science

Perfect lattice

Perfect lattice is a science 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 Perfect lattice rather than just read about it. In short: In mathematics, a perfect lattice (or perfect form) is a lattice in a Euclidean vector space, that is completely determined by the set S of its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. Perfect lattices were introduced by Korkine & Zolotareff (1877).

Key takeaways

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

Reference excerpt

In mathematics, a perfect lattice (or perfect form) is a lattice in a Euclidean vector space, that is completely determined by the set S of its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. Perfect lattices were introduced by Korkine & Zolotareff (1877). A strongly perfect lattice is one whose minimal vectors form a spherical 4-design. This notion was introduced by Venkov (2001). Voronoi (1908) proved that a lattice is extreme if and only if it is both perfect and eutactic. The number of perfect lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8 is given by 1, 1, 1, 2, 3, 7, 33, 10916 (sequence A004026 in the OEIS). Conway & Sloane (1988) summarize the properties of perfect lattices of dimension up to 7. Sikirić, Schürmann & Vallentin (2007) verified that the list of 10916 perfect lattices in dimension 8 found by Martinet and others is complete. It was proven by Riener (2006) that only 2408 of these 10916 perfect lattices in dimension 8 are actually extreme lattices.

References Conway, John Horton; Sloane, N. J. A. (1988), "Low-dimensional lattices. III. Perfect forms", Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 418 (1854): 43–80, Bibcode:1988RSPSA.418...43C, doi:10.1098/rspa.1988.0073, ISSN 0962-8444, JSTOR 2398316, MR 0953277 Conway, J. H.; Sloane, N. J. A. (1989). "Errata: Low-Dimensional Lattices. III. Perfect Forms". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 426 (1871): 441. Bibcode:1989RSPSA.426..441C. doi:10.1098/rspa.1989.0134. JSTOR 2398351. Korkine; Zolotareff (1877), "Sur les formes quadratique positives", Mathematische Annalen, 11 (2): 242–292, doi:10.1007/BF01442667, ISSN 0025-5831 Martinet, Jacques (2003), Perfect lattices in Euclidean spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 327, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-05167-2, ISBN 978-3-540-44236-3, MR 1957723 Riener, Cordian (2006), "On extreme forms in dimension 8", Journal de théorie des nombres de Bordeaux, 18 (3): 677–682, doi:10.5802/jtnb.565 Sikirić, Mathieu Dutour; Schürmann, Achill; Vallentin, Frank (2007), "Classification of eight-dimensional perfect forms", Electronic Research Announcements of the American Mathematical Society, 13 (3): 21–32, arXiv:math/0609388, doi:10.1090/S1079-6762-07-00171-0, ISSN 1079-6762, MR 2300003 Venkov, Boris (2001), "Réseaux et designs sphériques, Réseaux euclidiens, designs sphériques et formes modulaires", Monographie de l'Enseignement Mathématique, 37: 10–86 Voronoi, G. (1908), "Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier Mémoire: Sur quelques propriétés des formes quadratiques positives parfaites", Journal für die reine und angewandte Mathematik (in French), 1908 (133): 97–178, doi:10.1515/crll.1908.133.97, ISSN 0075-4102

External links List of perfect lattices in dimension 8

Worked examples

Example 1 — a first encounter with Perfect lattice

Start with the simplest possible case. Write down what Perfect lattice 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 Perfect lattice 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 Perfect lattice 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 Perfect lattice

In research: Perfect lattice 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 Perfect lattice 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: Perfect lattice is common in secondary-school and first-year university syllabi. It links to neighbouring topics Quadratic forms, so understanding it makes those chapters shorter.
In everyday life: Look for Perfect lattice 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 Perfect lattice in 20 minutes

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

Frequently asked questions

What is Perfect lattice in simple terms?

Why does Perfect lattice 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 Perfect lattice?

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 Perfect lattice.