Showing papers in "The Mathematical Gazette in 1974"
286 citations
123 citations
55 citations
38 citations
TL;DR: In fact, neither the passive contemplation of wallpaper patterns, nor abstract definitions, is mathematics: the latter is above all an activity in which definitions are used to obtain concrete results.
Abstract: Several modern mathematics courses contain a description of the 17 distinct ‘wallpaper patterns’. Others contain the definition of “group” and “isomorphism”, together sometimes with a vague statement that these concepts can be used to justify the fact that there are precisely 17 patterns. But neither the passive contemplation of wallpaper patterns, nor the passive contemplation of abstract definitions, is mathematics: the latter is above all an activity in which definitions are used to obtain concrete results. For this reason I have often been asked by teachers what is needed to give a rigorous proof that there are precisely 17 patterns.
37 citations
27 citations
16 citations
15 citations
TL;DR: The intersection of two circular cylinders of equal radius is not only of mathematical interest but also has application in both engineering and architecture as discussed by the authors, where the joining of pipes of circular cross-section at a variety of given angles is an obvious example.
Abstract: The intersection of two circular cylinders of equal radius is not only of mathematical interest but also has application in both engineering and architecture. The joining of pipes of circular cross-section at a variety of given angles is an obvious example. The Romans and Normans, in using the barrel vault to span their buildings, were familiar with the geometry of intersecting cylinders where two such vaults crossed one another to form a cross vault. Larger numbers of equal intersecting cylinders arise in the following way.
14 citations
13 citations
TL;DR: In this article, a new construction of the real numbers from the rationals is given, in which a real number is defined as an equivalence class of sets of natural numbers.
Abstract: There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.
TL;DR: In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal's triangle are reduced modulo m as mentioned in this paper.
Abstract: In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal’s triangle are reduced modulo m. For the case when m equals a prime number, p, the pattern formed by the zeros in the reduced triangle (corresponding to binomial coefficients divisible p) was completely described and the following result (stated by G. Gilbart-Smith) was proved: In the (n + l)th row of Pascal’s triangle, there are
TL;DR: The popular definition of a "straight line" as "the shortest distance between two points" is one of the few pieces of mathematical jargon in general circulation in the English language as discussed by the authors.
Abstract: The popular definition of a “straight line” as “the shortest distance between two points” is one of the few pieces of mathematical jargon in general circulation in the English language. This is unfortunate, for it is not a good definition, and one may well wonder who originated it. Euclid is not to blame, for he defined a straight line as “that which lies evenly between its points”. (This is indeed cryptic, and Sir Thomas Heath, in a lengthy analysis of the original Greek, had to confess that “the language is...hopelessly obscure”.)