Superformula

Key takeaways

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

Reference excerpt

The superformula is a generalization of the superellipse and was proposed by Johan Gielis in 2003. Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature. Gielis has filed a patent application related to the synthesis of patterns generated by the superformula, which expired effective 2020-05-10. In polar coordinates, with r {\displaystyle r} the radius and φ {\displaystyle \varphi } the angle, the superformula is:

r ( φ ) = ( | cos ⁡ ( m φ 4 ) a | n 2 + | sin ⁡ ( m φ 4 ) b | n 3 ) − 1 n 1 . {\displaystyle r\left(\varphi \right)=\left(\left|{\frac {\cos \left({\frac {m\varphi }{4}}\right)}{a}}\right|^{n_{2}}+\left|{\frac {\sin \left({\frac {m\varphi }{4}}\right)}{b}}\right|^{n_{3}}\right)^{-{\frac {1}{n_{1}}}}.}

By choosing different values for the parameters a , b , m , n 1 , n 2 , {\displaystyle a,b,m,n_{1},n_{2},} and n 3 , {\displaystyle n_{3},} different shapes can be generated. The formula was obtained by generalizing the superellipse, named and popularized by Piet Hein, a Danish mathematician.

2D plots In the following examples the values shown above each figure should be m, n1, n2 and n3.

A GNU Octave program for generating these figures

Extension to higher dimensions It is possible to extend the formula to 3, 4, or n dimensions, by means of the spherical product of superformulas. For example, the 3D parametric surface is obtained by multiplying two superformulas r1 and r2. The coordinates are defined by the relations:

x = r 1 ( θ ) cos ⁡ θ ⋅ r 2 ( ϕ ) cos ⁡ ϕ , {\displaystyle x=r_{1}(\theta )\cos \theta \cdot r_{2}(\phi )\cos \phi ,}

y = r 1 ( θ ) sin ⁡ θ ⋅ r 2 ( ϕ ) cos ⁡ ϕ , {\displaystyle y=r_{1}(\theta )\sin \theta \cdot r_{2}(\phi )\cos \phi ,}

z = r 2 ( ϕ ) sin ⁡ ϕ , {\displaystyle z=r_{2}(\phi )\sin \phi ,}

where ϕ {\displaystyle \phi } (latitude) varies between −π/2 and π/2 and θ (longitude) between −π and π.

3D plots 3D superformula: a = b = 1; m, n1, n2 and n3 are shown in the pictures.

A GNU Octave program for generating these figures:

Generalization The superformula can be generalized by allowing distinct m parameters in the two terms of the superformula. By replacing the first parameter m {\displaystyle m} with y and second parameter m {\displaystyle m} with z:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Superformula

Start with the simplest possible case. Write down what Superformula 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 Superformula 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 Superformula 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 Superformula

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

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

Frequently asked questions

What is Superformula in simple terms?

The superformula is a generalization of the superellipse and was proposed by Johan Gielis in 2003. Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature.

Why does Superformula 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 Superformula?

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 Superformula.