section, you refer to images 12 and 13 being side by side. In my browser, one appears on top of the other.
Iris(7/15) fixed this
If you wanted to, you could replace every reference throughout the page to "the image below" with a reference to an image number (it happens somewhere else in the page too), but don't worry about it if you don't want to. (Abram, 7/16)
Iris(7/19) I decided to leave it as it is
OK. (Abram, 7/19)
  • Something's not right with the first two images. When my window's taking up just part of my screen, the pictures smoosh the text around and make it very difficult to read
Iris(7/15) I fixed this for now, but maybe I'll use Xingda's table format later on
Looks good to me. (Abram, 7/16)
  • The sentences before and after image 3 both start with "for instance" Can you rephrase one or both so it's not so repeatative?
Iris(7/15) fixed this
Looks good to me. (Abram, 7/16)
  • There's a stray mark in this sentence: "First, we found out in the previous section that Eq. (1) is the polynomial for the diagonal that goes through the center 1 and has a slope of °+1"
Iris(7/15) fixed this
Looks good to me. (Abram, 7/16)
  • Your image numbering skips two sections--can you go back through and number those, and update the numbers in the last section?
Iris(7/15) fixed this
Looks good to me. (Abram, 7/16)
  • The "Goldbach's conjecture" bubble isn't working on my computer. I don't know what's up with that.
Iris(7/15) fixed this
Looks good to me. (Abram, 7/16)
  • 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)
Iris(7/8) I rearranged the images
Looks good to me. Anna? (Abram, 7/14)
I agree (Anna 7/14)

Edits for the sake of explanation

  • The first sentence of the basic description can be deleted, if the mouse-over definition of the prime numbers is moved to the next time those words are used.
Great. The mouse-over definition of prime-numbers is still placed over the *second* use of the term. This is a bigger problem in the difference tables section, where the link to difference tables is placed in the second use of the term, more than a paragraph after the first use of the term. (Abram, 7/14)
Iris (7/15) I moved the mouse-over definition of prime numbers to the very top, and I added a mouseover definition to difference tables the first time I used it
The mouse-over definition might be a little bit clearer if you said, "The difference table list the terms of a sequence in one row, and the differences between consecutive terms in the next row...", but you can leave your definition if you'd like. (Abram, 7/16)
Iris(7/19) I fixed this
Looks good. (Abram, 7/19)