mathematics

McShane's identity

McShane's identity 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 McShane's identity rather than just read about it. In short: In geometric topology, McShane's identity for a once punctured torus T {\displaystyle \mathbb {T} } with a complete, finite-volume hyperbolic structure is given by ∑ γ 1 1 + e ℓ ( γ ) = 1 2 {\displaystyle \sum _{\gamma }{\frac {1}{1+e^{\ell (\gamma )}}}={\frac {1}{2}}} where the sum is over all (unoriented) simple closed geodesics γ on the torus; and ℓ(γ) denotes the hyperbolic length of γ. This identity was general…

Key takeaways

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

Reference excerpt

In geometric topology, McShane's identity for a once punctured torus T {\displaystyle \mathbb {T} } with a complete, finite-volume hyperbolic structure is given by

∑ γ 1 1 + e ℓ ( γ ) = 1 2 {\displaystyle \sum _{\gamma }{\frac {1}{1+e^{\ell (\gamma )}}}={\frac {1}{2}}}

where

the sum is over all (unoriented) simple closed geodesics γ on the torus; and ℓ(γ) denotes the hyperbolic length of γ. This identity was generalized by Maryam Mirzakhani in her PhD thesis

References

Further reading Tan, Ser Peow; Wong, Yan Loi; Zhang, Ying (April 2006). "Necessary and Sufficient Conditions for Mcshane's Identity and Variations". Geometriae Dedicata. 119 (1): 199–217. arXiv:math/0411184. doi:10.1007/s10711-006-9069-9. S2CID 17575980. McShane, Greg (8 May 1998). "Simple geodesics and a series constant over Teichmuller space". Inventiones Mathematicae. 132 (3): 607–632. doi:10.1007/s002220050235. S2CID 16362716.

Worked examples

Example 1 — a first encounter with McShane's identity

Start with the simplest possible case. Write down what McShane's identity 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 McShane's identity 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 McShane's identity 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 McShane's identity

In research: McShane's identity 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 McShane's identity 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: McShane's identity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometric topology, so understanding it makes those chapters shorter.
In everyday life: Look for McShane's identity 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “McShane's identity” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study McShane's identity in 20 minutes

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

Frequently asked questions

What is McShane's identity in simple terms?

In geometric topology, McShane's identity for a once punctured torus T {\displaystyle \mathbb {T} } with a complete, finite-volume hyperbolic structure is given by ∑ γ 1 1 + e ℓ ( γ ) = 1 2 {\displaystyle \sum _{\gamma }{\frac {1}{1+e^{\ell (\gamma )}}}={\frac {1}{2}}} where the sum is over all (un…

Why does McShane's identity 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 McShane's identity?

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 McShane's identity.