mathematics

Ralph Fox

Ralph Fox 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 Ralph Fox rather than just read about it. In short: Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization of knot theory and of bringing it into the mainstream.

Key takeaways

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

Reference excerpt

Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization of knot theory and of bringing it into the mainstream.

Biography Ralph Fox attended Swarthmore College for two years, while studying piano at the Leefson Conservatory of Music in Philadelphia. He earned a master's degree from Johns Hopkins University, and a PhD degree from Princeton University in 1939. His doctoral dissertation, On the Lusternick–Schnirelmann Category, was directed by Solomon Lefschetz. (In later years he disclaimed all knowledge of the Lusternik–Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of John Milnor, John Stallings, Francisco González-Acuña, Guillermo Torres-Diaz and Barry Mazur, and supervised Ken Perko's undergraduate thesis. He was an Invited Speaker at the International Congress of Mathematicians held in 1950 in Cambridge, Massachusetts. His mathematical contributions include Fox n-coloring of knots, the Fox–Artin arc, and the free differential calculus. He also identified the compact-open topology on function spaces as being particularly appropriate for homotopy theory. Aside from his strictly mathematical contributions, he was responsible for introducing several basic phrases to knot theory: the phrases slice knot, ribbon knot, and Seifert circle all appear in print for the first time under his name, and he also popularized (if he did not introduce) the phrase Seifert surface. He popularized the playing of the game of Go at both Princeton and the Institute for Advanced Study.

Selected publications Introduction to Knot Theory, Richard H. Crowell and Ralph H. Fox, Reprint of the 1963 original, Graduate Texts in Mathematics, No. 57, Springer-Verlag, New York-Heidelberg, 1977. ISBN 0-387-90272-4 "A quick trip through knot theory", in: M. K. Fort (Ed.), Topology of 3-Manifolds and Related Topics, Prentice-Hall, New Jersey, 1961, pp. 120–167. MR 0140099 Fox, Ralph H. (1970). "Metacyclic invariants of knots and links". Canadian Journal of Mathematics. 22 (2): 193–201. doi:10.4153/CJM-1970-025-9. MR 0261584. S2CID 121427629. Fox, Ralph H. (1966). "Rolling". Bulletin of the American Mathematical Society. 72, Part 1: 162–164. doi:10.1090/s0002-9904-1966-11467-2. MR 0184221. Fox, Ralph H.; Smythe, Neville F. (1964). "An ideal class invariant of knots". Bulletin of the American Mathematical Society. 15 (5): 707–709. doi:10.1090/s0002-9939-1964-0165516-2. MR 0165516. Chen, Kuo Tsai; Fox, Ralph H.; Lyndon, Roger C. (1958). "Free differential calculus. IV. The quotient groups of the lower central series". Annals of Mathematics. 68 (1): 81–95. doi:10.2307/1970044. JSTOR 1970044. MR 0102539. Fox, Ralph H. (1945). "On topologies for function spaces". Bulletin of the American Mathematical Society. 51 (6): 429–432. doi:10.1090/s0002-9904-1945-08370-0. MR 0012224. Blankinship, William A.; Fox, Ralph H. (1950). "Remarks on certain pathological open subsets of 3-space and their fundamental groups". Proceedings of the American Mathematical Society. 1 (5): 618–624. doi:10.1090/s0002-9939-1950-0042120-8. MR 0042120. Fox, Ralph H. (February 1945). "Torus Homotopy Groups". Proceedings of the National Academy of Sciences of the United States of America. 31 (2): 71–74. Bibcode:1945PNAS...31...71F. doi:10.1073/pnas.31.2.71. PMC 1078755. PMID 16588687. Fox, Ralph H. (1943). "On fibre spaces. I". Bulletin of the American Mathematical Society. 49 (8): 555–557. doi:10.1090/s0002-9904-1943-07969-4. MR 0008702. Fox, Ralph H. (1943). "On fibre spaces. II". Bulletin of the American Mathematical Society. 49 (10): 733–735. doi:10.1090/s0002-9904-1943-08015-9. MR 0009109. Fox, Ralph H. (1942). "A characterization of absolute neighborhood retracts". Bulletin of the American Mathematical Society. 48 (4): 271–275. doi:10.1090/s0002-9904-1942-07652-x. MR 0006508. Fox, Ralph H. (January 15, 1940). "On Homotopy and Extension of Mappings". Proceedings of the National Academy of Sciences of the United States of America. 26 (1): 26–28. Bibcode:1940PNAS...26...26F. doi:10.1073/pnas.26.1.26. PMC 1078000. PMID 16577957. Fox, Ralph; Neuwirth, Lee (1962). "The braid groups". Mathematica Scandinavica. 10: 119–126. doi:10.7146/math.scand.a-10518. MR 0150755.

References

External links O'Connor, John J.; Robertson, Edmund F., "Ralph Fox", MacTutor History of Mathematics Archive, University of St Andrews [1] Jozef H. Przytycki, Notes to the early history of the Knot Theory in Japan, 2001.

Worked examples

Example 1 — a first encounter with Ralph Fox

Start with the simplest possible case. Write down what Ralph Fox 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 Ralph Fox 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 Ralph Fox 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 Ralph Fox

In research: Ralph Fox 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 Ralph Fox 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: Ralph Fox is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1913 births, 1973 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Ralph Fox 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 Ralph Fox in 20 minutes

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

Frequently asked questions

What is Ralph Fox in simple terms?

Why does Ralph Fox 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 Ralph Fox?

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 Ralph Fox.