From Math Images
Steve M's status 6/14: reread and edited the ms. See Active Comments below.
Becky's current status, 6/14: Editing according to both sets of comments
Abram's current status, 6/14: I have read the page and made a few last comments, but then I'm ready to sign off on the page for the summer.
Anna's current status:
Smaurer1 18:15, 14 June 2010 (UTC) Becky (indeed, all of you), I hope you have realized that no matter how much you polish this, I will think of things to change. That's the way math writing is - never finished. So make this the last time I read it!
I've mostly worked on the More Mathematical Explanation section.
I've marked most of my comments with the built-in author designator that I used just above, Smaurer1 in red with the Greenwich Mean Time. Of course, take my comments out once you understand them and respond to them.
Quick content questions
- The statement about geometric probability feels awkward in the basic description, partly because you don't explain what geometric probability is. I know it's been hard to find somewhere good to put it. You could always start a stub section on geometric probability, where you put the two sentences you know. Or you could drop it. Or, I don't know, whatever else you come up with.
- From their observations, researchers were able to draw the conclusion that scout ants must have a method of measuring areas. Is this because they noticed that ants generally chose the nest with the largest area, regardless of the shape, or...?
- Abram, 6/14
Small wording issues (it's fine to disagree with any of these)
- A needle will cut a line if the green arrow, d, is shorter than the leg opposite θ. d is the *length* of the green arrow, so it should say something like "a needle will intersect a line if the length of the green arrow, d, is less than length of the leg opposite theta."
- There are still a couple places where the word pi is in the text instead of the symbol.
- Because of the small angle, the center of the needle would have to fall very close. Maybe add to the end of the sentence, "...have to fall very close to one of the horizontal lines in order to intersect it."
- Add " = 1/2" to the integral equation, even though you say in the body of the text that it's .5.
- As a result, π cannot be written as an exact decimal. Maybe instead, "This also means that a normal decimal representation of pi is non-terminating and non-repeating."
- This method of random motion
- Abram, 6/14
Add a "geometrization of probability" section
This was our joint idea for how to be more clear about the fact, already somewhat described in this page, that part of what is interesting about this problem is that it began the idea of studying probability geometrically. (Abram, 6/8)
- We agreed to drop this section because it turns out we actually have very little to say about this. (Abram, 6/14, reporting on in-person meeting from 6/11)
Reorganize the "Needles in Nature" section and make it more concise
This section is really interesting. The material about what the ants do and their "reasoning" that currently (as of our meeting on 6/8) comes after the explanation of how this relates to Buffon should be moved before this explanation. This may also allow this section to be made more concise.
- I reorganized this section (Becky 6/10).
Also, the explanation of how the ant hills relate to Buffon should be slightly reworded to refer to the concept of area directly, not just lines being further apart.
- I think we decided not to do this Abram because it would just confuse the readers? I'm not sure if we really need it
- As discussed in person on 6/11, it was the number of intersections vs. probability issue that would confuse the readers, not this one. It seems like it's been addressed now, though. (Abram, 6/14)
Slightly reorganize the "Probability of an intersection" section
In the current version (as of our meeting on 6/8), the overall process is outlined twice. We've decided to rearrange it so that the process is outlined only once, but in a way that still provides very clear guidance for the reader. (Abram, 6/8)
- I took out the redundancy you noticed when I met with Dr. Maurer yesterday, and this solution allowed me to do so without rearranging the whole section. Dr. Maurer thought that the paragraph:
- The probability of a hit can be calculated by taking the number of total ways an intersection can occur over the total number of possible outcomes (the number of trials). For needle drops, the probability is proportional to the ratio of the two areas in this case because each possible value of theta and d is equally probable.
- was too filled with probability language. So since I needed to take this out anyway, I didn't think the reorganization was still necessary (because i had already taken out the redundancy. Dr. Maurer thought the section was pretty clear.
- I agree. This section looks good now. (Abram, 6/14)
Links to other pages
We agreed that it doesn't make sense to link to any other pages that currently exist, but may think about creating a related page later in the summer. (Abram, 6/8)
Various other small comments
We discussed a long list of other small changes. (Abram, 6/8)
- I changed all the comments from my papers. Thanks Abram!
- Looks great. I'm about to add my last round of small comments. (Abram, 6/14)
Conference with Becky and Abram about general layout suggestions
Comments: Abram, 5/27 early morning
Becky, these revisions are great. I have a couple of questions/ideas, but we can talk when I get in later today.
In response to your comments, yeah, it would probably look good for the first formula to be centered. I don't have anything useful to say about hiding the "Why It's Interesting" part. I haven't yet come up with useful thoughts about these organizational issues, so for the moment, umm, yeah, sounds great.
Comments: Becky 5/25
- Revised changes suggested by Dr. Maurer.
- Do you think the first formula should be centered? I think it looks a little squished next to the main picture...
- I hid the 'Why It's Interesting' part
Comments: Becky 5/20 Afternoon
I think I improved all the things you mentioned.... Hopefully they look OK. I sent the link to Dr. Maurer too, so hopefully he'll have some additional feedback. Things that I changed:
- Added the paragraph in the Basic Description part beginning with "The idea behind..."
- Added description of Case 1 and Case 2 pictures. New paragraph begins with "See case 1,..."
- Added paragraph on sample space beginning with "The sample space is useful..."
- Added another diagram to the ants section and additional explanation. Do you think the italics work?
Comments: Abram, 5/20
Great things about the page:
- Clear development and explanation of mathematical content
- Well-explained connection to other branches of mathematics and even to nature
Things to work on:
- Beefing up the basic description a bit, without making the page redundant
- A description of what the reader should notice in the Case 1 and Case 2 diagrams.
- More elaboration on the "The estimated area of a flat surface is inversely proportional to the number of intersections between the set of lines randomly scattered across the surface" .
- More elaboration of how this sample space calculation works
- Explanation of what the numbers in the "accuracy" column of the table represent
Comments: Becky 5/19
If you have a chance to look at this, Steve and Gene seemed to think that it was looking close-ish to ready. Let me know what needs work... Thanks!!
Comments: Abram, 5/17
Great things about the page:
- Good flow (each idea is clearly connected to the preceding ideas and the following ideas)
- Good explanation of concepts (Careful explanations, not glossing over anything)
Things to work on:
- Exploring the relationship between this topic and other Monte Carlo / probabilistic approaches to non-probability problems
- Expanding the basic description
- Adding a contrasting image to the current page