mathematics

Wythoff array

Wythoff array 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 Wythoff array rather than just read about it. In short: In mathematics, the Wythoff array is an infinite matrix of positive integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array.

Key takeaways

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

Reference excerpt

In mathematics, the Wythoff array is an infinite matrix of positive integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array. The Wythoff array was first defined by Morrison (1980) using Wythoff pairs, the coordinates of winning positions in Wythoff's game. It can also be defined using Fibonacci numbers and Zeckendorf's theorem, or directly from the golden ratio and the recurrence relation defining the Fibonacci numbers.

Values The Wythoff array has the values

1 2 3 5 8 13 21 ⋯ 4 7 11 18 29 47 76 ⋯ 6 10 16 26 42 68 110 ⋯ 9 15 24 39 63 102 165 ⋯ 12 20 32 52 84 136 220 ⋯ 14 23 37 60 97 157 254 ⋯ 17 28 45 73 118 191 309 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ {\displaystyle {\begin{matrix}1&2&3&5&8&13&21&\cdots \\4&7&11&18&29&47&76&\cdots \\6&10&16&26&42&68&110&\cdots \\9&15&24&39&63&102&165&\cdots \\12&20&32&52&84&136&220&\cdots \\14&23&37&60&97&157&254&\cdots \\17&28&45&73&118&191&309&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \\\end{matrix}}} (sequence A035513 in the OEIS).

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Wythoff array

Start with the simplest possible case. Write down what Wythoff array 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 Wythoff array 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 Wythoff array 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 Wythoff array

In research: Wythoff array 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 Wythoff array 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: Wythoff array is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fibonacci numbers, Triangles of numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Wythoff array 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 Wythoff array in 20 minutes

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

Frequently asked questions

What is Wythoff array in simple terms?

Why does Wythoff array 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 Wythoff array?

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 Wythoff array.