Talk:Compass & Straightedge Construction and the Impossible Constructions

• Excellent choice of material and scope
• Good overall structure: presentation of basic postulates, examples of construction and an example of a theorem, section on algebra
• Good attention to some of the more confusing points in the algebra section (especially definition of "1")
Abram, 6/11

Read the page word-by-word, out loud

A lot of really good editing on this page, both with the math and the explanations. I'll have some more specific suggestions later, but try once reading the page word-by-word, out loud, and making sure that everything you are saying is correct and is exactly what you want to say. As you go, reviewing the bullet points from the draft checklist for finishing pages, might help you think about what to pay attention to.

This may sound like a lot, but I'm walking the walk, so to speak: I try to read pages word for word *every time* I read them. If I do skip over words, I end up making stupid comments. (Abram, 7/8)

Will do (Xingda 7/12)
Cool. Once you have done this, send your thoughts about the draft checklist.

Layout Issues

• Consider making the basic construction pictures smaller. Each one fills up about 40% of the vertical viewing space on my laptop. (Abram, 7/21)
• The picture that goes with the construction of a x b now is too small. The letters (a, b, l) are readable, but it's kind of painful to read them. The subscripts are totally unreadable. a / b is starting to push it also. (Abram, 7/19)
I enlarge them but there is a risk that small screen may have to scroll right to see them (Xingda 7/20)
Hmm, it's still hard to see. Options I can think of include: Making the table one column, rearranging the table so that the left column is always the picture and the right column is always the text description (it seems like this might allow you to resize things, though you would know better than me), renaming the lines so they don't have subscripts, consulting with Becky or Iris. (Abram, 7/20)

Add helper text for "congruent triangles"

Look at the line, "The proofs for these constructions are relatively simple and only require the knowledge of congruent triangles." It's not clear whether you mean that the reader only has to know the definition of congruent triangles or whether the reader has to know postulates (Euclid considers them theorems, but I think he's full of it) that establish that triangles are congruent. It seems that you mean the latter. Maybe clarify this wording and add a link to a helper page on triangle congruence postulates (at some later date, you or someone else can actually create that page, but the link can be there now). (Abram, 7/11)

Will do (Xingda 7/12)
The wording "knowledge of congruent triangles" is still vague, though thanks for creating the link to a congruent triangles page. (Abram, 7/19)

Reduce "prerequisites" for the Why It's Interesting Section

Don't assume that readers of the "Why It's Interesting" section have read the more mathematical section. It might be a good idea to include a restatement of the conditions under which a number is constructible in basically plain English (it's fine to refer to square roots).

Also, all the mathematical details should be hidden, though it's great to have them there. (Abram, 6/24)

Issue addressed. What do you think? (Xingda 6/28)
It's pretty good. The part about pi should be described at least once without reference to transcendental numbers or rational polynomials. You can include a second sentence that says something like, "In fact, pi is transcendental", with a mouse-over definition of transcendental. See below for some mathematical and formatting details for this section. (Abram, 6/28)
I really do not know how to water down the current version any further. (Xingda, 7/8)
I don't know a simple way of defining what it means for a number to be transcendental either. But you *can* start write something sentence like, "pi obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots. In fact, \pi is an especially "weird" kind of number called a transcendental number", with a mouse-over definition of transcendental. This way, you a) make this understandable to someone who doesn't know the word transcendental, b) signal to this someone that it's ok if they don't understand the word "transcendental", and c) still have all the information about pi being transcendental there for anyone who wants it. (Abram, 7/12)
We've talked about this, and your change is good. It still might be nice to put the definition of transcendental in a mouse-over. (Abram, 7/21)
done (Xingda 7/22)
• Drop the example of doubling a cube of volume 5, because the argument is not valid. The problems with this example is it says, "Given a cube of volume 5...." If you are given a cube of volume 5, then your whole argument that cube root of 2 is constructible doesn't necessarily work any more, because that proof depends on being given only a length of measure 1 unit.
There is still a reference to a cube of volume 5. (Abram, 7/6)
I thought that we agreed to emphasize that the given circle or cube has be constructible in the first place or you still want to delete it? (Xingda, 7/8)
Again, it seems like the most straight-forward chain of reasoning is to begin with a specific constructible cube and show that it cannot be doubled. The sentence "For example, we cannot even have a cube with sides equal to sqrt(5) and thus volume 5, let alone doubling it" is also problematic because it sounds like you are saying that given *any* cube that is not constructible, it is certainly impossible to construct a cube with twice the volume. But that statement is not true. (Abram, 7/11).
Yeah, you could make it a bit shorter by dropping "because we could not construct cube root of 2" from the first sentence. There's also no need to hide the rest of the proof. It's no more complicated than the unhidden text in the squaring of the circle. (Abram, 7/12)
Looks good. (Abram, 7/21)
• The layout will probably be easier to follow if the proofs for examples 2 and 3 are hidden, as you have done, but still included "within" the example.
I tried but it will mess up the automatic numbering unless you don't want the numbers. (Xingda 6/29) Issue addressed (Xingda 6/29)
It looks like you and Maria have worked out a solution. For doubling the cube, there's no real sense in hiding the proof. It's only one sentence. For trisecting the angle, maybe it should say "Click to see proof" instead of "Click to see more". (Abram, 7/6)
Abram, 6/28
Cool. The sentence "Prove that 60 degrees is impossible to trisect" is confusing. Who are you talking to? (Abram, 7/12)
What about deleting the sentences "See below for proof" and "Proof of 60 degrees is impossible to trisect" (this latter one isn't even a grammatically correct sentence)? (Abram, 7/21)

Mathematical details about the constructions and Compass Equivalence proof

• Phrasing like, "Draw a circle at point B" is not quite precise. "Draw a circle centered at point B" is precise. See if there are other places where your wording is not quite precise.Issue addressed. (Xingda 6/23). Nice changes. I think there are a few more, but I'll look for them later. (Abram, 6/23)
• In a couple of places, you refer to points that haven't actually been defined at all. See if you can find them.
I cannot find them...... (Xingda 6/23)
They're all fixed by now. (Abram, 7/12)

(Abram, 6/18)

Presentation details for the constructions and Compass Equivalence proof

• When you are defining a letter or a symbol, use wording that emphasizes that you are providing a definition. For instance, instead of saying, " These two circles intersect at points C and D", say, "Define C and D as the points of intersection of these two circles" or "Call these circles' two points of intersection C and D".Will do. (Xingda 6/23)
I may have missed one or two places that are still a problem, but basically, it looks good. (Abram, 7/12)
• You now indicate at the beginning that final products are given in red. Complete the color-coding explanation: blue represents... and black dotted lines/curves represent... (Abram, 7/6)
Looks good. (Abram, 7/12)
• In the line "From the center B, at the distance BC", "at the distance" should be "with radius", or something like that. (Abram, 7/12)

(Abram, 6/18)

Parts of this page seem extremely rigorous (outlining Euclid's postulates, pointing out that you don't get to just choose any two points at the beginning of the algebraicization section). Other parts (for example, the proof of the Compass Equivalence Theorem) omit a large number of steps. Be more consistent, or at least be explicit in the text about where you are not being rigorous. (Abram, 6/11) I will talk to Prof. Maurer before addressing this. (Xingda 6/17)Which part of it is not so rigorous? (Xingda 6/23)

At this point, it's pretty good. The only place where the page fails to be rigorous now is that you mention that the constructions are provable using basic propositions/postulates about congruent triangles, but don't say where those come from. I'm not sure this is a big deal. (Abram, 6/23)
Agreed. (Abram, 6/28).

Move the Compass Equivalence Proof

Put it after the basic constructions. (Abram, 6/11)

Great. You may want to remind the reader what the Compass Equivalence Theorem says, because you first mention it quite a bit earlier in the page, and it doesn't take long to state. (Abram, 6/18)Issue addressed. (Xingda 6/23)
The wording could be changed just a bit, basically, yes, well done. (Abram, 6/23)
I still want to change the wording just a bit, but that can wait for a final pass through. (Abram, 6/28)

Color-code the basic constructions

Choose one color for the given information, a different one - three colors for the intermediate construction steps, and a different color for the final product. (Abram, 6/11)

This may be a little troublesome. I will try if time permits (Xingda 6/17)
I'm a bit confused. How would time not permit? (Abram, 6/17)

Really nicely done. You just somehow lost the labels A and B from your construction of a perpendicular. (Abram, 6/23)
What? The perpendicular construction still just has point M labeled. (Abram, 6/28)
I did not mention A or B. I simply said that where the circle intersects the line, draw perp bisector." It renders the point A and B obsolete and the diagram much simpler.(Xingda 6/29)
It looks like A and B are back in there, but are also referred to appropriately. (Abram, 7/6)

Make the things about why it's interesting more accessible to a non-math person

It seems like there are two complementary interesting messages that arise from this page. 1) You can construct an awful lot of stuff with a straightedge and compass (hey, you can even treat your collapsing compass as though it didn't collapse!). 2) Despite how much you can do, there are very simple tasks that not only seem impossible, but can be *proven* impossible. You bring these themes out well, but only within the more mathematical section. Is there any way you can talk about this a little bit in a "Why It's Interesting" section? If you have to be a little bit redundant, that's ok. (Abram, 6/17)We need to talk about this. (Xingda 6/23)

Agreed. (Note: there is a separate conversation about details around the Why It's Interesting section elsewhere on this discussion page). (Abram, 6/28)

Slightly edit presentation of Euclid's postulates

His wording for the first few postulates makes it sound like you can choose to construct a line segment of any desired measure, rather than you can end a line segment wherever you want, but you don't know how long it will be. Also, the second set of postulates listed aren't actually necessary for the page. (Abram, 6/11)

Based on what you have written, it seems like you are saying that the first postulate *implies* that the alternate (wrong) interpretation of the 2nd postulate is incorrect. I don't think that's true. Anyway, it would probably be good to specifically state that the 2nd and 3rd *don't* mean that you can specify the length of the line segment or the measure of the circle's radius. (Abram, 6/17)Issue addressed. (Xingda 6/23)
There are still a couple sentences that are less helpful than they could be, but this can wait for a final pass-through. (Abram, 6/28)

Mathematical details about the constructions and Compass Equivalence proof

• For line bisection, specify that the radius of circle A and the radius of circle B have to be the same (in fact, you may want to simply require that the radius be the length of segment A and B so we don't have to worry about the compass collapsing).Issue addressed. (Xingda 6/23). Agreed (Abram, 6/23).
• For the perpendicular, you left out a few steps of the construction.How so? (Xingda 6/23) I didn't read it carefully enough. The only problem is that you refer to constructing a perpendicular bisector, but you don't point out that the instructions for creating a bisector in fact create a perpendicular bisector. (Abram, 6/23).
This section still as a final step says "construct a perpendicular bisector" but you don't have any construction titled "perpendicular bisector" nor does the "bisector" construction explicitly point out that a perpendicular bisector is being constructed (Abram, 6/28).
In the last sentence of line segment bisection, I said that "CD is the perpendicular bisector of AB" so I think it should not be a problem if I say construct "perpendicular bisector" in the subsequent sections. (Xingda 6/29)
You can still make it a lot easier for readers to find this information. The final step could say "construct a perpendicular bisector (see the line segment bisector construction above)", and that would take care of it. (Abram, 7/6).
• When you say that triangle DAB is constructed the same way as constructing a segment bisector, this is only true if you require the radius of the circle to be equal to the length of segment AB (this is another reason for making the change I suggest above).
I don't think it is. The instructions for constructing a bisector still don't require the two circles to have radius equal to length AB. (Abram, 6/23)
No, it's still just a suggestion in the bisector section to make the radius equal AB. It's not yet a requirement. (Abram, 6/28)
I am confused. Which part of the instruction in the line bisection section even hints that radius = AB is a mere suggestion? It say exactly this "1. Draw a circle centered at point A with radius equaling AB. 2. Next, draw a circle centered at point B with the same radius." and the reader should follow the instruction exactly. It has been emphasized in the previous section that whatever points, lengths and angles one uses, they have to be given or derived. Therefore, the reader should be aware that using radius = AB is absolutely the only option they have without me saying it. (Xingda 6/29)
Yeah, sorry, it's fine. (Abram, 7/6)

Presentation details for the constructions and Compass Equivalence proof

• In the parallel line construction, you first state what is given and what is supposed to be constructed, but you do this nowhere else. It might be a good idea to include a statement of what is given and what is to be constructed for each construction. Issue addressed. (Xingda 6/23). Nicely done. In the first two examples, you name the given line/angle, etc. Do the same in the later constructions. (Abram, 6/23).
Issue addressed. (Xingda 6/24). Yes, except for the perpendicular through a point. (Abram, 6/28).
Looks good. (Abram, 7/6)
• It's confusing that numbered steps (1., 2., etc.) sometimes refer to the steps of a construction, sometimes refer to a statement of the problem, and sometimes refer to the steps of a proof. Changing the formatting could help.Will do. (Xingda 6/23). Mostly good. The tangent to a circle construction still gives step numbers to the proof as though the proof were part of the construction. (Abram, 6/23).
Issue addressed. (Xingda 6/24). Step 4 still says, "To see that AC..." I understand that this part of the proof requires drawing a new line, but this step is still only necessary for the proof, not for the basic construction, so it probably doesn't deserve a numbered step. (Abram, 6/28).
Looks good. (Abram, 7/6).
• Drop the use of the word "I" from the page. Wikis are meant to be collaborative, so you don't want to separate the author from the reader by using words like "I". Also, other people may edit the page later, so it's not even clear who "I" would refer to.Issue addressed. (Xingda 6/23). At the beginning of the Compass Equivalence Proof, you still refer to "my comments". Otherwise, nicely done. (Abram, 6/23)Issue addressed. (Xingda 6/24). Agreed (Abram, 6/28).

Layout issues

• Echoing what I said on the Straight line page, I'd encourage you to make diagrams smaller. This really does help the reader connect the text to the image, and if they want to see a larger version, they can always click on the image itself.
• Also, I see the same repetition of the first image going on here. Is there an issue with the template?
What do you mean by repetition? I don't see any repetition of the first image. (Xingda 6/28)
• For some of the constructions, it might help to have separate images for each step.
It will overcrowd the page. I tried to have a step by step construction animation for Compass Equivalence Theorem and it did not work out so well. (Xingda 6/28)
• In the latter half of the page, definitely shorten paragraph lengths. This is the area in particular where separate images for the different steps would be very useful.(Anna 6/25)
I think there should be a balance between good explanation and paragraph length. I should not compromise the integrity of the explanation just for shorter and more attractive page. What do you think? (Xingda 6/28)
• Make more of the material hidden by default. For instance, each individual construction could start hidden by default, both in the basic constructions and in the algebraicization section. This will make the page feel less sprawling. (Abram, 7/7)
I was thinking it might make sense to hide the illustrations as well as the text (to make it easier for the the reader to "flip through" to whatever section they want). This is especially true in the algebraicization section, where the illustrations are really large. It might also help to give each construction in the algebraicization section a bold-faced heading, like in the basic constructions. (Abram, 7/11)
I have reservations about this. There are already many things hidden and two levels of hiding. I will give them bold face heading but I really don't want to hide them. (Xingda 7/12)
Hiding the illustrations wouldn't add an additional hidden layer -- it would just be hidden in the same layer as the explanatory text. That said, if you really don't want to hide them, that's fine. But before you make that decision, look at the page once with the browser window shrunk down to laptop size, and then ask yourself, "Is it a pain to scroll through all of this?" If you decide the answer is "no", then leave it as is. (Abram, 7/12)
I hid the pictures in algebraicization. However, the first picture cannot be hidden. (Xingda 7/12)
The table takes care of this problem. (Abram, 7/19)
• Color-code / highlight the "final products" in the algebraicization constructions, so that the reader doesn't have to search for a+b, a/b, etc. In the sqrt(a) construction, the "final product" isn't even labeled on the construction. (Abram, 7/11)
There is a difficulty here. I cannot color code the a+-b on the same picture because they are on the same line. However, I can color code the points if that is ok. For the rest of the two picture, I can color code the line segment if you want. (Xingda 7/12)
Hmm, good point. How about just color coding the labels (a+b, a-b, ab, etc.) throughout? Just color-coding the point itself probably won't be enough for anyone to notice, and using a different scheme for a+/-b than for the others is confusing. (Abram, 7/12)
Looks good. (Abram, 7/19(

Yet a few more mathematical details.

• There are two errors in the construction of a x b, where "error" can mean undefined term/symbol, as well actual mistakes. See if you can find them.
Looks good. (Abram, 7/20)
• There is an error within the first two sentences of "A simple derivation". See if you can find it.
Now it reads "Firstly, we define "1" on a straight line as stated previously. Then, once you have chosen that point to be (1,0), you have to stick to this specification throughout your construction. (Xingda 7/20)
Looks good. (Abram, 7/20)
• In the simple derivation section, you talk about finding the intersection of two lines and about finding the intersection of a line and a circle, but not about finding the intersection between two circles.
Looks good. (Abram, 7/20)

(Abram, 7/19)

Reorganize the explanation of Euclid's postulates

Explain better what is meant by starting with stuff you have and creating stuff you don't. The problem with this paragraph is that it seems like the main purpose of it is basically to explain what is meant by this statement, but the way it does so is kind of disorganized. It may help to actually re-write this paragraph specifically from the point of view of explaining this one idea. (Abram, 7/7).

I made a some change. I deleted two sentences. I don't know if that is ok. Anyway, it is also emphasized in the later section what is meant by "construction". (Xingda, 7/7)
Nicely done. The main thing that still leaves the reader a bit off-balance is that first you state Euclid's postulates (which tell you what kind of actions ARE allowed), then you state in plain language actions that are NOT allowed, *then* you translate into plain language Euclid's postulates what kinds of actions ARE allowed, then explain what kinds of tools are used. You can decide where it makes sense to describe the tools, but it seems like it would be helpful at least to translate the language about what *is* allowed before explaining what isn't allowed. (Abram, 7/11)
I think you are overreading this. The first two sentences after the postulates explain in plain language what you are allowed to do and the next sentence gives an example what you cannot do and the last sentence gives a summary. Anyway, I did make a little change and break up the paragraph (which was my original attention but did not show on the formatting). I hope that helps. (Xingda 7/12)
It has gotten much clearer. When you read the page word by word, see if you are totally happy with it. I may also make an attempt to slightly reword it at some point, and you can tell me if you like my suggestions. (Abram, 7/12)

The reference to the picture in the basic description is problematic

Looks good. (Abram, 7/12).
• The wording should make it clear that the reader doesn't have to follow every step to get the basic idea.
what do you mean? do you mean i should state the purpose of this page more explicitly (it is pretty explicit anyway)? (Xingda, 7/7)
Sorry, no, I was thinking that readers could get a little bit scared off by trying to understand the main image, especially because it doesn't quite provide all the needed information to replicate it. I'm wondering if there's a way to relieve this possible fear. (Abram, 7/12)
I have a sentence that says "In the main image of this page, we want to divide the circle into six equal arcs and then connect consecutive points to form the hexagon." Doe that help? (Xingda 7/12)
Yeah, looks great. (Abram, 7/20)
• Consider adding the intermediate question "are other polygons constructible" along with "is every polygon constructible, etc." (Abram, 7/7)
What are the difference between the two questions. (Xingda, 7/7)
"Are other polygons constructible" means do there exist ANY other constructible polygons. To you, this is obvious. To a reader who hasn't thought about this topic, this question could be useful. (Abram, 7/12)
Looks great. (Abram, 7/20)

The page title is problematic

"Compass&Straightedge" should be "Compass & Straightedge" or "Compass and Straightedge" or... (Abram, 7/19)

Looks good. (Abram, 7/20)

Cut some text, hide some text

There are places where material that would be appropriate for a book chapter feel either too auxiliary for this kind of a page or where the material deserves a sentence but gets a paragraph. In these places, it may make sense to cut or hide text. There are also places Examples of material that maybe should be shortened or cut include: spending a whole paragraph on the word "algebraicization",shortened(Xingda, 7/7) the reference to the Band of Brothers episode,Deleted (Xingda, 7/7) the three paragraphs beginning with Hudson's quote,greatly shortened (Xingda, 7/7) the reference to the metric system or British customary system,deleted (Xingda, 7/7) the initial statement early on that pi is not constructible,deleted (Xingda, 7/7) the paragraph explaining Euclid's postulates,what is wrong with that explanation? that is where most confusion occurs and I think it is necessary to clarify things in the beginning (Xingda, 7/7) the very first paragraph of the page,deleted (Xingda, 7/7) etc. (Abram, 7/7)

Really nicely done on a lot of this. I'm not sure the first paragraph of the page had to be deleted entirely. It might be that it was too long, but as I think about it, it might have been some other problem. I'll take a look at this later. Similarly, a reference to how crude compasses can be seen in military operations is actually kind of cool. It was mostly spending a long sentence referring to a particular episode of a TV show that many people haven't seen that I was concerned about. (Abram, 7/12)
I think it is good without it. i think it is much simpler and succinct now and i like it. (Xingda 7/12)
I might have a couple wording suggestions at the end, but yes, overall, the information you choose to include and leave out seems great. (Abram, 7/20)

Move and slightly reword the entire description of the compass equivalence theorem

The current text states that "It stated that from a given point, it was possible to construct a straight line equal to a given straight line using collapsible compass." For one thing, this wording doesn't technically imply that a collapsible compass is equivalent to a non-collapsing one (see if you can figure out why).

I don't get it. Do you want the wording to imply that being collapsible does not matter or does matter ? (Xingda, 7/8)
It should imply that compass being collapsible does not matter (don't you agree), or that the compass can effectively be treated as non-collapsing. Your current wording, though, doesn't actually imply that. Let me know if you don't see why. (Abram, 7/11)
Issue addressed. I emphasized the word directly in "so may not be directly used to transfer distances" and stated that you can treat the compass as non-collapsible.(Xingda 7/12)
Basically, what you are saying is clear. There is still a slight error in your wording that is easier to explain person. (Abram, 7/12)
We went over this, and it looks good now. (Abram, 7/20)

Also, it doesn't make sense that *first* you spend several sentences talking about the detail of the collapsible compass, and *then* you lay out the postulates. That's like talking about the strategy of a game before you lay out the rules. (Abram, 7/7)

I switched the order and put the postulates on top. (Xingda, 7/7)
Looks good. (Abram, 7/12)

Constructibility is about coordinates, not lengths

The page defines a constructible number as a constructible length, but then constructibility of negative integers has no meaning. The actual definition is that given two starting points with coordinates (0,0) and (1,0), a real number r is constructible if a point can be constructed that has r as either its x or its y coordinate.

This mistake is in some ways minor, thanks to the following theorem: "A real number r is constructible [given this coordinate-based definition] if and only if a line segment of length |r| is constructible." I suspect it's also a lot of work to fix this mistake. One or two sentences is almost certainly not going to be enough. On the other hand, we do have the wrong definition. What are your thoughts? (Abram, 7/7)

I realized that when I was editing the algebraicization part. Now the whole explanation is new and I carefully defined constructibility. (Xingda, 7/8)
Great, nicely done. The only slight error in your definition is in the informal description you describe "constructing numbers representing the coordinates of all the points on them." In fact, the only numbers being constructed are the coordinates of points of intersection. (Abram, 7/11)
Yeah, that is exactly what i mean. when you construct a circle say centered $(0,0)$ with radius $1$. then that construction corresponds to $x^2+y^2=1$. but you cannot pinpoint a point, that is to say you cannot claim you know can mark out $(\frac{\sqrt {2}}{2},\frac{\sqrt {2}}{2})$. the only times we can do that is when we have intersections whose coordinates we define as constructible which is the whole point of my next few sentence. you must have misread it. let me know if there are any more problems.(Xingda 7/12)
Yeah, I know what you mean. The small problem is with the word "constructing" in the sentence "we think of it instead as constructing numbers representing the coordinates of all the points on them." The problem with this word is that later, we specifically use the word "constructing" to *mean* "pinpointing", as you call it. (Abram, 7/12)

I changed it into "Then, every time we construct a straight line or a circle, we think of it instead as adding a new equation into a system of equations. These equations represent the coordinates of all the points on the line or circle, but that is easy since we all know the expression for a line and a circle as..." (Xingda 7/12)

This has looked good for a while now. (Abram, 7/21)

Another mathematical detail

• The line "However, by intuition, we know that the possibility could not be infinite" is not correct. The possibilities are infinite (for instance, you can construct any given integer). It seems like you are trying to say something like, "You can't construct every possible shape."

(Abram, 7/11)

infinite means "every possibility" right? if that is the definition, then we cannot claim that there are infinite things we can construct because we later prove that it is indeed no possible (all integers possible but not all numbers). or maybe you thought infinite means something else? (Xingda 7/12)
"Infinite" does not mean "every possibility". It just means "not finite". For example, the curves $y = \cos(x)$ and $y = \sin(x)$ have an infinite number of intersections, but they don't intersect everywhere. Similarly, there an infinite number of positive integers that are prime, but clearly not every positive integer is prime. (Abram, 7/12)
Sentence deleted. (Xingda 7/12)
Looks good. (Abram, 7/21)

Formatting and mathematical details for the "Why It's Interesting" section

• On number 2, it actually might be possible to construct $\sqrt{\pi}r$, depending on r (for instance, r could equal $\frac{1}{\sqrt{\pi}}$). An accurate argument goes something like: "A circle of radius 1 and area pi is constructible. Squaring the circle would create a side of length $\sqrt{\pi}$. But $\sqrt{\pi}$ is not constructible, so this task is impossible." Let me know if you have questions about this reasoning.
There is still a reference to a radius of $\frac{1}{\sqrt{\pi}}$? (Abram, 7/6)
I thought that we agreed to emphasize that the given circle or cube has be constructible in the first place or you still want to delete it? (Xingda, 7/8)
Correct, the circle has to be constructible, so nothing you are saying is wrong. It's just that I don't think giving an example of what *doesn't* work is actually helpful in this case. It works to simply say something like "A circle with radius 1 (and area pi) is constructible. In order to construct a square of area pi, we would have to construct a length of \sqrt(pi), but this is not possible. Therefore, because this one case is impossible, there can't be a general method for squaring the circle." Feel free to disagree on this point, though. (Abram, 7/11)
Nicely done. (Abram, 7/12)

Edit the introduction to the algebraicization

Hilda's quote is a bit confusing because it's not clear yet what this "exact" correspondence is. I'm going to give an example / suggestion for what the introduction might look like instead. (Abram, 6/11) I like the quote and want to keep it. Will address it after talking to Prof. Maurer about it. (Xingda 6/17)Prof. Maurer seems to have some confusion about the quote. We need to talk about this(Xingda 6/23)

At this point, the introduction is too long, between Hudson's quote, the next paragraph (where you explain what Hudson means), and the paragraph after that (where I explain what Hudson means). Your paragraph does a good job referring directly to Hudson's quote. On the other hand, my paragraph gives the reader an idea of what this "translation" or "correspondence" actually is. Somehow, this whole block of text should be shortened. For the record, part of the reason I still object to including Hudson's quote at all is because it's a LONG quote that ALSO requires a whole paragraph of explanation. It seems like a lot for a brief introduction. (Abram, 6/24)
I think we agreed that Hudson's old-timey language is nice, but I think you also believed me that as this stands, it's too long without hiding some of the text. (Abram, 7/6)
I deleted a bunch of sentences (Xingda, 7/8)
What you have done is really good. When you read the page word-by-word, see if you are happy with the exact wording. I may also take a pass at it later. (Abram, 7/12)

Be a bit more clear about the purpose of the page

When the basic description describes how Euclid wanted to base his proofs on as few axioms as possible, it suggests that you are going to talk about how a straightedge and compass can be used to prove theorems about Euclidean geometry. Make it clearer that this page is going to be about what kinds of things can and cannot be constructed. (Abram, 6/17)Issue addressed. (Xingda 6/23)

I really like what you have done with the introduction. We just need to figure out how to deal with Prof. Maurer's concerns about this not being a legitimate page. (Abram, 6/23)