# Talk:Steiner's Chain

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.

### 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.

### 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?

## 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"?

#### 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.

#### 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.

#### 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.
##### 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.