mathematics

Holditch's theorem

Holditch's theorem 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 Holditch's theorem rather than just read about it. In short: In plane geometry, Holditch's theorem states that if a chord of fixed length is allowed to rotate inside a convex closed curve, then the locus of a point on the chord a distance p from one end and a distance q from the other is a closed curve whose enclosed area is less than that of the original curve by π p q {\displaystyle \pi pq} . The theorem was published in 1858 by Rev.

Key takeaways

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

Reference excerpt

In plane geometry, Holditch's theorem states that if a chord of fixed length is allowed to rotate inside a convex closed curve, then the locus of a point on the chord a distance p from one end and a distance q from the other is a closed curve whose enclosed area is less than that of the original curve by π p q {\displaystyle \pi pq} . The theorem was published in 1858 by Rev. Hamnet Holditch. While not mentioned by Holditch, the proof of the theorem requires an assumption that the chord be short enough that the traced locus is a simple closed curve.

Observations The theorem is included as one of Clifford Pickover's 250 milestones in the history of mathematics. Some peculiarities of the theorem include that the area formula π p q {\displaystyle \pi pq} is independent of both the shape and the size of the original curve, and that the area formula is the same as for that of the area of an ellipse with semi-axes p and q. The theorem's author was a president of Caius College, Cambridge.

Extensions Broman gives a more precise statement of the theorem, along with a generalization. The generalization allows, for example, consideration of the case in which the outer curve is a triangle, so that the conditions of the precise statement of Holditch's theorem do not hold because the paths of the endpoints of the chord have retrograde portions (portions that retrace themselves) whenever an acute angle is traversed. Nevertheless, the generalization shows that if the chord is shorter than any of the triangle's altitudes, and is short enough that the traced locus is a simple curve, Holditch's formula for the in-between area is still correct (and remains so if the triangle is replaced by any convex polygon with a short enough chord). However, other cases result in different formulas.

References

Further reading B. Williamson, FRS, An elementary treatise on the integral calculus : containing applications to plane curves and surfaces, with numerous examples (Longmans, Green, London, 1875; 2nd 1877; 3rd 1880; 4th 1884; 5th 1888; 6th 1891; 7th 1896; 8th 1906; 1912, 1916, 1918, 1926); Ist 1875, pp. 192–193, with citation of Holditch's Prize Question set in The Lady's and Gentleman's Diary for 1857 (appearing in late 1856), with extension by Woolhouse in the issue for 1858; 5th 1888; 8th 1906 pp. 206–211 J. Edwards, A Treatise on the Integral Calculus with Applications, Examples and Problems, Vol. 1 (Macmillan, London, 1921), Chap. XV, esp. Sections 478, 481–491, 496 (see also Chap. XIX for instantaneous centers, roulettes and glisettes); expounds and references extensions due to Woolhouse, Elliott, Leudesdorf, Kempe, drawing on the earlier book of Williamson. Kılıç, Erol; Keleş, Sadık (1994), "On Holditch's theorem and polar inertia momentum", Communications Faculty of Sciences University of Ankara, Series A1: Mathematics and Statistics, 43 (1–2): 41–47 (1996), MR 1404786 Cooker, Mark J. (July 1998), "An extension of Holditch's theorem on the area within a closed curve", The Mathematical Gazette, 82 (494): 183–188, doi:10.2307/3620400, JSTOR 3620400, S2CID 123443685 Cooker, Mark J. (March 1999), "On sweeping out an area", The Mathematical Gazette, 83 (496): 69–73, doi:10.2307/3618685, JSTOR 3618685, S2CID 125103358 Apostol, Tom M.; Mnatsakanian, Mamikon A. (2012), "9.13 Remarks concerning Holditch's theorem", New Horizons in Geometry, The Dolciani Mathematical Expositions, vol. 47, Washington, DC: Mathematical Association of America, pp. 291–294, ISBN 978-0-88385-354-2, MR 3024916

External links

Weisstein, Eric W., "Holditch's Theorem", MathWorld

Illustrations

Worked examples

Example 1 — a first encounter with Holditch's theorem

Start with the simplest possible case. Write down what Holditch's theorem 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 Holditch's theorem 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 Holditch's theorem 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 Holditch's theorem

In research: Holditch's theorem 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 Holditch's theorem 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: Holditch's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in plane geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Holditch's theorem 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 Holditch's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Holditch's theorem 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 Holditch's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Holditch's theorem in simple terms?

Why does Holditch's theorem 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 Holditch's theorem?

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 Holditch's theorem.