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Parbelos

Parbelos is a science 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 Parbelos rather than just read about it. In short: The parbelos is a figure similar to the arbelos but instead of three half circles it uses three parabola segments. More precisely the parbelos consists of three parabola segments, that have a height that is one fourth of the width at their bases.

Parbelos — main illustration
Parbelos — illustration

Key takeaways

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

Reference excerpt

The parbelos is a figure similar to the arbelos but instead of three half circles it uses three parabola segments. More precisely the parbelos consists of three parabola segments, that have a height that is one fourth of the width at their bases. The two smaller parabola segments are placed next to each other with their bases on a common line and the largest parabola is placed over the two smaller ones such that its width is the sum of the widths of the smaller ones (see graphic). The parbelos has a number of properties which are somewhat similar or even identical to the properties of the Arbelos. For instance, the following two properties are identical to those of the arbelos:

The arc length of the outer parabola is equal to the sum of the arc lengths of the inner parabolas. In a nested arbelos construction with the inner parabola segments being arbeloses themselves the two innermost parabola segments being adjacent to the cusp of the outer arbelos are congruent, that is of equal size. The quadrilateral B M 2 M M 1 {\displaystyle BM_{2}MM_{1}} formed by the inner cusp B {\displaystyle B} and the midpoints M , M 1 , M 2 {\displaystyle M,M_{1},M_{2}} of the three parabola arcs is a parallelogram the area of which relates to the area of the parbelos as follows:

F parallelogram = 3 4 F parbelos {\displaystyle F_{\text{parallelogram}}={\frac {3}{4}}F_{\text{parbelos}}}

The four tangents at the three cusps of the parabola intersect in four points, which form a rectangle being called the tangent rectangle. The circumcircle of the tangent rectangle intersects the base side of the outer parabola segment in its midpoint, which is the focus of the outer parabola. One diagonal of the tangent rectangle lies on a tangent to the outer parabola and its common point with it is identical to its point of intersection with perpendicular to the base at the inner cusp. For the area of the tangent rectangle the following equation holds:

F rectangle = 3 2 F parbelos {\displaystyle F_{\text{rectangle}}={\frac {3}{2}}F_{\text{parbelos}}}

References

Further reading Emmanuel Tsukerman: "Solution of Sondow’s Problem: A Synthetic Proof of the Tangency Property of the Parbelos". In: The American Mathematical Monthly, Vol. 121, No. 5 (May 2014), pp. 438-443 Antonio M. Oller-Marcén: "The f-belos". In: Forum Geometricorum, 13 (2013), pp. 103–111 (online copy) Viktorija Ternar: Arbelos, parabelos in f-belos (master thesis, University of Maribor, 2015)

Illustrations

Parbelos: parbelos with parallelogram B M 2 M M 1 {\displaystyle BM_{2}MM_{1}} outer cusps A , C {\displaystyle A,C} and inner cusp B {\displaystyle B} F parbelos = 4 3 F parallogramm {\displaystyle F_{\text{parbelos}}={\frac {4}{3}}F_{\text{parallogramm}}}
parbelos with parallelogram B M 2 M M 1 {\displaystyle BM_{2}MM_{1}} outer cusps A , C {\displaystyle A,C} and inner cusp B {\displaystyle B} F parbelos = 4 3 F parallogramm {\displaystyle F_{\text{parbelos}}={\frac {4}{3}}F_{\text{parallogramm}}}
Parbelos: nested parbelos with congruent half discs in grey
nested parbelos with congruent half discs in grey
Parbelos: parbelos with tangent rectangle B D E F {\displaystyle BDEF}
parbelos with tangent rectangle B D E F {\displaystyle BDEF}

Worked examples

Example 1 — a first encounter with Parbelos

Start with the simplest possible case. Write down what Parbelos claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Parbelos 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 Parbelos 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 Parbelos

In research
Parbelos appears in science 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 Parbelos 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
Parbelos is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometric shapes, so understanding it makes those chapters shorter.
In everyday life
Look for Parbelos 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 Parbelos in 20 minutes

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

Frequently asked questions

What is Parbelos in simple terms?

The parbelos is a figure similar to the arbelos but instead of three half circles it uses three parabola segments. More precisely the parbelos consists of three parabola segments, that have a height that is one fourth of the width at their bases.

Why does Parbelos matter?

Because it connects several science 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 Parbelos?

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 Parbelos.

Tags

  • Geometric shapes

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