# Talk:Steiner's Chain

## Messages to the Future

• Provided precise suggestions for what could be added in the future
• The applet on the Inversion Page does something similar to the applet that you want. Did you put up a request on the Math Tools Requests page?

## References and Footnotes

• Provided references of all websites and articles used at the bottom of the page
• Gave credit to image creators in all circumstances
• Images throughout the page (speaking of those excluding the main image) show how they were created by me by clicking on the image

## Good Writing

• Went through numerous drafts and had the page reviewed by many students as well as my professor

### Context

• Full of context, especially at the beginning of each section as well as between steps in mathematical equations
• I didn't include a "Why it's Interesting" section because I was unable to find any real-world applications of this topic, even though one was suggested to me on this discussion page by a fellow student, I wasn't able to locate any verifiable sources to prove this.

### Quality of Prose and Page Structuring

• Provided a purpose for each section and included it in the first few sentences of the section.
• In the "Basic Description" I tried to layout exactly what I was doing for the reader
• The paragraph that begins "With these measurements we can say that" is confusing. I suggest breaking all equations onto different lines, since they run into each other right now. Also, try to provide the justification for the equation before you write the equation instead of after.

*I'm having trouble understanding the relevance of the following sentence to the rest of the page: "When two circles are concentric, the area of the annulus in between is the area of the large circle minus the area of the small circle: $\text{Area of Annulus} = \pi(R^2 - r^2)$"

• Rebecca 13:05, 22 July 2011 (UTC) “Now, a Steiner chain does not always have to consist of a circle inside the other” This is unclear to me. Do you mean “Now, a Steiner chain does not always consist of a collection of circles within one larger circle”?
• “The image on the left, Figure 3, represents a closed annular chain whereas the image on the right, represents the Steiner chain that is obtained by inverting Figure 3 over a circle.” No comma after right.

### Integration of Images and Text

• Used many many images to explain my topic
• Constantly encouraged my reader to refer to my figures

### Connection to other Mathematical topics

• Provided one link to another page for readers to refer to for a better explanation of a specific term
• We have a page on Inversion that definitely should be linked by your page. Also, since your algebraic discussion very closely mirrors Problem of Apollonius, that page should also be linked.

### Examples, Calculations, Applications, Proofs

• Provided numerous thorough examples
• Included proofs where necessary
• Included context between difficult steps in algebra
• Is there a way to construct other annual steiner's chains by using polygons other than triangles? It'd be great to have a note about whether or not you can use the same process with other regular polygons.
I didn't see any change related to this comment.
• You need to explain a bit more of how you go from this statement "1. We can see that C, B, C', B' form a quadrilateral and one of the most basic theorems about quadrilaterals says that their opposite angles are supplementary." to the angle equality below it. You definitely skip a step of reasoning that should be stated explicitly.

### Mathematical Accuracy and Precision of Language

• Tried to clearly explain topics in the most simple words
• Included mouse-overs very often
• In your basic description, can you clarify if Steiner's Chains have to have one circle totally inside another? That seems to be the case, but you don't say it explicitly
• This sentence: "points C' and B' are concyclic as are points C and B." is confusing given your definition of concyclic. Points C' and B' don't lie on the same circle as you show in the image, rather one is the center and the other is on the outside. Can you clarify what you mean?

### Layout

• Organized well, with little white space
• Utilized features such as hide/show as well as mouse-overs
• Organized into understandable sections
• I'd encourage making this sentence: "By inverting points along all the circles of the Steiner chain, another can be formed that differs slightly from the original but still maintains the properties specific to Steiner chains (which are mentioned above under "Basic Description")". it's own paragraph. That will make the reader pause more after the link to inversion and click the link if they need more information

Kate 18:20, 28 June 2011 (UTC): You've got some great pictures, and it looks like you know what you're talking about, but you need to do a much better job of defining they key terms (Steiner's Porism, Steiner chains) and of explaining what you're doing and why you're doing it. I read the whole page carefully more than once, and I'm still confused about what's going on.

Hey Anna!

Cool page! One comment that I got a lot from the people who worked on this last summer, was that when I have math writing, like you do with your proofs, to explain what happens from one line to the next. Like this:

$\sin \frac{\pi}{n}= \frac{R-r}{R+r}$

Multiplying both sides by $R+r$ gives us

$\sin \frac{\pi}{n} (R+r)= R-r$

Distribution gives us

$R \sin \frac{\pi}{n} + r\sin \frac{\pi}{n} = R - r$

After we subtract $R, r$ from both sides, we have

...and so on and so forth.

Becky left a ton of comments on the the discussion page for Law of Sines about this in hot pink if you need any reference. That page has a lot of math writing.

Richard 6/13

## Intro

You mention circles, but your main picture has spheres. Maybe mention spheres in this part? Richard 7/18

## Basic Description

Kate 18:17, 28 June 2011 (UTC):

• You should link to Inversion in this section. It's not a perfect page, but what's there will definitely help people understand inversion.
• A Steiner chain is a figure of tangent circles - sounds awkward. Might be better to say it's "made of" tangent circles.
• This section really doesn't help me understand what a Steiner chain or Steiner porisms are at all. I see that you have a bit more of an explanation in the original caption, but you should really move that down here or at least say the same thing with different words here. Since the caption comes before the TOC, a lot of people tend to skip over it.
• Also, you need to offer some sort of explanation as to what a "porism" is - is "Steiner's porism" synonymous with "a Steiner chain"? Because the only term you've used so far is "Steiner chain", and I'm very confused as to why the page is called "Steiner's Porism" and not "Steiner chains".
• You end this bit by saying something about the "properties of a Steiner chain" - what are these properties? You should at least touch on them somewhere in the basic description. This will also help give context to the two construction sections that follow - give me something so that I can kind of see why we're doing what we're doing, and how we know when we've succeeded in creating a Steiner chain. To be honest, I read this whole section and still didn't really know what a Steiner chain was besides a picture involving circles. You need to give some sort of definition, and not just a description of how to make one example.

I'm not sure I understand your definition of inversion in this section. And you should say something about reflection fixing the points the same distance from the line as they start out. Richard 7/18

### Creating a Steiner Chain

Kate 18:17, 28 June 2011 (UTC):

• Unsure why you would say "regular triangle" - I think "equilateral triangle" is a much more familiar term for most people.
• Using the points of $\triangle XYZ$ as centers, construct tangent circles $X,Y,Z$
Upon re-reading, I'm pretty sure you mean that XYZ are tangent to each other, but at first I thought you meant that they were tangent to the original circle, which they're not. Consider rewording this sentence.
• Is the picture at the end of this section a Steiner chain? If so, please say so. If not, what is it? As it is, the section ends rather abruptly.

• By your definition, the main image is not concentric circles. You may want to make sure that this is just one particular way of making a Steiner chain. Richard 7/18

### Creating a Steiner Chain using Inversion

Kate 18:17, 28 June 2011 (UTC):

• Construct an inversion circle to reflect over the three tangent circles and the two concentric circles.
This sentence confused me. The way you've phrased it sounds as though the new circle you're creating is the one that's going to be reflected - "Construct an inversion circle that will then be reflected over the three tangent circles and the two concentric circles," whereas I'm pretty sure you mean to say "Construct an inversion circle to reflect the three tangent circles and the two concentric circles over." If you're uncomfortable with ending the sentence with a preposition, try "Construct an inversion circle over which the three tangent circles and the two concentric circles will be reflected." Also, there's no need to call that figure "the three tangent circles and the two concentric circles" - call it "the Steiner chain" or "our previous figure" or something.

• The last image in this section is kind of huge. Make sure it's on a different line than your text - it splits the paragraph up weird. And maybe make it a tiny bit smaller?
• This may just be part of my general confusion over what Steiner chains actually are, but why do we want to create a new Steiner chain by inversion? What is the purpose of doing this? Just for fun?
• I'm not sure that I understand all of the elements of this last picture. Nothing is tangent to the big circle. I assuming you can make a Steiner chain by inversion without being tangent? (I thought they were supposed to be) And why are there different sized circles within the smaller one? Is it a proportional thing? Richard 7/18

## A More Mathematical Explanation

### Formulas

Kate 18:17, 28 June 2011 (UTC):

• I don't understand why you have this section heading. First of all, it's the wrong size - it's hidden inside the MME, but if you look at the TOC it's not listed as a sub-section of the MME - to fix that you need to add another equals sign on either side of the heading. But there's nothing in the MME that's not also inside Formulas, so it seems to me that all you've done is re-name your MME, which doesn't seem necessary - all of the subheadings say "Formula" in them…
• So Steiner's Porism is a theorem about Steiner chains? You NEED to explain this above, even if you don't want to state the full thing. You just can't have a page about something and not even show what that thing is except for inside a hidden section.
• Also, you need to actually explain the Porism - I'm not stupid, and I have almost no idea what it's actually saying. What is the "starting circle" of a Steiner chain? What does it mean for a Steiner chain to "close"? What are the "loops" of a Steiner chain? Try to rephrase this statement using just the terms you've introduced in the basic description (and then move that simpler statement to the basic description!). Also, why is it called a "Porism" and not a "theorem"?
I second Kate here. What's a Porism? It's sort of like this is just hanging out here. Richard 7/18

#### Tangent Circles Formula

Kate 18:17, 28 June 2011 (UTC):

• In your first image in this section, the ordered pairs for the points aren't close enough to the arrows showing which points they belong to.
• What is the point of this section? How are these equations about tangent circles relevant to the rest of the page? If you're just establishing them here so that you can use them later, you need to say that.

#### Concentric Circles Formula

• Kate 18:17, 28 June 2011 (UTC): Again, what on earth is the purpose of this section? Give me some context so I'm not so confused! :(

#### Circle to Circle Inversion

Kate 18:17, 28 June 2011 (UTC):

• First of all, context! Context, context, context! I am so very very confused. Why are we doing any of this????
• Points (C, C'), (B, B') are inverses with respect to J \Rightarrow points C, C', B, B' are concyclic
I've gotta question your notation here - I thought that C, C', B, and B' were all points in their own right. Why are you making ordered pairs out of points? Or are they not points after all? If they're not points, what are they?
• In this whole section, I think you need a little more explanation than just all these arrows - what you've got is technically correct as a proof, but it'd be a lot more readable if you sort of embedded it in explanatory sentences. (This is what Richard was talking about in his earlier comment.)
• Why did we stop where we did? What is the significance of that result? Also, fix that period that's just sorta chillin by itself at the end here.
• Try to add overlines to all of your formulas in this section?

Richard 7/18

#### Steiner Chain Construction via Inversion Formulas

Kate 18:17, 28 June 2011 (UTC):

• An inversion can be done on a symmetrical arrangement of n circles (shown in Figure 3) in a region between two concentric circles, one with radiusR and the other with radius r.
I have no idea what this sentence means. Also, what is the purpose of this section? What are we trying to accomplish in it?
• This arrangement is represented by: [math stuff]
What arrangement? And where did that math come from? :( :( so confused.
• Again, you need to provide some explanation with your math. I have no idea what you're trying to do or how you're doing it, and also I don't like this format where you say "X=Y! Therefore, Z=W!" and then have an expandable section showing how you get from X=Y to Z=W - I think the explanation should come in the middle of the two statements.

Do you have a figure 1 and 2? Richard 7/18

See my comment from the last time. And also where did you get these formulas? I'm not exactly sure how you got them.

You've got an "is" "are" problem in the sentence starting with "The lengths of the radii of the tangent circles..." Richard 7/18

#### Types of Steiner Chains

• Kate 18:17, 28 June 2011 (UTC): This section needs to be at the top of the page! These sections define important terms that I need to understand before I can understand the statement of Steiner's Porism. They should not be hidden down here! It should go at the top of the MME or even the bottom of the Basic Description - this is actually more important to understand what's going on with the page than the steps to constructing a Steiner Chain are, imo.
Agreed 100%. Richard 7/18
##### Annular Steiner Chains
• Kate 18:17, 28 June 2011 (UTC): I'd like to see an example of a non-annular Steiner Chain in this section, just for contrast.
• This type reminds me of a gun cylinder/barrel thing!...Why It's Interesting maybe????? Steiner Chain in real life? Richard 7/18