Showing papers in "The Mathematical Gazette in 2007"

Intriguing integrals : an Euler-inspired odyssey

Abstract: This article began life as a talk given at the 2006 Mathematical Association Conference at Loughborough University. Many of the examples discussed had their roots in the work of Euler and the 'Euler boxes' below explore these connections. We use the standard references to Euler's oeuvres by 'Enestrbm numbers' (from Enestrom's catalogue of Euler's works) and dates of publication: a very convenient source is the almost complete Euler Archive, freely accessible at http:// www.math.dartmouth.edu/~euler/.

91.13 A new proof of a curious identity

Abstract: We give a short WZ-proof of a binomial coefficient identity due to Zhi-Wei Sun. In [3], [2], and [1] the identity m ∑ i=0 (x+m+ 1)(−1) ( x+ y + i m− i )( y + 2i i ) − m ∑ i=0 ( x+ i m− i ) (−4) = (x−m) ( x m ) (1) was proved using generating functions, double recursions and the concept of Riordan Arrays respectively. Here we use the WZ-method to give yet another proof. First we divide both sides of (1) by (x + m + 1) and then try to write the second indefinite sum on the LHS of (1) without the running index m under the summation sign. This is automated in EKHAD in the procedure zeillim and can be run by the command: zeillim(SUMMAND , i,m,M, 0, 0). But we include here the mathematics behind zeillim for the sake of clarity. Consider the indefinite sum

Euler and triangle geometry

Abstract: There is a very easy way to produce the Euler line, using transformational arguments. Given a triangle ABC, let AʹBʹ'C be the medial triangle, whose vertices are the midpoints of the sides. These two triangles are homothetic: they are similar and corresponding sides are parallel, and the centroid, G, is their centre of similitude. Alternatively, we say that AʹBʹC can be mapped to ABC by means of an enlargement, centre G, with scale factor –2.

Applied geometry for computer graphics and CAD (2nd edn.), by Duncan Marsh. Pp. 350. £18.95. 2005. ISBN 1 85233 801 6 (Springer Verlag).

Abstract: Applied geometry for computer graphics and CAD (2nd edn.), by Duncan Marsh. Pp. 350. £18.95. 2005. ISBN 1 85233 801 6 (Springer Verlag). The reviewed item is a well-written textbook aimed for undergraduate mathematics or non-mathematics students. It contains 11 chapters of the same and clear structure: a short theoretical introduction illustrated by well-chosen examples, then facts describing the considered problems deeper and finally exercises to solve (at the end of the book the reader will find solutions to the majority of the exercises).

Trigonometry and Fibonacci numbers

Abstract: This article sets out to explore some of the connections between two seemingly distinct mathematical objects: trigonometric functions and the integer sequences composed of the Fibonacci and Lucas numbers. It establishes that elements of Fibonacci/Lucas sequences obey identities that are closely related to traditional trigonometric identities. It then exploits this relationship by converting existing trigonometric results into corresponding Fibonacci/Lucas results. Along the way it uses mathematical tools that are not usually associated with either of these objects.

91.69 Can you ‘bend’ a truncated octahedron?

Abstract: The parallelograms Now K is also the midpoint of FF\" (the join of the two Fermat points which also lie on opposite branches of the Kiepert hyperbola), so FOPS', FHFT and GHS'T are all bridging parallelograms. That the first two parallelograms have the same area is easily demonstrated in the Cartesian format [4] where the Kiepert hyperbola has equation xy = 1. Here the relevant coordinates are F(p, £), F\"{-p, -£), G(g, | ) , H(h, £), K (0,0), S (3g, j), S' {-g, -J), T {-h, -I) where g, h, p > 0, g * h, p = gh. Now

91.40 An amusing sequence of trigonometrical integrals

Abstract: u-shaped and takes its maximum value over the domain of allowed values of t at the endpoint t 1 sin \\A, corresponding to the isosceles triangle withx = \\(n A) = B = C. If JI tan \\A 1 > 0 or A > 2 tan £, the quadratic graph of Q (?) is n-shaped, but the value of t giving its global maximum occurs at sin A t = r^ which (check!) is always greater than 1 siniA, the n tan \\A 1 largest allowed value of t. So in this case also, there is an endpoint maximum corresponding to an isosceles triangle with apex angle A We now use a lovely old argument (which may be traced back to Simon Lhuilier (1750-1840), [1]) to deduce that, among all triangles, the largest value of Q is given by an equilateral triangle. Certainly, since the expression for Q is continuous in (A, B, C), it attains its global maximum on the compact region 0 < A, B, C < n, A + B + C = n. And, starting with an arbitrary triangle with angles (A, B, C), the isosceles triangle with apex angle A and angles (A, \\(B + C),\\{B + C)) has larger Q-value; repeating this argument shows that the isosceles triangle with angles {\\{B + C), \\{A + \\{B + C)), \\{A + \\{B + C))) has even larger Qvalue. Iterating this argument generates a sequence of isosceles triangles with ever-increasing Q-values which always converge to the equilateral triangle, since the greatest difference between the angles of the n th isosceles triangle in the sequence is \ 3A|/2\" which tends to zero as n tends to infinity. Among all triangles, the equilateral triangle thus has the largest Qvalueof^ \\ = 0.1635. Finally, it is worth reflecting on why the proof is a bit fiddly. First, since f(x) = In sinx is concave on (0, n), ^E In sin A < In sin (3EA) or Ft sin A < sinf and |Z In sin^A < In sin(^EA) or II sin^A < sinf so the individual constituents of Q, %H sin A and 1611 sin^A, are themselves separately maximised for an equilateral triangle. Second, as in the STEP question, it is easy to generate relatively tight upper bounds for Q. For example, writing S = Tl sin \\A and C = Tl cos \\A we have Q = %CS 16S= %S(C nS) < ^ r on maximising as a quadratic in S. But, as above, C < cosf = ^ , so that Q <, -^ = 0.1710.

91.11 Explicit formula for power sums of an arithmetic sequence

Abstract: In this expression, a and fc are assumed to be arbitrary constant numbers, possibly integers, with n a positive integer. For notational simplicity, Sm (n; a, b) will be abbreviated to Sm in what follows. In Simons' approach, an infinite series first has to be inverted and the result then multiplied by another infinite series; the desired sums are identified with the coefficients of the final power series expansion. Because inverting an infinite series is a very arduous task, to say the least, Simons remarks that 'exact results for Sm cannot readily be found for general m...' and the author then proceeds to a derivation of an asymptotic formula whose validity is restricted by the condition n\a\ » m. The interested reader should consult [1] for further details. The sums of the m th powers of n consecutive terms of an arithmetic sequence have been of interest before and different methods have been proposed to obtain them; see [2] and [3], for example. References [1] and [3] use an exponential generating function to address the issue at hand. The purpose of the present Note is to propose a different approach, that leads to exact explicit expressions for any required sum, Sm. The method makes use of the repeated action of a simple differential operator on a monic geometric polynomial and the solution is arrived at by means of matrix methods. The method can be used effectively to show, for example, that the sum, Z"Io rTM, where 0 < m, is a polynomial of degree (m + 1) in n, with no constant term. The present approach is simple and could be used at an early level of a course in calculus or in number theory.

Cevian axes and related curves

Abstract: When exploring some triangle geometry, we stumbled upon a set of Cevian axes: lines having certain properties in common with the Euler line. These axes yielded interesting results that we had not seen before. However, we have since learnt that many of them were discovered in the 19th Century. We were also unaware that John Rigby had covered much of this ground in unpublished work in the 1990s (private communication). We are therefore reluctant to claim any of our results as new, even though we hope that some may be.

Proof and disproof in formal logic: an introduction for programmers, by Richard Bornat. Pp. 243. £83 (hbk) £34 (pbk). 2005. ISBN 0-19-8530277 (Oxford University Press).

Abstract: being flexible and universal in both syntax and execution. The choices of MAPLE and Java seem appropriate. Indeed, in order to acquire some skill on Web computing one has to master object-oriented programming, and Java has excellent error-handling capabilities during compilation and execution. Particularly for a subject such as scienctific computation, knowledge and expertise can only be obtained from a hands-on approach, and the book does deliver an interactive working environment for students. Indeed the only realistic way to learn about computers and computing is to make plenty of mistakes and be able to recover from the errors, regardless of whether they are trivial or fundamental. The author is well aware of this and much of the text is written with this in mind. The book thus provides a wellorganised introduction for its intended readers to a subject usually given only to students on computing science and mathematics courses.

91.33 Fermat’s last theorem in the case n = 4

Abstract: The LHS shows that dn is an integer and, using the fact that 1 < | ekp e'kq | < | p | + | q | for all k, we can bound the RHS in much the usual way to give the contradiction that 0 < | d„ \\ < 1 for all sufficiently large n. Finally, it is easy to classify the pairs x, x for which: ^ = 1 : this requires that 2+ + + £2 + ... = 0. This (check!) / « H 1! 2! only occurs if £i = e\\ = -1 and ek + e'k = 0 for all k > 2, i.e. x + x' = \\. As suggested by Figure 1 then, the graph of y = f(x) really does have an element of rotational symmetry about the point (4,0).

91.30 Fibonacci meets Chebyshev

Abstract: 91.30 Fibonacci meets Chebyshev The continuing stream of articles on the identities satisfied by the Fibonacci numbers is a testament to the widespread interest in this topic. However, as these articles usually exclude other recurrence relations, they . tend to give the impression that the identities are restricted to the Fibonacci numbers alone. The purpose of this article is to show that they are not, and that, even for the general recurrence relation, many (and perhaps all) of these identities are consequences of simple trigonometric identities. The Fibonacci numbers Fn are defined for all integers n by

91.65 A question of balance: an application of centroids

Abstract: The following surprising geometry result provides a nice challenge to high school or undergraduate students. Given a quadrilateral ABCD with equilateral triangles ABP, BCQ, CDR and DAS constructed on the sides so that say the first and third are exterior to the quadrilateral, while the second and the fourth are interior to the quadrilateral, prove that quadrilateral PQRS is a parallelogram. This result appears to be reasonably well-known and can be found as an exercise in [1], and the special case when two of the vertices of ABCD coincide to form a triangle, apparently also appears in some high school textbooks in Korea and other Eastern countries [2]. Less well known appears to be the following generalisation and proof of the result in [3].

The Hindu method for completing the square

Abstract: In a September 2005 note in The Mathematics Teacher , Gordon [1] gave a variation on the method of completing the square that fraction-phobic students are likely to find easier, along with the comment: ‘This variation is so obvious it must have been discovered many times over, but I have never seen it in print and thought it would be useful to disseminate it more widely.’ The method Gordon [1] gave is to multiply by 4 a , and then add b 2 , to both sides of ax 2 + bx = − c to produce 4 a 2 x 2 + 4 abx + b 2 = b 2 −4 ac . The left side of this last equation can be factorised as (2 ax + b ) 2 , and now we have a pure quadratic whose solution is straightforward.

Elementary number theory in nine chapters (2nd edn.), by James J. Tattersall. Pp. 430. £19.99 (pbk), £55 (hbk). 2005. ISBN 0 521 61524 0 (pbk), 0 521 85014 2 (Cambridge University Press).

Abstract: The word 'elementary' is one of the most used, often abused, title words in the Gazette reviews. On this occasion, I am happy to say, the use is honest. The mathematical content is unremarkable induction, integers, primes, perfect numbers, modular arithmetic, cryptology, continued fractions, partitions, quadratic reciprocity, Pythagorean triples . . . . It will be familiar to anyone who has studied any reasonably serious first course in number theory. A very minor exception is perhaps the brief account of p-adic analysis and some exercises designed to explore its elementary properties. It rarely strays above the first year undergraduate level, and much of it is a good read for pre-university students. Quadratic reciprocity is about the most advanced topic treated in detail.

91.06 On the Lagrange method for constrained critical points

Abstract: References 1. M. D. Hirschhorn, Comments on 'Triangular numbers and perfect squares', Math. Gaz. 88 (November 2004) pp. 500-503. 2. T. Beldon and A. Gardiner, Triangular numbers and perfect squares, Math. Gaz. 86 (November 2002) pp. 423-431. 3. H. E. Rose, A course in number theory, Oxford University Press, Oxford (1988). 4. C. F. Gauss, Disquisitiones Arithmeticae, (tr. Arthur A. Clarke), Springer-Verlag, New York (1986). 5. H. Cohen, A course in computational algebraic number theory, Springer-Verlag, Berlin (1993). ROBIN CHAPMAN Department of Mathematics, University of Bristol, Bristol BS8 1TW

Euler's contribution to number theory

Abstract: Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

91.52 Similarity properties of the Bride’s Chair

Abstract: 91.52 Similarity properties of the Bride's Chair Roger Webster, [1], has drawn attention to an unexpected area property of the 'Bride's Chair' configuration of three squares erected externally on the sides of a triangle which is associated with Pythagoras' Theorem. In this note we look at the problem of whether, in Figure 1, the large triangle A'B'C can be similar to the initial triangle ABC. A' (x+y,a+y-x)

91.19 The Basel problem and the zeros of

Abstract: References 1. http://mathworId.wolfram.com/ArchimedesRecurrenceFormula.html 2. G. M. Phillips, Archimedes the numerical analyst, Amer. Math. Monthly: 88 (1981) pp.165-169. 3. L. Bergrren, J Borwein, P Borwein : Pi: a source book, Springer (1997). 4. J. Scott: Even more series for n, Math. Gaz. 85 (July 2001) pp. 299-301. 5. Stephen Vincent and David Forsey, Fast and accurate parametric curve length computation, Journal of Graphics Tools 6 (4) (2002) pp. 29-40. S. P. VINCENT 315-6366 Cassie Ave, Burnaby, BC, Canada V5H2W5 e-mail: spvincent®shaw.ca

Man versus Computer

Abstract: A few years ago, Shail [1] described earlier computer-assisted work on the Lester circle. Shail then showed how a Cartesian proof was ‘just feasible by hand’ but that it was more convenient to use computer algebra systems. However, not relying on computer assistance does have the advantage that it forces us to search for improvements in our methods. In this instance, using vectors referenced to a special set of three coplanar axes enables us to avoid involved computations.

The Hypercubical Dance - a solution to Abbott’s problem in Flatland ?

Abstract: Abbott’s Flatland (1884) [1] is the story of the encounter between a two-dimensional creature, a Square, and a visitor from space, a Sphere. The Sphere tries to convince the Square of the reality of the third dimension, but with little success at first. Finally, after several long conversations and a few other worldly demonstrations, the Sphere succeeds in making the Square see the light (or the height?), with rather unfortunate consequences for the poor Square. Flatland became a big hit in the years and decades after it came out, and continues to enjoy a robust reputation today. Much of the fallout generated by Flatland in the more than hundred years since it was written is documented in the annotated version of the book brought out by Stewart [2].

Hagge Circles and Isogonal Conjugation

Abstract: Let Q be a point on the circumcircle of triangle ABC. The reflections of Q in the three triangle sides are known to be collinear, and the line thus defined contains the orthocentre H. This fact can form the basis of a proof of the existence of the Simson line, or alternatively can be deduced from the existence of the Simson line by enlarging the Simson line by a factor of 2 from the centre Q.

Reflections on an exponential mirror

Abstract: It has been known for some time that when the sun is reflected in a circular cylinder, the illuminated part of the cylinder's base is bounded by part of a nephroid (the curve obtained by rolling a circle around another circle with twice the radius).

'Catch-up' numbers

Abstract: It never ceases to amaze me how often an apparently innocent-looking problem can lead on to something altogether more deep and interesting. As an example of this, I had recently set one of my Lower Sixth classes, as part of a statistics revision lesson, the task of devising some original questions on probability when the following problem emerged: ‘A person tosses a fair coin six times. Given that 4 heads and 2 tails result, what is the probability that the proportion of tails obtained at each point before the last toss is less than one third?’