Reuleaux tetrahedron

Key takeaways

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

Reference excerpt

The Reuleaux tetrahedron is the intersection of four balls of radius s centered at the vertices of a regular tetrahedron with side length s. The spherical surface of the ball centered on each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron. Thus the center of each ball is on the surfaces of the other three balls. The Reuleaux tetrahedron has the same face structure as a regular tetrahedron, but with curved faces: four vertices, and four curved faces, connected by six circular-arc edges. This shape is defined and named by analogy to the Reuleaux triangle, a two-dimensional curve of constant width; both shapes are named after Franz Reuleaux, a 19th-century German engineer who did pioneering work on ways that machines translate one type of motion into another. One can find repeated claims in the mathematical literature that the Reuleaux tetrahedron is analogously a surface of constant width, but it is not true: the two midpoints of opposite edge arcs are separated by a larger distance,

( 3 − 2 2 ) ⋅ s ≈ 1.0249 s . {\displaystyle \left({\sqrt {3}}-{\frac {\sqrt {2}}{2}}\right)\cdot s\approx 1.0249\,s.}

Volume and surface area The volume of a Reuleaux tetrahedron is

s 3 12 ( 3 2 − 49 π + 162 tan − 1 ⁡ 2 ) = s 3 12 ( 32 π − 81 cos − 1 ⁡ ( 1 3 ) + 3 2 ) ≈ 0.422 s 3 . {\displaystyle {\frac {s^{3}}{12}}{\big (}3{\sqrt {2}}-49\pi +162\tan ^{-1}{\sqrt {2}}{\big )}={\frac {s^{3}}{12}}\left(32\pi -81\cos ^{-1}\left({\tfrac {1}{3}}\right)+3{\sqrt {2}}\right)\approx 0.422\,s^{3}.}

The surface area is

[ 8 π − 18 cos − 1 ⁡ ( 1 3 ) ] s 2 ≈ 2.975 s 2 . {\displaystyle \left[8\pi -18\cos ^{-1}\left({\tfrac {1}{3}}\right)\right]s^{2}\approx 2.975\,s^{2}.}

Meissner bodies Ernst Meissner and Friedrich Schilling showed how to modify the Reuleaux tetrahedron to form a surface of constant width, by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. According to which three edge arcs are replaced (three that have a common vertex or three that form a triangle) there result two noncongruent shapes that are sometimes called Meissner bodies or Meissner tetrahedra.

Bonnesen and Fenchel conjectured that Meissner tetrahedra are the minimum-volume three-dimensional shapes of constant width, a conjecture which is still open. In 2011 Anciaux and Guilfoyle proved that the minimizer must consist of pieces of spheres and tubes over curves, which, being true for the Meissner tetrahedra, supports the conjecture. In connection with this problem, Campi, Colesanti and Gronchi showed that the minimum-volume surface of revolution with constant width is the surface of revolution of a Reuleaux triangle through one of its symmetry axes. Man Ray's painting Hamlet was based on a photograph he took of a Meissner tetrahedron, which he thought of as resembling both Yorick's skull and Ophelia's breast from Shakespeare's Hamlet.

References

External links Lachand-Robert, Thomas; Oudet, Édouard. "Spheroforms". Archived from the original on 2006-10-02. Retrieved 2006-09-12. Weber, Christof. "Bodies of Constant Width". There are also films and even interactive pictures of both Meissner bodies. Roberts, Patrick. "Spheroform with Tetrahedral Symmetry". Includes 3D pictures and link to mathematical paper showing proof of constant width.

Illustrations

Reuleaux tetrahedron: Animation of a Reuleaux tetrahedron, showing also the tetrahedron from which it is formed — Animation of a Reuleaux tetrahedron, showing also the tetrahedron from which it is formed

Worked examples

Example 1 — a first encounter with Reuleaux tetrahedron

Start with the simplest possible case. Write down what Reuleaux tetrahedron 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 Reuleaux tetrahedron 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 Reuleaux tetrahedron 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 Reuleaux tetrahedron

In research: Reuleaux tetrahedron 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 Reuleaux tetrahedron 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: Reuleaux tetrahedron is common in secondary-school and first-year university syllabi. It links to neighbouring topics Constant width, Euclidean solid geometry, Tetrahedra, so understanding it makes those chapters shorter.
In everyday life: Look for Reuleaux tetrahedron 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.

How to study Reuleaux tetrahedron in 20 minutes

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

Frequently asked questions

What is Reuleaux tetrahedron in simple terms?

The Reuleaux tetrahedron is the intersection of four balls of radius s centered at the vertices of a regular tetrahedron with side length s. The spherical surface of the ball centered on each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron.

Why does Reuleaux tetrahedron 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 Reuleaux tetrahedron?

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 Reuleaux tetrahedron.