biology

Von Neumann neighborhood

Von Neumann neighborhood is a biology 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 Von Neumann neighborhood rather than just read about it. In short: In cellular automata, the von Neumann neighborhood (or 4-neighborhood) is classically defined on a two-dimensional square lattice and is composed of a central cell and its four adjacent cells. The neighborhood is named after John von Neumann, who used it to define the von Neumann cellular automaton and the von Neumann universal constructor within it.

Key takeaways

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

Reference excerpt

In cellular automata, the von Neumann neighborhood (or 4-neighborhood) is classically defined on a two-dimensional square lattice and is composed of a central cell and its four adjacent cells. The neighborhood is named after John von Neumann, who used it to define the von Neumann cellular automaton and the von Neumann universal constructor within it. It is one of the two most commonly used neighborhood types for two-dimensional cellular automata, the other one being the Moore neighborhood. This neighbourhood can be used to define the notion of 4-connected pixels in computer graphics. The von Neumann neighbourhood of a cell is the cell itself and the cells at a Manhattan distance of 1. The concept can be extended to higher dimensions, for example forming a 6-cell octahedral neighborhood for a cubic cellular automaton in three dimensions.

Von Neumann neighborhood of range r An extension of the simple von Neumann neighborhood described above is to take the set of points at a Manhattan distance of r > 1. This results in a diamond-shaped region (shown for r = 2 in the illustration). These are called von Neumann neighborhoods of range or extent r. The number of cells in a 2-dimensional von Neumann neighborhood of range r can be expressed as r 2 + ( r + 1 ) 2 {\displaystyle r^{2}+(r+1)^{2}} . The number of cells in a d-dimensional von Neumann neighborhood of range r is the Delannoy number D(d,r). The number of cells on a surface of a d-dimensional von Neumann neighborhood of range r is the Zaitsev number (sequence A266213 in the OEIS).

See also Moore neighborhood Neighbourhood (graph theory) Taxicab geometry Lattice graph Pixel connectivity Chain code

References

External links Weisstein, Eric W. "von Neumann Neighborhood". MathWorld. Tyler, Tim, The von Neumann neighborhood at cell-auto.com

Illustrations

Worked examples

Example 1 — a first encounter with Von Neumann neighborhood

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

In research: Von Neumann neighborhood appears in biology 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 Von Neumann neighborhood 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: Von Neumann neighborhood is common in secondary-school and first-year university syllabi. It links to neighbouring topics Cellular automata, Theoretical computer science stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Von Neumann neighborhood 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Von Neumann neighborhood” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Von Neumann neighborhood in 20 minutes

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

Frequently asked questions

What is Von Neumann neighborhood in simple terms?

Why does Von Neumann neighborhood matter?

Because it connects several biology 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 Von Neumann neighborhood?

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 Von Neumann neighborhood.