# Steiner's Chain

Steiner's Chain in Third Dimension
In the image on the right, the Steiner chain consists of a sphere inside another, with a ring-like region in between. This space contains spheres of different diameters but each is tangent to the previous and succeeding spheres as well as to the two non-intersecting spheres.

# Basic Description

We will begin first with the definition of a Steiner chain and follow this description with geometric visuals that will help aid you in the construction of a specific Steiner chain, known as an Annular Steiner Chain. We will also include instructions on how to construct a Steiner chain using inversion. Lastly, we will provide numerous formulas, all of which represent the algebraic proofs of the geometric Steiner chain.

A Steiner chain is:

A set, or chain, of $n$ circles that are tangent to each other as well as tangent to two non-intersecting circles.

Many types of Steiner chains exist and below we have included visual representations accompanied with short descriptions. Although a wide range exists, all Steiner chains share specific properties. For a chain to be considered a Steiner chain, it MUST have:

• Two circles that are NOT tangent to one another
• A region that is made up of additional circles all of which are tangent to the two non-intersecting circles

## Types of Steiner Chains

#### Closed Steiner Chains

Steiner chains are "closed" when the first and last circles in the set (the region between the red and blue circles above) are tangent to one another.

#### Open Steiner Chains

Steiner chains are "open" when the first and last circles in the set are not tangent to one another. Above you can see two of the black circles intersect each other, but both of these circles still remain tangent to the two concentric circles.

#### Multicyclic Steiner Chains

Steiner chains are "multicyclic" when the set of $n$ circles wrap continuously around each other before closing, therefore the first and last circles are not tangent to one another, but overlap. This is similar to an Open Steiner chain but instead of having only a few intersecting circles in the region, all intersect.

#### Annular Steiner Chains

Annular Steiner chains are the most simple, they are closed chains consisting of $n$ circles of equal size that surround the inscribed circle. This also means that the inscribed and circumscribed circles are concentric.

A Steiner chain doesn't always have to be made with one of the non-intersecting (not tangent) circles within the larger as the images above show. Below (We have hidden it due to its necessary size) is an example where one of the non-intersecting circles is outside the other. It is considered to be a Steiner chain since it follows the two properties that say to be a Steiner chain:

• There must be two circles that are not tangent to each other
• There must be a region where there are additional circles that are all tangent to both the non-tangent circles.

# Creating a Steiner Chain From Scratch

Below are the visuals that show you how to construct an Annular Steiner chain starting with an equilateral triangle and ending with circles with Steiner chain properties. By using an equilateral triangle, we are going to have three tangent circles within the annulus (since there are three vertices on a triangle) If we wanted an annulus with eight symmetrical tangent circles, we would use an equilateral octagon to construct the Steiner chain. So, the number of tangent circles in the annulus is determined by the equilateral polygon we choose.

1. Construct a regular triangle$\triangle XYZ$ with center $C$.

2. Using the points of $\triangle XYZ$ as centers, construct tangent circles $X,Y,Z$. Each with a radius of $\frac{1}{2}$ the length of a side of the regular triangle.

3. Construct two concentric circles (red and blue in the image below) so that each is tangent to the three circles $X, Y, Z$.

Now that you have seen how to create an Annular Steiner chain, we will show you how to construct a Steiner chain from an already existing one.

# Creating A Steiner Chain Using Inversion

A Steiner chain can also be constructed by reflecting, or inverting, another Steiner chain. Reflection is taking points to the other side of a line so that they are the same distance from the line as they were before. Inversion is taking points to the "other side" of a circle. For a better understanding of inversion as well as for an active applet where you can invert a circle over another circle, I highly suggesting looking at the page titled Inversion.

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").

1. Construct an inversion circle over which the present Steiner chain will be reflected.

2. To obtain another Steiner chain, invert the already existing Steiner chain over the inversion circle.

Below is a more accurate inversion, this figure was obtained by using the program Cabri Plus II. You can see that after inverting the original Steiner chain, a different Steiner chain was formed. The circles within the ring between the two non-intersecting circles now have different diameters than their related non-inverted circles, meaning that they are not symmetric anymore (this also suggests that the two non-tangent circles are no longer concentric). The large size of the image helps you to notice these mentioned points.

# A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra and Trigonometry

## Formulas

Within this section we will provide numerous formulas with the purpose of supplying you [...]

## Formulas

Within this section we will provide numerous formulas with the purpose of supplying you with the algebraic proofs of Steiner chain construction. By applying these formulas geometrically, you can verify that what you have created is indeed a Steiner chain.

#### Tangent Circles Formula

The purpose of this section is to remind you of the formulas used to determine whether two circles are tangent. When constructing a Steiner chain, you can choose radii that in fact will produce tangent circles. This is the algebraic proof for the geometric image of tangent circles.

Two circles as pictured above, are tangent if:

$(x_{1}-x_{2})^2 + (y_{1}-y_{2})^2 = (r_{1}\pm r_{2})^2$

Further explanation of the above equation can be found on the page Problem of Apollonius

Figure 1 represents two circles that are externally tangent; their centers are separated by a distance:

$d = r_1+r_2$

Figure 2 represents two circles that are internally tangent; their centers are separated by a distance:

$d = \left\vert r_1-r_2 \right\vert$

#### Concentric Circle Formula

The purpose of this section is again algebraic, this formula will verify two circles to be concentric. When constructing an annular Steiner chain, it is important to make sure the two non-intersecting circles are indeed concentric. If they aren't concentric, the tangent circles within the annulus won't be symmetrical.

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:

$\text{Area of Annulus} = \pi(R^2 - r^2)$

#### Circle to Circle Inversion

The purpose of this section is to show you the algebra associated with the geometric property inversion. We have created an image that represents an inverted circle, below the figure is an algebraic explanation. This section will help you better understand inversion and therefore will help you be able to find the inverse of circles in the future. As you already know, a Steiner chain can be formed by taking another Steiner chain and inverting it over a circle. The figure below shows the inverse of circle $E$ with respect to circle $J$.

The figure above illustrates the following relationship:

• The point $C'$ is the inverse of the point $C$ with respect to circle $J$
• The point $B'$ is the inverse of the point $B$ with respect to circle $J$

1. We can see that $C, B, C', B'$ form a quadrilateral. One of the most basic theorems about quadrilaterals says that their opposite angles are supplementary, meaning

$\angle B'C'C + \angle B'BC = 180^\circ$

Therefore, $\angle ABC$ must equal $\angle B'C'C$ since

$\angle B'BC + \angle ABC=180$

and

$\angle B'BC + \angle B'C'C=180$

2. If you reorient$\triangle AC'B'$ you will see that it is a larger version of $\triangle ABC$. The angle measurements of both triangle are the same, the only thing that differs are the lengths of the sides.

$\triangle ABC\sim\triangle AC'B'$

3. Now that we know these two triangles are similar, we can solve for the length of any side of the triangle. ${C B}$ is proportional to ${B' C'}$ and ${A B}$ is proportional to ${A C'}$

$\frac{CB}{B'C'}=\frac{AB}{AC'}$

Multiplying each side by the common denominator gives us

$(CB)(AC')=(B'C')(AB)$

Dividing both sides by $AB$ gives us

$(B'C')=\frac{(CB)(AC')}{AB}$

Now we know the radius of the new circle. You can find the lengths of the other segments simply by solving for them in the equation.

#### Steiner Chain Construction via Inversion Formulas

Below are two figures of Steiner chains. 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. The new Steiner chain is no longer annular or concentric. Noticeably, Figure 3 has more than just a Steiner chain. There is also an enlarged triangle shown and I will describe how to find the measurements of its legs. These measurements are helpful when looking at the formula which represents:

• The radii ratio for the two concentric circles of the Steiner chain

Since the radius of the large red circle is $R$ and the radius of the small blue circle is $r$ (and since both these circles are concentric) we can agree that the diameter of each tangent circle located in the annulus is

$R-r$.

The radius is half the diameter, thus expressed as

$\frac{R-r}{2}$

By looking at Figure 3, we see that $AD$ is the radius of each tangent circle.

$AD=\frac{R-r}{2}$

Since we know the length of $AD$, we can easily find the length of $BA$ by adding together the radius of the inner concentric circle and the radius of the tangent circle, $AD$

 $BA$ $=$ $r+AD$ $BA$ $=$ $r+\frac{R-r}{2}$ $r+\frac{R-r}{2}$ $=$ $\frac{R+r}{2}$ $BA$ $=$ $\frac{R+r}{2}$
.

Knowing the lengths of $AD$ and $BA$ will help us find a trigonometric equation representing the radii ratio of the concentric circles. So, if we label the triangle in figure 3 we will see that

$\sin\theta=\frac{R-r}{R+r}$

Look at the triangle in Figure 3, as you can see $\angle B$ opens directly into the yellow circle. Well, a circle is $360^\circ$ and in radians is expressed as $2\pi$. Therefore, we can say that the measure of this angle is

$\angle B = 2\pi$

But, we need to account for all the circles, because this angle will be determined by the number of circles. So, we can write this as follows

$\angle B =\frac{2\pi}{n}$

Therefore,

$\sin B = \sin (\frac{2\pi}{n})$

But, we want the sine of theta which is half the measurement of $\angle B$. To find the sine of theta, just multiply $\frac{2\pi}{n}$ by a half

$\frac{2\pi}{n}\cdot\frac{1}{2}=\frac{\pi}{n}$

Therefore,

$\sin\theta=\sin\frac{\pi}{n}$

And therefore,

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

With the above equation we can find that the ratio of radii for the non-intersecting, concentric circles is:

$\frac{R}{r}=\frac {1-\sin\left(\frac{\pi}{n}\right) }{1+ \sin\left(\frac{\pi}{n}\right)}$

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

Multiply both sides by $R+r$

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

On the left side of the equation, distribute $\sin\frac{\pi}{n}$

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

Collect like-variables

$Rsin\frac{\pi}{n}-R=-r-rsin\frac{\pi}{n}$

Factor out the like-variables on each side of the equation

$R[sin\frac{\pi}{n}-1]=r[-1-sin\frac{\pi}{n}]$

Therefore, the relationship between $R$ and $r$ is

$\frac{R}{r}=\frac {1-\sin\left(\frac{\pi}{n}\right) }{1+ \sin\left(\frac{\pi}{n}\right)}$ .

# About the Creator of this Image

Fdecomite contributes pictures and self-created images to the website http://www.flickr.com/. He has created many interesting images of things other than Steiner's chain as well.

# References

[2] Chien-Hsun Lu. Exploring Steiner's Porism with Cabri Geometry. Retrieved from http://sylvester.math.nthu.edu.tw/d2/lue-thesis-inversion/Exploring_Steiner_s_Porism_with_Cabri_Geometry.pdf

[5] Davis, T. Inversion in a Circle. (2011, March 23). Retrieved from http://geometer.org/mathcircles/inversion.pdf

Explore the elliptical and hyperbolic properties of the Steiner chain. Also, including an applet displaying circle inversion would be extremely helpful.