mathematics

Henneberg surface

Henneberg surface 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 Henneberg surface rather than just read about it. In short: In differential geometry, the Henneberg surface is a non-orientable minimal surface named after Lebrecht Henneberg. It has parametric equation x ( u , v ) = 2 cos ⁡ ( v ) sinh ⁡ ( u ) − ( 2 / 3 ) cos ⁡ ( 3 v ) sinh ⁡ ( 3 u ) y ( u , v ) = 2 sin ⁡ ( v ) sinh ⁡ ( u ) + ( 2 / 3 ) sin ⁡ ( 3 v ) sinh ⁡ ( 3 u ) z ( u , v ) = 2 cos ⁡ ( 2 v ) cosh ⁡ ( 2 u ) {\displaystyle {\begin{aligned}x(u,v)&=2\cos(v)\sinh(u)-(2/3)\cos(3…

Key takeaways

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

Reference excerpt

In differential geometry, the Henneberg surface is a non-orientable minimal surface named after Lebrecht Henneberg. It has parametric equation

x ( u , v ) = 2 cos ⁡ ( v ) sinh ⁡ ( u ) − ( 2 / 3 ) cos ⁡ ( 3 v ) sinh ⁡ ( 3 u ) y ( u , v ) = 2 sin ⁡ ( v ) sinh ⁡ ( u ) + ( 2 / 3 ) sin ⁡ ( 3 v ) sinh ⁡ ( 3 u ) z ( u , v ) = 2 cos ⁡ ( 2 v ) cosh ⁡ ( 2 u ) {\displaystyle {\begin{aligned}x(u,v)&=2\cos(v)\sinh(u)-(2/3)\cos(3v)\sinh(3u)\\y(u,v)&=2\sin(v)\sinh(u)+(2/3)\sin(3v)\sinh(3u)\\z(u,v)&=2\cos(2v)\cosh(2u)\end{aligned}}}

and can be expressed as an order-15 algebraic surface. It can be viewed as an immersion of a punctured projective plane. Up until 1981 it was the only known non-orientable minimal surface. The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem.

References

Further reading E. Güler; Ö. Kişi; C. Konaxis, Implicit equations of the Henneberg-type minimal surface in the four-dimensional Euclidean space. Mathematics 6(12), (2018) 279. doi:10.3390/math6120279. E. Güler; V. Zambak, Henneberg's algebraic surfaces in Minkowski 3-space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68(2), (2019) 1761–1773. doi:10.31801/cfsuasmas.444554.

Illustrations

Worked examples

Example 1 — a first encounter with Henneberg surface

Start with the simplest possible case. Write down what Henneberg surface 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 Henneberg surface 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 Henneberg surface 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 Henneberg surface

In research: Henneberg surface 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 Henneberg surface 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: Henneberg surface is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential geometry, Minimal surfaces, so understanding it makes those chapters shorter.
In everyday life: Look for Henneberg surface 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 Henneberg surface in 20 minutes

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

Frequently asked questions

What is Henneberg surface in simple terms?

In differential geometry, the Henneberg surface is a non-orientable minimal surface named after Lebrecht Henneberg. It has parametric equation x ( u , v ) = 2 cos ⁡ ( v ) sinh ⁡ ( u ) − ( 2 / 3 ) cos ⁡ ( 3 v ) sinh ⁡ ( 3 u ) y ( u , v ) = 2 sin ⁡ ( v ) sinh ⁡ ( u ) + ( 2 / 3 ) sin ⁡ ( 3 v ) sinh ⁡…

Why does Henneberg surface 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 Henneberg surface?

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 Henneberg surface.