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Pappus chain

Pappus chain is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Pappus chain rather than just read about it. In short: In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. Construction Given two circles CU and CV, let the inner circle CU be enclosed by the outer circle CV, and let the two circles be tangent to each other at point A.

Pappus chain — main illustration
Pappus chain — illustration

Key takeaways

  • Pappus chain belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Pappus chain to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Pappus chain from memory before moving on to harder problems.

Reference excerpt

In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD.

Construction Given two circles CU and CV, let the inner circle CU be enclosed by the outer circle CV, and let the two circles be tangent to each other at point A. Let the radii of these two circles be denoted as rU, rV, respectively, and let their respective centers be the points U, V. The Pappus chain consists of the circles in the shaded grey region, which are externally tangent to CU (the inner circle) and internally tangent to CV (the outer circle). Let the radius, diameter and center point of the nth circle of the Pappus chain be denoted as rn, dn, Pn, respectively. The Pappus chain is often considered with respect to an arbelos, a circular triangle whose three sides are semicircles of the two given tangent circles and of the circle in the chain whose center is collinear with the two given circles.

Properties

Centers of the circles

Ellipse All the centers of the circles in the Pappus chain are located on a common ellipse, for the following reason. The sum of the distances from the nth circle of the Pappus chain to the two centers U, V of the arbelos circles equals a constant

P n U ¯ + P n V ¯ = ( r U + r n ) + ( r V − r n ) = r U + r V {\displaystyle {\overline {P_{n}U}}+{\overline {P_{n}V}}=(r_{U}+r_{n})+(r_{V}-r_{n})=r_{U}+r_{V}}

Thus, the foci of this ellipse are U, V, the centers of the two circles that define the arbelos; these points correspond to the midpoints of the line segments AB, AC, respectively.

Coordinates If r = A C ¯ A B ¯ , {\displaystyle r={\tfrac {\overline {AC}}{\overline {AB}}},} then the center of the nth circle in the chain is:

( x n , y n ) = ( r ( 1 + r ) 2 [ n 2 ( 1 − r ) 2 + r ] , n r ( 1 − r ) n 2 ( 1 − r ) 2 + r ) {\displaystyle (x_{n},y_{n})=\left({\frac {r(1+r)}{2[n^{2}(1-r)^{2}+r]}}~,~{\frac {nr(1-r)}{n^{2}(1-r)^{2}+r}}\right)}

Radii of the circles If r = A C ¯ A B ¯ , {\displaystyle r={\tfrac {\overline {AC}}{\overline {AB}}},} then the radius of the nth circle in the chain is:

r n = ( 1 − r ) r 2 [ n 2 ( 1 − r ) 2 + r ] {\displaystyle r_{n}={\frac {(1-r)r}{2[n^{2}(1-r)^{2}+r]}}}

Circle inversion

… excerpt ends here. Continue reading the full article.

Illustrations

Pappus chain: A Pappus chain
A Pappus chain
Pappus chain: Under a particular inversion centered on A, the four initial circles of the Pappus chain are transformed into a stack of four equally sized circles, sandwiched between two parallel lines. This accounts for the height formula hn = ndn and the fact that the original points of tangency lie on a common circle.
Under a particular inversion centered on A, the four initial circles of the Pappus chain are transformed into a stack of four equally sized circles, sandwiched between two parallel lines. This accounts for the height formula hn = ndn and the fact that the original points of tangency lie on a common circle.

Worked examples

Example 1 — a first encounter with Pappus chain

Start with the simplest possible case. Write down what Pappus chain claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pappus chain before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pappus chain ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pappus chain

In research
Pappus chain appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Pappus chain in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Pappus chain is common in secondary-school and first-year university syllabi. It links to neighbouring topics Ancient Greek mathematics, Arbelos, Circle packing, so understanding it makes those chapters shorter.
In everyday life
Look for Pappus chain outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Pappus chain in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Pappus chain means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Pappus chain out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pappus chain in simple terms?

In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. Construction Given two circles CU and CV, let the inner circle CU be enclosed by the outer circle CV, and let the two circles be tangent to each other at point…

Why does Pappus chain matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pappus chain?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pappus chain.

Tags

  • Ancient Greek mathematics
  • Arbelos
  • Circle packing
  • Inversive geometry

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