# Talk:Dandelin Spheres Theory

## Contents

- 1 Response to Checklist
- 2 Discussion
- 3 General Comments
- 4 Section-specific comments

# Response to Checklist

## Messages to the Future

- Have left message for future directions.

## References and footnotes

- All images are properly attributed in the page you see when you click on the image. Attributions include original source and remarks if you've modified.
- Direct quotes from textual sources are cited.
- References for text are at the end of the page, with option links to the footnotes within the text.

## Good writing

### Context (aka Generating interest aka Who cares?)

- This page is specifically for college students who have interest of math, since it contains bunch of proof and not a lot fun.
- I don't know if I need to move my application section to
**why interesting**

### Quality of prose and page structuring

- The beginning paragraph(s) of the page clearly define the topic or purpose of the page as a whole, and may outline the page or preview conclusions that will be reached later in the page.
- The purpose of each section is clearly relevant to the purpose of the page as a whole.
- Still, I don't have a thesis but I follow the section titles as thesis.
- Do not need a helper page.
- My order is definition > 1 proof > 2 proof > application.

### Integration of Images and Text

- Because my images are used be facilitate proofs, I use a lot of words to explain. Since the images I found are from an applet, they are not clear enough, I'm still looking for better ones.
- The text explicitly points out what the reader should observe in a picture.

Connections to other mathematical topics

- Wherever possible, relationships between the content of this page and of other topics/ideas in mathematics are identified and explicitly described.
- I made a
**see also**section to leave links to related topics.

### Examples, Calculations, Applications, Proofs

- Do not have difficult concepts. And always introduce first.
- Proofs are included wherever they would be of interest (this probably means anytime you have a statement whose truth is not totally self-evident and which is important to the page as a whole), but ONLY if writer feels comfortable with the proof (otherwise, it is perfectly acceptable to leave a proof for others).
- Applications of equations, theorems, etc, either to other branches of math or to the world outside of math, whenever these applications seem interesting and/or are needed to set a context (see context section, above).

### Mathematical Accuracy and precision of language

- No error for equations.
- My page has a lot of proof and symbols, I hope they are not overwhelming.
- Any mathematical term that the reader can't be expected to know is defined (err on the side of defining too many terms), either in the body text or via a mouse-over or link to another web resource or a helper page.

## Layout

- Text is in short paragraphs, and broken up by relevant images throughout.
- Hide and mouse-over features are used as appropriate to reduce clutter and scariness. Proofs and large masses of equations in particular should be hidden, and terms should often be defined by mouse-over, rather than in text, if the definitions are short and readers may already know them. But if people cannot be expected to know them, then they should be defined in proper mathematical style, that is, the word being defined should be boldfaced to announce that what follows is a definition, not a rough description.
- To whatever extent possible, pages do not have large, awkward chunks of white space.
- No image in one section of a page vertically aligns with the text or a heading of a different section.
- In hidden text, none of the preview text appears as weird computer code (see Wiki Tricks for help on this).
- The page has been viewed at a few different window sizes to make sure funky things don't happen.
- Much better with less hidden text

# Discussion

Flora 23:41, 24 June 2011 (UTC) Finish most of the page, still need to work on the application of dandelin spheres, and partial work of the spheres tangent to a circle.

* Rebecca 04:26, 6 July 2011 (UTC)Hi, my name is Becky and I'm one of the people responsible for giving feedback on the project. I've been working primarily with Swarthmore, but I'll be reading a few of your pages as well.

- This is a great idea for a page. Very visual and well presented!

Kate 18:42, 6 July 2011 (UTC): Hey Flora, I was just coming back to see how this page was going and it looks like you've made a lot of changes! The things you've fixed definitely help - those sections are a lot easier to read. Like Becky pointed out, there's a little more re-organization left to do, but the content is great, and by the time you're done, this is gonna be a kickass page!

# General Comments

Kate 17:46, 29 June 2011 (UTC):

- First off- your table of contents is frighteningly large. There are so many sections, and your section titles are really long, and just
*looking*at the table of contents makes me feel exhausted. See if you can shorten the section titles, or if you can merge any sections. Maybe some of the proofs can be introduced without having a section title?

- I changed it. Is it much better?
- Kate 23:22, 29 June 2011 (UTC): THIS IS SO MUCH BETTER!!! :D

- Second, as I mentioned on Anna's & Phoebe's pages, a lot of people will miss the content that you have before the table of contents. What they've told us here is that it's better to give just like a one-sentence caption for your picture there, and to move the rest of the information into the Basic Description section, because that's where most people start reading.

- Flora 00:29, 30 June 2011 (UTC)I left 3sentences for this part, I hope this works.

- In general, you need to provide a lot more context for what you're doing. Nearly every section could use a bit more explanation for what you're going to do in that section and why. Try and make sure each section has a "topic sentence" or thesis.

- This page is very hard to read. There are a lot of proof-reading type mistakes, but there's also a lack of interest. I had no real sense of what you were doing or why I should care. The organization of this page where you just have to click and click and click to expand more and more dense and confusing proofs was very discouraging. See if you can streamline your organization and find ways to make it clear why I should care about each of your proofs.

- Rebecca 04:35, 6 July 2011 (UTC) I think overall you have too many "hide/shows" on your page. I understand why you would use them because you have so much information, but it seems like I'm getting lost as I'm reading the page. I would suggest cutting out at the very least half of them.
- I also think you need less sections in the page. It's generally a good idea to divide the text up using section headings, but I think you've got too many on this page.
- I also have to agree with Kate. A lot of the content on this page is GREAT- really informative and interesting, but it's hard to stay interested because I keep losing track of why I'm reading about these things. Would you consider cutting out some of the information and providing more context for the sections you keep? You could discuss this with your prof. at Sweet Briar and see what she thinks as well.

# Section-specific comments

Kate 17:46, 29 June 2011 (UTC): About the image caption before the table of contents -

*a ball floats in the cone with a touch of the ocean surface.*

- First, you forgot to capitalize this sentence! Second, the ball is the guy's head, right? I think it might be clearer what you're talking about if you say that it's his head. Third, I'm not really sure what you mean by "with a touch of the ocean surface".

*This image can be described as using a plane to cut a cone to create an ellipse, and two different balls tangent to the cone and the ellipse.*

- Awkward wording. I'd try "We can view the water level inside the cone as being an ellipse created by the intersection of a cone and a plane, and the fish and the man's head as being spheres that are tangent to the cone and the ellipse."

Flora 00:32, 30 June 2011 (UTC)Fixed this part

## Basic Description

Kate 17:46, 29 June 2011 (UTC):

*As Figure 1 shows.*

- Awkward wording, try "See Figure 1". Also, this link didn't work for me - is there a picture that goes with it anywhere?

- The sentence beginning with "In addition" shouldn't be indented.
*Since the Dandelin Spheres are created by Conic Sections*- don't capitalize "conic sections".*(However, parabola can only create one sphere, instead of two.) The inner and outer spheres tangent internally to a cone and also to a plane intersecting the cone are called Dandelin spheres.*

- I think the part in parentheses should go next to "Parabola" in your bulleted list, not after it. The other sentence is redundant with the definition you gave above - you can probably just delete it.

Flora 00:09, 30 June 2011 (UTC)Have fixed this part. very appreciate your comments

- I would consider removing the third paragraph of this section. The third paragraph introduces new definitions like "foci and focal constants" that might be overwhelming, and it seems like you might not need to discuss these topics yet.
- I would also recommend adding more images to this basic description. You're writing is very clear, but especially for someone who hasn't encountered these ideas before, it might be difficult to pick out which parts of the main image relate to your explanations. For example, you could make another image of a cone sitting in water with an arrow pointing to the sphere and a label that says "an example of a Dandelin sphere".
- Another image might help in the area where you list the types of conic sections. You could make an image that shows a cone standing upright with a circular conic section as well as a cone tilted sideways with an ellipse for a conic section.

### History of Dandelin Spheres

- Kate 17:46, 29 June 2011 (UTC):
*He also gave a elegant proof that the spheres intersect the conic section at its foci.*- I think you should say that the spheres "are tangent to" the conic section, not that they intersect.

*(Also, it's "an elegant", not "a elegant"!)*

Flora 00:09, 30 June 2011 (UTC)Fixed. But not update yet. XD

- Rebecca 04:29, 6 July 2011 (UTC) Good history. You hit the key points, and keep it short.

## A More Mathematical Explanation

### Two Theorems Proved by the Dandelin Spheres Theory

Kate 17:46, 29 June 2011 (UTC):

- I don't think you need to use bullet points in this section - you can just have two paragraphs.

- Fixed

**Focus-Directrix Property**, The first theorem is that a closed conic section (i.e. an ellipse) is the locus of points such that the sum of the distances to two fixed points (the foci) is constant.

- Some grammar mistakes, and I think just "the first theorem" is a little unclear. Try either
**Focus-Directrix Property:**The first theorem proven by Dandelin Spheres is that a closed conic section…

- or
- The
**Focus-Directrix Property**is the first theorem proven by Dandelin Spheres. It states that a closed conic section…

- The
- or
- The
**Focus-Directrix Property, the first theorem proven by Dandelin Spheres, is that a closed conic section…**

- The

- Some grammar mistakes, and I think just "the first theorem" is a little unclear. Try either

- Fixed

- "locus" might be an unfamiliar word for some people - you might want to use a green definition balloon here.

- I have added a balloon~

*That the intersection of the plane with the cone is symmetric about the perpendicular bisector of the line through F_1 and F_2 may be counterintuitive, but this argument makes it clear.*

- This sentence is confusing - it's connection with what went before isn't obvious.

- I deleted this sentence

*Sum of Distances to Foci Property, the second theorem…*

- Same issues as the sentence about the first theorem, same suggestions for fixing it.

- Fixed

- The paragraph that starts "The directrix of a conic section can be found…" is hard to understand without a picture.

- I will try to find a image for it, but it kinda hard

- I don't think the hidden part here should be hidden. Also, be careful with your spacing near the end of this section.

- Fixed

*This will be shown in the part of Applied Dandelin Spheres.*and*These two Theorems will be proved in the part of Explore different Dandelin Spheres.*

- You don't need the "of"s - you should just say "This will be shown in the section Section Title" or "See Section Title". Also, don't capitalize "theorems".

- Fixed

- I found it confusing that in this section you introduce the two theorems and seem to give a brief outline of the proofs, but then leave the proof to a later section. Is there a reason you decided not to do the proofs here?

- I just state the two theorems in this part, but did not prove them. Because I need to prove these two theorems for each conic section, so I put the proof in explore section
- Kate 13:26, 30 June 2011 (UTC): Ok, that makes sense. It wasn't clear to me as I was reading though, so you might want to say just that: "The proofs of these two theorems are different for each conic section, and will be given below" or something.

- Rebecca 04:31, 6 July 2011 (UTC) I have a few formatting suggestions...
- I would use a colon not a comma after "sum of distances to foci property" and the other small headings in this section.
- I think we've been trying to use bold to indicate words that we are about to define, so I wouldn't use it for "Pappus of Alexandria" and things like that.
- I think mouseover would work well for "the focus," "the directrix," and "eccentricity." That way you can use the word normally in your sentences and put the definition in the bubble, and you can avoid parentheses.
- Nice use of the mouseover for "locus"

### Mathematical Description of Two Theorems and Notation

Kate 18:13, 29 June 2011 (UTC):

- I don't understand the purpose of this section. As far as I can tell, you're making believable but unjustified claims about a picture involving Dandelin Spheres for no apparent reason. Is this part of a proof for one of the two theorems? Can this section be integrated into either the previous or the following sections?
- The picture in this section is kind of hard to see - can you find one that's bigger and less blurry?
*Germinal Pierre Dandelin emloys spheres inscribed to a cone which touch the intersecting plane in two points which are foci of the conic section. In what follows all the three Dandelin's proofs are presented.*

- Why is this sentence hidden? It's very short and not too mathematical. It's not obvious why the sentence is
*relevant*, but you should fix that by giving it context, not by hiding it. Also, what are these three proofs? We were just talking about using Dandelin Spheres to prove two theorems, not three theorems. Are these three proofs the proofs published by Dandelin to show that the spheres do in fact touch the conic section at its foci?

- Why is this sentence hidden? It's very short and not too mathematical. It's not obvious why the sentence is

- Also, "all the three Dandelin's proofs" is bad English - say "all three of Dandelin's proofs"

I deleted this section, and I'll try to find a better image.

## Explore Different Dandelin Spheres

Kate 18:29, 29 June 2011 (UTC):

- Don't capitalize "conic section"
*And it can also help to prove that the intersection(s) are focus(foci), after we could prove the shape of the plane.*

- Try to reword this so that it either doesn't start with "And" or is connected to the preceding sentence.
- I know you're having a difficult time with whether or not to make things plural because not all conic sections have two foci, but I think you can just go with plural all the time - even if one specific parabola has only one intersection and one focus, you're making statements about
*all*the tangencies between Dandelin spheres and conic sections. Your sentences will be much easier to read that way. - I have no idea what "after we could prove the shape of the plane" is supposed to mean.

*All we need to do is to show, that a conic section under consideration, satisfies the definition of circle, ellipse, hyperbola, or parabola.*

- Unnecessary commas and missing or wrong determiners - should be "All we need to do is to show that the conic section under consideration satisfies the definition of a circle, an ellipse, a hyperbola, or a parabola."
- Or, if you don't like listing all the conic sections out, you can say "the definition of some conic section."

- Linking to an applet like that is fine, but
*Using the*isn't a complete sentence. Try "Use this**Applet**to play with Dandelin Spheres**applet**to play with Dandelin Spheres" or "This**applet**lets you play with Dandelin Spheres."

Fixed these problems

- Rebecca 04:37, 6 July 2011 (UTC) Nice inclusion of an applet, and Kate's suggestion for the way to introduce it works well!

### Spheres Tangent To A Circle

Kate 18:46, 29 June 2011 (UTC):

*Here we will show some detail about Dandelin Spheres Tangent To A Circle.*

- This sentence is awkward. Try "More information about Dandelin Spheres that are tangent to a circle" or use a Hide/Show template without a preview or don't hide this section at all.

*In*~~the image~~Figure 2, when the plane \pi intersects all generators of the cone,~~as in down figure,~~it is possible to inscribe two spheres which will touch the conical surface and the plane.

- The labels in the picture are really hard to see - I can't find the label π for the plane.

- Make sure the labels you use in the picture and the text match up - the picture seems to have k and k' , not k
_{1}and k_{2}. Actually, I don't see most of the labels you're using in the picture. This is a problem.

- You should say "The upper sphere, G
_{1},…" and "The lower sphere, G_{2},…" and "The arbitrarily chosen…"

*We see that points, P_1 and F are the tangency points of the upper sphere and points, P_2 and F the tangency points of the lower sphere of the tangents drawn from the point P exterior to the spheres.*

- The first part should be:
- We see that points P_1 and F are the tangency points of the upper sphere, and points P_2 and F are the tangency points of the lower sphere

- and I have no idea what the second part of this sentence is supposed to mean.

- The first part should be:

I fixed this part

**Distances From P To Tangent Points Of The Spheres Are Equal**
Kate 19:14, 29 June 2011 (UTC):

- This section was really hard to follow, mostly because I can't see what's going on in the picture. Can you get a bigger, clearer picture.
*Showing in the image Proof of the Sphere a sphere centered at point O.*

- "Proof of the Sphere" isn't really a good title for this image. Why not just call it by a number? Also, this isn't a complete sentence. Try:
- "Shown in the image Image Title is a sphere centered at point O."

- or
- "Image Title shows a sphere centered at point O."

- "Proof of the Sphere" isn't really a good title for this image. Why not just call it by a number? Also, this isn't a complete sentence. Try:

~~Besides, s~~**S**ide**s**OA and OB are the radi**i**of the sphere, so OA=OB.- I don't think this hidden section should be hidden.
- I don't think the last couple sentences here should be italicized.
- What is N? I don't see an N in either picture.

Fixed

**Proof of Conic Section Focal Properties**
Kate 19:26, 29 June 2011 (UTC):

*By intersecting either of the two right circular conical surfaces (nappes) with the plane perpendicular to the axis of the cone the resulting intersection is a circle.*

- This sentence doesn't make sense. What is a "nappe"? "By intersecting… the resulting intersection is…" isn't a valid grammatical construction. I think maybe you're trying to say "When either of the two right circular conical surfaces are intersected with the plane perpendicular to the axis of the cone, the resulting intersection is a circle," but I still don't really know what that sentence means.

- Don't italicize the sentence beginning with "Also, we can assume…"
*We can say that PF is the radius of the circle. The conic section is a circle.*

- I would say "Since PF is the distance from a point to the edge of the conic section, we know that the conic section has a constant radius, and is therefore a circle."

Fixed

**Proof of The Tangent Point Is The Center**

- Kate 19:26, 29 June 2011 (UTC): What you've said here doesn't seem to be long enough to be worth being it's own section, nor does it seem like an adequate proof of anything. Can't you just use that the spheres are tangent at the point that's the endpoint of the constant radius to show that it's the center?

Fixed

### Spheres Tangent To An Ellipse

Kate 19:57, 29 June 2011 (UTC):

*In*~~the image~~Figure 3, when the plane \pi intersects all generators of the cone,~~as in down figure,~~…- Again, you've gotta say "The [adjectives] sphere" and "The [adjectives] line". If there's no adjectives, then it's okay to say "sphere G" or "line SP", but when you add adjectives like "upper", "lower", or "arbitrarily chosen generating", you've gotta put a "the" in front. And no comma in "points P and F".

Fixed
**Distances From P To Tangent Points Of The Spheres Are Equal**

*Since sphere A is tangent to the cone and the tangent line k_1 is a circle, and point P_1***is**on tangent line k_1, line P_1 P is tangent to sphere A.**T**he sphere is also tangent to the ellipse at point F_1, so the line F_1 P on the ellipse is**also**tangent to the sphere.

- Also, I don't see anything labeled A in the picture.

*Using the same method.*

- First, this isn't a complete sentence. You should use it like this:
- "Using the same method, statement X is true" or "Using the same method, X=Y."

- Second, it doesn't read well to say "Using the same method" and then give the whole argument again. Either cut the argument and just present Eq. 2, or cut the "Using the same method" sentence.

Fixed

**Proof of Conic Section Focal Properties**

*When the cutting plane is inclined to the axis of the cone at a greater angle than that made by the generating segment or generator (the slanting edge of the cone), i.e., when the plane cuts all generators of a single cone, the resulting curve is the ellipse.*

- This sentence doesn't make sense.

- That "thus" is unnecessary, because of the "since". Say either "Since X is true, Y is true." or "X is true. Thus, Y is true."

- You need more of an explanation as to how the picture and the numbered points below it connect to what was going on before.

- Calling it the "fish's belt" was very confusing - what fish? what belt? "Hat band" was less so, because a hat band is a thing, and I could go look for something in the picture that looked like a hat band, but a fish's belt isn't a thing.

*Thus, the intersection curve is the locus of points in the plane for which sum of distances from the two fixed points F_1 and F_2, is constant, the curve E is an ellipse.*

- Should be "Since the intersection curve is the locus of points in the plane for which the sum of the distances from the fixed points F_1 and F_2 is constant, the intersection curve is an ellipse."

Fixed

**Proof of The Tangent Points Are The Foci**
Kate 19:57, 29 June 2011 (UTC):

- This subject heading is incorrect, it should be "Proof that the Tangent Points Are the Foci"
- There should be a comma after "major diameter"
*By using the definition of ellipses, point P can be any point on the ellipse, and the only points F_1 and F_2, satisfied that P_1 P_2 = P F_1 + P F_2 is constant, are the foci of the ellipse.*

- Try "By definition, any two points F_1 and F_2 that make P F_1 + P F_2 constant for all points P on the ellipse are the foci of that ellipse."

Fixed

### Spheres Tangent To A Hyperbola

Kate 20:23, 29 June 2011 (UTC):

*When the intersecting plane is inclined to the vertical axis at a smaller angle than***is**the generator of the cone,- Again, the labels in the picture are hard to read, and not all of the letters you use in the text seem to be there.

Fixed
**Distances From P To Tangent Points Of The Spheres Are Equal**
**Proof of Conic Section Focal Properties**
Kate 20:23, 29 June 2011 (UTC):

- No commas around "are parallel"
- that sentence should end with a period: "…of equal length."
- Since/thus issues again.
**Since**the intersection curve is the locus of points in the plane for which**the**difference of**the**distances from the two fixed points F' and F [no comma here] is constant, the conic section curve is a hyperbola.

Fixed
**Proof of The Tangent Points Are The Foci**

- Kate 20:23, 29 June 2011 (UTC): Section title should be "Proof that the Tangent Points Are the Foci"

Fixed

### Sphere Tangent To A Parabola

Kate 21:02, 29 June 2011 (UTC):

- Your first paragraph is really hard to follow without a picture. Please please please add a picture!
- Don't capitalize "parabola"
- Again, you're using letters that aren't in your picture.
*Inscribed sphere*~~s~~A, centered at C, touch**es**the plane~~on the same side~~at point F and the cone surface at circle k.- There shouldn't be a comma between "k" and "at point T" in the sentence that begins "The generator intersects…"
- You should probably write "V" (for the vertex) in math font, to be consistent with all your other letters. I think you need to fix it in the hyperbola section too.

Fixed
**The Distances From P To The Tangent Points Of The Spheres Are Equal**
Kate 21:02, 29 June 2011 (UTC):

- Incomplete sentence:
*The sphere A centered at point C situated below the cone vertex V.*(Should say "is centered") - Having two things called p is confusing to me.
- Typo!
*This creates two tiangles*(Should be "triangles") *As we shown the proof in Proof of the circle. Since point F and point T are on the sphere, Similarly as in the case of circle it holds true that:*

- Should be: "As we showed in the proof Proof Title, because points F and T are on the sphere, we know that: "

Fixed
**Proof of Conic Section Focal Properties**
Kate 21:02, 29 June 2011 (UTC):

*In this case, it is a little different that we can not use*

- Should be "This is different from the previous cases, in that we cannot use…"

- I don'tt think this last bit about the angles should be hidden. Also, don't start those lines with a space, the text shouldn't be in a box.

Fixed

**The Distances From P To The Tangent Points Of The Spheres Are Equal**
Kate 21:02, 29 June 2011 (UTC):

- The hidden paragraphs just before this section seems to be redundant with the information at the top of the paragraph on parabolas. Also, it pains me to suggest making the table of contents longer, but I think if you're going to do the same thing over again a different way, which is what it sounds like, this whole hidden section needs to have another heading.
- Yet again, your labels don't seem to match your picture. Which plane is plane π?
*The segments, PF and PT belong to tangents drawn from P to the sphere, we can get:*

- Should be "Since the segments PF and PT belong to tangents drawn from P to the sphere, we can get:"

*Since***the'****planes of the circles[no comma here] k and k' are parallel to each other and perpendicular to the section through the cone axis, and**since**the plane \pi is parallel to the slanting edge VB,**PN is the~~then~~the intersection d, of planes E and K, is also perpendicular to the section through the cone axis.*perpendicular from P to the line d.***Thus**,

Fixed

**Proof of Conic Section Focal Properties**
Kate 21:02, 29 June 2011 (UTC):

*Since we get Eq. 2 and Eq. 3 from above proof,*

- Should be "We get Eq. 2 and Eq. 3 from the above proof. Therefore, for any point P on the intersection curve the distance from the fixed point F is the same as it is from the fixed line d,
**proving**that the intersection curve is the parabola.

- Should be "We get Eq. 2 and Eq. 3 from the above proof. Therefore, for any point P on the intersection curve the distance from the fixed point F is the same as it is from the fixed line d,

Fixed

**Proof of The Tangent Point Is The Focus**

- Should be called "Proof that the Tangent Point Is the Focus"

Fixed

## Applied Dandelin Spheres

### Conic Section Eccentricity

**Ellipse's Eccentricity**

- Rebecca 04:39, 6 July 2011 (UTC) When you talk about a "hinge line" in the Ellipse's Eccentriciy section, you should either bold "hinge line" and keep the definition the way you have it in the next sentence, or put the definition in a mouseover.
- "Hinge line is a leg ofboth blue and red right triangles." TYPO. You need to add a space "of both"

**Hyperbola's Eccentricity**
**Parabola's Eccentricity**

### Newton's Astronomy Theory

- Rebecca 04:41, 6 July 2011 (UTC) When you talk about a "hinge line" in the Ellipse's Eccentriciy section, you should either bold "hinge line" and keep the definition the way you have it in the next sentence, or put the definition in a mouseover.
- "Hinge line is a leg ofboth blue and red right triangles." TYPO. You need to add a space "of both"

### Interesting Math problems

- Rebecca 04:41, 6 July 2011 (UTC) Again, I don't think "De Villiers" should be bold.

## Why It's Interesting

- Rebecca 04:47, 6 July 2011 (UTC) I think this video could be moved much higher up in the page. It could even be in one of the first sections.
- Also, I think it might be a good idea for you to work on the organization of this page before you add to the why it's interesting section. I'm sure you'll have great info for this section eventually, but for now you should focus on explaining why things are interesting as you give proofs/explain topics. This will keep readers interested as they go. Then you can use this section to elaborate on ideas you introduced earlier. Again, this is just a suggestion, but I think it will help give your page more direction and ultimately make it better.
- Finally, I want to reiterate how impressive this is as a page. I think you've got your work cut out for you as far as organization, but it's evident that you've conquered a tough topic. You have a great start here, but I think the next step is to work on combining sections and removing some of the hide shows, adding more pictures (although you do a have a good number and the applets/movies help!), and providing context so that people have incentive to keep reading. Nice work Flora!