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Hinged dissection

Hinged dissection 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 Hinged dissection rather than just read about it. In short: In geometry, a hinged dissection, also known as a swing-hinged dissection or Dudeney dissection, is a kind of geometric dissection in which all of the pieces are connected into a chain by "hinged" points, such that the rearrangement from one figure to another can be carried out by swinging the chain continuously, without severing any of the connections. Typically, it is assumed that the pieces are allowed to overlap…

Hinged dissection — main illustration
Hinged dissection — illustration

Key takeaways

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

Reference excerpt

In geometry, a hinged dissection, also known as a swing-hinged dissection or Dudeney dissection, is a kind of geometric dissection in which all of the pieces are connected into a chain by "hinged" points, such that the rearrangement from one figure to another can be carried out by swinging the chain continuously, without severing any of the connections. Typically, it is assumed that the pieces are allowed to overlap in the folding and unfolding process; this is sometimes called the "wobbly-hinged" model of hinged dissection.

History

The concept of hinged dissections was popularised by the author of mathematical puzzles, Henry Dudeney. He introduced the famous hinged dissection of a square into a triangle (pictured) in his 1907 book The Canterbury Puzzles. The Wallace–Bolyai–Gerwien theorem, first proven in 1807, states that any two equal-area polygons must have a common dissection. However, the question of whether two such polygons must also share a hinged dissection remained open until 2007, when Erik Demaine et al. proved that there must always exist such a hinged dissection, and provided a constructive algorithm to produce them. This proof holds even under the assumption that the pieces may not overlap while swinging, and can be generalised to any pair of three-dimensional figures which have a common dissection (see Hilbert's third problem). In three dimensions, however, the pieces are not guaranteed to swing without overlap.

Other hinges

Other types of "hinges" have been considered in the context of dissections. A twist-hinge dissection is one which use a three-dimensional "hinge" which is placed on the edges of pieces rather than their vertices, allowing them to be "flipped" three-dimensionally. As of 2002, the question of whether any two polygons must have a common twist-hinged dissection remains unsolved.

References

Bibliography Frederickson, Greg N. (26 August 2002). Hinged Dissections: Swinging and Twisting. Cambridge University Press. ISBN 978-0521811927. Retrieved 19 December 2013.

External links An applet demonstrating Dudeney's hinged square-triangle dissection A gallery of hinged dissections An applet with 11 hinged dissections

Illustrations

Hinged dissection: Loop animation of hinged dissections from triangle to square, then to hexagon, then back again to triangle. Notice that the chain of pieces can be entirely connected in a ring during the rearrangement from square to hexagon.
Loop animation of hinged dissections from triangle to square, then to hexagon, then back again to triangle. Notice that the chain of pieces can be entirely connected in a ring during the rearrangement from square to hexagon.
Hinged dissection: Dudeney's hinged dissection of a triangle into a square.
Dudeney's hinged dissection of a triangle into a square.
Hinged dissection: Animation of hinged dissection from hexagram to triangle to square
Animation of hinged dissection from hexagram to triangle to square
Hinged dissection: Hinged square to pentagon
Hinged square to pentagon

Worked examples

Example 1 — a first encounter with Hinged dissection

Start with the simplest possible case. Write down what Hinged dissection 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 Hinged dissection 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 Hinged dissection 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 Hinged dissection

In research
Hinged dissection 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 Hinged dissection 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
Hinged dissection is common in secondary-school and first-year university syllabi. It links to neighbouring topics Discrete geometry, Euclidean plane geometry, Geometric dissection, so understanding it makes those chapters shorter.
In everyday life
Look for Hinged dissection 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 Hinged dissection in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Hinged dissection 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 Hinged dissection out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hinged dissection in simple terms?

In geometry, a hinged dissection, also known as a swing-hinged dissection or Dudeney dissection, is a kind of geometric dissection in which all of the pieces are connected into a chain by "hinged" points, such that the rearrangement from one figure to another can be carried out by swinging the chai…

Why does Hinged dissection 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 Hinged dissection?

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 Hinged dissection.

Tags

  • Discrete geometry
  • Euclidean plane geometry
  • Geometric dissection
  • Recreational mathematics

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