Archive for the ‘Uncategorized’ Category

Steiner Chains

June 28, 2018

Steiner Chains (named for Jakob, the Swiss geometer not Rudolph, the Austrian educationalist) are a long-standing project which, thankfully, I have at last completed.  My original intention was to find an alternative method to ‘inversive geometry’ for deriving the closure condition.  A derivation using Cartesian coordinate geometry however defeated me and the pdf here uses a mapping of the plane to itself which is adapted from the inversive geometry method.

Magic Squares

March 11, 2017

A magic square is a square (n x n) array of integers in which each of the integers 1 to n2 appears exactly once, and such that the sum of the integers in each row, each column and each diagonal is the same (the ‘magic number’).

The 3×3 problem was presented to me as a party game.  Participants were told that the magic number was 15, and asked to construct the square.  My method of solving the problem was by writing out the possible sums without regard to the order of their components (there are only 8) and noting that the central element must appear in 4 sums, and the corner elements in 3.  5 is the only number that appears in four sums, 2, 4, 6 and 8 appear in three, and 1, 3, 7 and 9 in only 2.  Thus the only possible solution (together with its variants resulting from the symmetry of the square, which can be produced by rotation, reflection and inversion of the elements) is as below

8      1      6

3      5      7

4      9      2

No magic square exists for n = 2 but squares do exist for all n > 2 and the number of possibilities increase very rapidly with n.  It happens, however, that all squares of a given n have the same magic number, given by

M = (n2 +1)n/2

Methods for constructing magic squares for any given n > 2, but without demonstration of their validity for all n, are given on page 148 et seq. of Mathematical Recreations by Maurice Kraitchik (1942). Kraitchik’s book can be read for free and without breach of copyright online (google the author, title and ‘free ebook’). The purpose of the pdf is to validate the methods Kraitchik describes.

Lambert’s proofs that e and pi are irrational

May 10, 2016

The last few weeks I’ve spent revising, in response to an emailed comment, a pdf (revised version here) I attached to a 2010 post about Lambert’s proofs. I’ve started a new page, ‘Rational or Irrational?’, in which I’ve put a link to the revised version and I will add to this page over time material from posts I made some time ago about Hermite’s, Niven’s and Apéry’s work on irrational numbers.

Descartes’ Circle Theorem

April 9, 2016

Finding a proof of Steiner’s Porism not using inversive geometry proved a tough task, so I decided to look at Descartes’ Circle Theorem. If we include both the inscribed and circumscribed circles this actually represents the simplest case of a Steiner Chain, with three circles between the inner and outer circles. For more, see the Geometry Page.

Steiner Chains

January 29, 2016

On the way from Pappus Chains (previous post) to Clifford’s Circle Theorem I was diverted by the thought that the equations governing Pappus Chains should also govern, possibly with slight modification, Steiner Chains.

Steiner Chains were briefly discussed in Martin Gardner’s 1979 Scientific American article and they are a generalisation of Pappus Chains (or to be more accurate Pappus Chains are a limiting case of Steiner Chains).

A distorted annulus is formed by two circles, one within the other. A Steiner Chain is an infinite sequence of circles tangent (externally) to the circles forming the distorted annulus and to each other, using a particular inscribed circle as the first in the sequence.

In certain circumstances the chain reproduces itself after one ‘circuit’ of the distorted annulus. In these cases the chain will close itself regardless of where the first circle is placed (see diagram below reproduced from Gardner’s article).

This result, referred to as ‘Steiner’s Porism’, is usually explained using inversion geometry (see for example the relevant chapter of ‘Geometry Revisited’ by H S M Coxeter and S L Greitzer).

Pappus Chains are obviously a limiting case of Steiner Chains, where the circles forming the distorted annulus are tangent. The centres of the circles forming Steiner Chains are known to lie on an ellipse (see relevant Mathworld article) so quite possibly the parameters of Steiner Chains, like those of Pappus Chains, are determined by the parameters of the ellipse. In that case, the cases when the chain closes will be determined by a diophantine equation involving those parameters.

Of course it is quite possible that if the chain does not close after one ‘circuit’ of the distorted annulus, it will close after two or more.

The Arbelos and Pappus Chains

January 17, 2016

These have been studied since antiquity but in more recent times were popularised in 1979 by Martin Gardner in his Scientific American recreational maths column. The properties can be elegantly explained using inversion geometry, but at the moment I am using Cartesian coordinates to explore them. The results and some references will be shown at the Geometry page.

The Tusi Couple

December 27, 2015

The Tusi Couple was a brief diversion on the way from Bernoulli Numbers and the Euler Maclaurin Sum Function to some new destination, as yet undecided.

I came across reference to Tusi while reading Lost Enlightenment.  Central Asia’s Golden Age from the Arab Conquest to Tamerlane by S Frederic Starr.  Nasir al-Din al-Tusi (1201 – 1274) was a polymath who founded what was then the world’s largest astronomic observatory at Maragha or Maragheh in what is now Azerbaijan.

What is now referred to as the Tusi Couple, and naturally was studied by Tusi, is a planar geometric construct consisting of two circles, the smaller of which is half the diameter of the larger and inside it, touching at a single point.  The smaller circle rolls around the circumference of the larger without sliding in a fixed direction, anticlockwise say.

The singular feature of the Tusi Couple is that each point on the circumference of the smaller circle exhibits a periodic motion along a fixed diameter of the larger.

Bernoulli Numbers and the Euler Maclaurin Sum Formula

December 7, 2015

I have been revising and adding to the notes on this topic here

Long Division Method of Calculating Square Roots

February 21, 2014

In the 1950s and 1960s schools taught a method of calculating square roots by hand, that resembled long division. A number of demonstrations of the method, via Youtube clips, can be found on the web. My favourite is this one, by Anita Govilkar. Not only is the exposition clear, but Ms Govilkar’s carefully enunciated Indian English is a delight to listen to.

Explanations of just why the method works are less frequently encountered. In this pdf, I give my take on the workings of the method.

Fermat’s Last Theorem for the case where the index is 4

March 23, 2012

Fermat’s Last Theorem, proved in 1995, is that the diophantine equation

$x^n + y^n = z^n$

has no solution in positive integers x, y, z when n > 2. Knowing the solution for n = 2 (see previous post) we can show that there is no solution for n = 4.  If

$x^4 + y^4 = z^4$   …………………………… (1)

has a solution x1, y1, z1 then

$x^4 + y^4 = z^2$   …………………………….   (2)

has a solution x1, y1, z12 and a solution x1/d, y1/d, z12/d2 where d = (x1, y1), the greatest common divisor of x1 and y1. This implies that (2) has a solution where (x, y) = 1. If we can show there is no such solution then by the contrapositive there is no solution to (1).

Suppose to the contrary there is such a solution x1, y1, z1. It is easy to see that these numbers have greatest common divisor 1. We use the method of infinite descent by showing that there exist x2, y2, z2 such that z2 < z1.

By the properties of Pythagorean Numbers (previous post) we know that

$z_{1} = a^{2} + b^2$  ;  $x_{1}^{2} = a^2 - b^2$  ;  $y_{1}^{2} = 2ab$

with a > b> 0, (a,b) = 1, and a, b of opposite parity.

x12 + b2 = a2 and it is not hard to show that that x1, b and a have the remaining properties of Pythagorean numbers. Thus

$x_1 = p^2 - q^2$  ;  $b = 2pq$  ;  and  $a = p^2 + q^2$

with (p,q) = 1, p > q > 0, and p,q of opposite parity. Consequently

$y_{1}^2 = 2ab = 4pq(p^2 + q^2)$

Furthermore p, q and p 2 + q2 have greatest common divisor one, so by the Unique Factorisation Theorem each is a square and we can write:

$p = x_{2}^2$  ;  $q = y_{2}^2$  ;  and $p^2 + q^2 = z_{2}^2$ and thus

$x_{2}^4 + y_{2}^4 = z_{2}^2$

with x2, y2, z2 > 0 and z1 = a2+b2 = (p2+q2)2+b2=z24+b2 > z2

To prove Fermat’s Last Theorem it suffices to prove it for every prime index.   If n > 2 is composite and has no prime factor other than 2 then it is a multiple of 4 and the Theorem is true for this index because if

$x^n + y^n = z^n$ then $\left( x^{n/4}\right)^4 + \left( y^{n/4} \right)^4 = \left( z^{n/4} \right)^4$

which is impossible.   If n is composite and has an odd prime factor p and if

$x^n + y^n = z^n$ then $\left( x^{n/p} \right) ^p + \left( y^{n/p} \right) ^p = \left( z^{n/p} \right) ^p$

thus by the contrapositive if the Theorem is true for p it is true for n.

A copy of this post in pdf form is here.